Space-Time Coding in Statistical MIMO Radar

Marco LopsDAEIMI- University of CassinoENSEEIHT ndash University of Toulouse

EURASIP one-day Seminar onrdquoRadar Signal Processing Hot Topics and New Trendsrdquo ndash Pisa

September 25th 2009

MIMO Detection context

Multiple Input Multiple Output Radar with widely spaced AntennaeLinear Space-Time Coding helps us shed light on the achievable detection performanceSTC allows shaping the transmit waveform to achieve

Diversity (through different aspect angles)Energy Integration (ie power multiplexing)A compromise between the two

RADAR Basic Operation

Pulse (duration τp bandwidth B energy E)

Range cell ldquokrdquo

Delay τ

c=3times108 ms

Last cell Rmax=cT2

t=0 t=T

Size=c2B[m]Distance=cτ2

Pulsed Radars

T

τp

Pulse 1 Pulse N

N echoes from a target in cell k+N ldquoclutterrdquo echoes exhibiting time correlation

Energy E

Basic Receiver

1

0

1E

1

H

HH

α= +=

⎡ ⎤⎢ ⎥⎡ ⎤ = =⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

r 1 wr w

ww M 1

E

Test on the signal received in the k-th cell

N-dimensionalvectors

Matched Filter

1

21

0

H

H

T

H

minus gtlt

r M 1

Pfa=PDeclare H1|H0 fixed (T)

Pd=PDeclare H1|H1 maximum

Performance

If α is complex Normal with variance σa2 (Swerling-I)

( )

infinrarrsdot

minusasymp=

sdot==

sdot+

minus

SCRSCRNP

PP

SCRNSCR

faSCRNfad

aT

T

ln11

1

21 σ1M1E

SCR Signal-to-Clutter Ratio per pulse

N Coherent Integration Gain

Angle Diversity

V ldquotarget extensionrdquoλ carrier wavelength

Different aspects at distance R

dgt λRV (full decorrelation)

Pulsed MIMO Radar Scheme

Pulsed Waveforms

Scheme 2

Underlying Assumption

ldquoNarrow-bandrdquo assumption on the transmitted pulses The target is seen in the same range-cell by all of the antennae

TX 1

TX 2

TX 3

TX s

dmax

RX 1

RX r

hmax

dmax+hmaxltltcB

Ex B=1MHz we have

dmax+hmaxltlt300m

Degrees of Freedom

The number of degrees of freedom is

( ) (tall code matrix)min

(fat code matrix)sr

s r NsN

⎧sdot = ⎨

⎩

Why Because the transmitted signal space has at most dimension N whereby a ldquofatrdquo code matrix means that the codewords are not linearly independent

MIMO Radar The Model

1

0

1 1

i i i

i i

H i rH i r

= + == =

r Aα wr w

helliphellip

receivedat antenna

ldquoirdquoCode-Matrix

(Ntimess)

Coefficientsfrom the TX to ldquoirdquo RX ant

1

0

HH

= +=

R A H WR W

Clutter withCovariance M

Parameters

Designing the test

( )( )

[ ]Hii

r

rrT

H

H

Hf

Hfr

wwM

Mrr

Mααrrαα

E

max

0

1

01

1111

=

ltgt

hellip

helliphelliphellip (GLRT)

GLR Test Statistics

M-12Projector onto

Signal Subspace

||middot||2ri zi

12

1

22 12

1 1

0

r r

i ii i

H

T

H

minusminus

= =

gt=

ltsum sum M Az P M r

GLRT TestStatistics

PV=projector onto the rangespan of V

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

MIMO Detection context

Multiple Input Multiple Output Radar with widely spaced AntennaeLinear Space-Time Coding helps us shed light on the achievable detection performanceSTC allows shaping the transmit waveform to achieve

Diversity (through different aspect angles)Energy Integration (ie power multiplexing)A compromise between the two

RADAR Basic Operation

Pulse (duration τp bandwidth B energy E)

Range cell ldquokrdquo

Delay τ

c=3times108 ms

Last cell Rmax=cT2

t=0 t=T

Size=c2B[m]Distance=cτ2

Pulsed Radars

T

τp

Pulse 1 Pulse N

N echoes from a target in cell k+N ldquoclutterrdquo echoes exhibiting time correlation

Energy E

Basic Receiver

1

0

1E

1

H

HH

α= +=

⎡ ⎤⎢ ⎥⎡ ⎤ = =⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

r 1 wr w

ww M 1

E

Test on the signal received in the k-th cell

N-dimensionalvectors

Matched Filter

1

21

0

H

H

T

H

minus gtlt

r M 1

Pfa=PDeclare H1|H0 fixed (T)

Pd=PDeclare H1|H1 maximum

Performance

If α is complex Normal with variance σa2 (Swerling-I)

( )

infinrarrsdot

minusasymp=

sdot==

sdot+

minus

SCRSCRNP

PP

SCRNSCR

faSCRNfad

aT

T

ln11

1

21 σ1M1E

SCR Signal-to-Clutter Ratio per pulse

N Coherent Integration Gain

Angle Diversity

V ldquotarget extensionrdquoλ carrier wavelength

Different aspects at distance R

dgt λRV (full decorrelation)

Pulsed MIMO Radar Scheme

Pulsed Waveforms

Scheme 2

Underlying Assumption

ldquoNarrow-bandrdquo assumption on the transmitted pulses The target is seen in the same range-cell by all of the antennae

TX 1

TX 2

TX 3

TX s

dmax

RX 1

RX r

hmax

dmax+hmaxltltcB

Ex B=1MHz we have

dmax+hmaxltlt300m

Degrees of Freedom

The number of degrees of freedom is

( ) (tall code matrix)min

(fat code matrix)sr

s r NsN

⎧sdot = ⎨

⎩

Why Because the transmitted signal space has at most dimension N whereby a ldquofatrdquo code matrix means that the codewords are not linearly independent

MIMO Radar The Model

1

0

1 1

i i i

i i

H i rH i r

= + == =

r Aα wr w

helliphellip

receivedat antenna

ldquoirdquoCode-Matrix

(Ntimess)

Coefficientsfrom the TX to ldquoirdquo RX ant

1

0

HH

= +=

R A H WR W

Clutter withCovariance M

Parameters

Designing the test

( )( )

[ ]Hii

r

rrT

H

H

Hf

Hfr

wwM

Mrr

Mααrrαα

E

max

0

1

01

1111

=

ltgt

hellip

helliphelliphellip (GLRT)

GLR Test Statistics

M-12Projector onto

Signal Subspace

||middot||2ri zi

12

1

22 12

1 1

0

r r

i ii i

H

T

H

minusminus

= =

gt=

ltsum sum M Az P M r

GLRT TestStatistics

PV=projector onto the rangespan of V

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

RADAR Basic Operation

Pulse (duration τp bandwidth B energy E)

Range cell ldquokrdquo

Delay τ

c=3times108 ms

Last cell Rmax=cT2

t=0 t=T

Size=c2B[m]Distance=cτ2

Pulsed Radars

T

τp

Pulse 1 Pulse N

N echoes from a target in cell k+N ldquoclutterrdquo echoes exhibiting time correlation

Energy E

Basic Receiver

1

0

1E

1

H

HH

α= +=

⎡ ⎤⎢ ⎥⎡ ⎤ = =⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

r 1 wr w

ww M 1

E

Test on the signal received in the k-th cell

N-dimensionalvectors

Matched Filter

1

21

0

H

H

T

H

minus gtlt

r M 1

Pfa=PDeclare H1|H0 fixed (T)

Pd=PDeclare H1|H1 maximum

Performance

If α is complex Normal with variance σa2 (Swerling-I)

( )

infinrarrsdot

minusasymp=

sdot==

sdot+

minus

SCRSCRNP

PP

SCRNSCR

faSCRNfad

aT

T

ln11

1

21 σ1M1E

SCR Signal-to-Clutter Ratio per pulse

N Coherent Integration Gain

Angle Diversity

V ldquotarget extensionrdquoλ carrier wavelength

Different aspects at distance R

dgt λRV (full decorrelation)

Pulsed MIMO Radar Scheme

Pulsed Waveforms

Scheme 2

Underlying Assumption

ldquoNarrow-bandrdquo assumption on the transmitted pulses The target is seen in the same range-cell by all of the antennae

TX 1

TX 2

TX 3

TX s

dmax

RX 1

RX r

hmax

dmax+hmaxltltcB

Ex B=1MHz we have

dmax+hmaxltlt300m

Degrees of Freedom

The number of degrees of freedom is

( ) (tall code matrix)min

(fat code matrix)sr

s r NsN

⎧sdot = ⎨

⎩

Why Because the transmitted signal space has at most dimension N whereby a ldquofatrdquo code matrix means that the codewords are not linearly independent

MIMO Radar The Model

1

0

1 1

i i i

i i

H i rH i r

= + == =

r Aα wr w

helliphellip

receivedat antenna

ldquoirdquoCode-Matrix

(Ntimess)

Coefficientsfrom the TX to ldquoirdquo RX ant

1

0

HH

= +=

R A H WR W

Clutter withCovariance M

Parameters

Designing the test

( )( )

[ ]Hii

r

rrT

H

H

Hf

Hfr

wwM

Mrr

Mααrrαα

E

max

0

1

01

1111

=

ltgt

hellip

helliphelliphellip (GLRT)

GLR Test Statistics

M-12Projector onto

Signal Subspace

||middot||2ri zi

12

1

22 12

1 1

0

r r

i ii i

H

T

H

minusminus

= =

gt=

ltsum sum M Az P M r

GLRT TestStatistics

PV=projector onto the rangespan of V

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Pulsed Radars

T

τp

Pulse 1 Pulse N

N echoes from a target in cell k+N ldquoclutterrdquo echoes exhibiting time correlation

Energy E

Basic Receiver

1

0

1E

1

H

HH

α= +=

⎡ ⎤⎢ ⎥⎡ ⎤ = =⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

r 1 wr w

ww M 1

E

Test on the signal received in the k-th cell

N-dimensionalvectors

Matched Filter

1

21

0

H

H

T

H

minus gtlt

r M 1

Pfa=PDeclare H1|H0 fixed (T)

Pd=PDeclare H1|H1 maximum

Performance

If α is complex Normal with variance σa2 (Swerling-I)

( )

infinrarrsdot

minusasymp=

sdot==

sdot+

minus

SCRSCRNP

PP

SCRNSCR

faSCRNfad

aT

T

ln11

1

21 σ1M1E

SCR Signal-to-Clutter Ratio per pulse

N Coherent Integration Gain

Angle Diversity

V ldquotarget extensionrdquoλ carrier wavelength

Different aspects at distance R

dgt λRV (full decorrelation)

Pulsed MIMO Radar Scheme

Pulsed Waveforms

Scheme 2

Underlying Assumption

ldquoNarrow-bandrdquo assumption on the transmitted pulses The target is seen in the same range-cell by all of the antennae

TX 1

TX 2

TX 3

TX s

dmax

RX 1

RX r

hmax

dmax+hmaxltltcB

Ex B=1MHz we have

dmax+hmaxltlt300m

Degrees of Freedom

The number of degrees of freedom is

( ) (tall code matrix)min

(fat code matrix)sr

s r NsN

⎧sdot = ⎨

⎩

Why Because the transmitted signal space has at most dimension N whereby a ldquofatrdquo code matrix means that the codewords are not linearly independent

MIMO Radar The Model

1

0

1 1

i i i

i i

H i rH i r

= + == =

r Aα wr w

helliphellip

receivedat antenna

ldquoirdquoCode-Matrix

(Ntimess)

Coefficientsfrom the TX to ldquoirdquo RX ant

1

0

HH

= +=

R A H WR W

Clutter withCovariance M

Parameters

Designing the test

( )( )

[ ]Hii

r

rrT

H

H

Hf

Hfr

wwM

Mrr

Mααrrαα

E

max

0

1

01

1111

=

ltgt

hellip

helliphelliphellip (GLRT)

GLR Test Statistics

M-12Projector onto

Signal Subspace

||middot||2ri zi

12

1

22 12

1 1

0

r r

i ii i

H

T

H

minusminus

= =

gt=

ltsum sum M Az P M r

GLRT TestStatistics

PV=projector onto the rangespan of V

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Basic Receiver

1

0

1E

1

H

HH

α= +=

⎡ ⎤⎢ ⎥⎡ ⎤ = =⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

r 1 wr w

ww M 1

E

Test on the signal received in the k-th cell

N-dimensionalvectors

Matched Filter

1

21

0

H

H

T

H

minus gtlt

r M 1

Pfa=PDeclare H1|H0 fixed (T)

Pd=PDeclare H1|H1 maximum

Performance

If α is complex Normal with variance σa2 (Swerling-I)

( )

infinrarrsdot

minusasymp=

sdot==

sdot+

minus

SCRSCRNP

PP

SCRNSCR

faSCRNfad

aT

T

ln11

1

21 σ1M1E

SCR Signal-to-Clutter Ratio per pulse

N Coherent Integration Gain

Angle Diversity

V ldquotarget extensionrdquoλ carrier wavelength

Different aspects at distance R

dgt λRV (full decorrelation)

Pulsed MIMO Radar Scheme

Pulsed Waveforms

Scheme 2

Underlying Assumption

ldquoNarrow-bandrdquo assumption on the transmitted pulses The target is seen in the same range-cell by all of the antennae

TX 1

TX 2

TX 3

TX s

dmax

RX 1

RX r

hmax

dmax+hmaxltltcB

Ex B=1MHz we have

dmax+hmaxltlt300m

Degrees of Freedom

The number of degrees of freedom is

( ) (tall code matrix)min

(fat code matrix)sr

s r NsN

⎧sdot = ⎨

⎩

Why Because the transmitted signal space has at most dimension N whereby a ldquofatrdquo code matrix means that the codewords are not linearly independent

MIMO Radar The Model

1

0

1 1

i i i

i i

H i rH i r

= + == =

r Aα wr w

helliphellip

receivedat antenna

ldquoirdquoCode-Matrix

(Ntimess)

Coefficientsfrom the TX to ldquoirdquo RX ant

1

0

HH

= +=

R A H WR W

Clutter withCovariance M

Parameters

Designing the test

( )( )

[ ]Hii

r

rrT

H

H

Hf

Hfr

wwM

Mrr

Mααrrαα

E

max

0

1

01

1111

=

ltgt

hellip

helliphelliphellip (GLRT)

GLR Test Statistics

M-12Projector onto

Signal Subspace

||middot||2ri zi

12

1

22 12

1 1

0

r r

i ii i

H

T

H

minusminus

= =

gt=

ltsum sum M Az P M r

GLRT TestStatistics

PV=projector onto the rangespan of V

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Matched Filter

1

21

0

H

H

T

H

minus gtlt

r M 1

Pfa=PDeclare H1|H0 fixed (T)

Pd=PDeclare H1|H1 maximum

Performance

If α is complex Normal with variance σa2 (Swerling-I)

( )

infinrarrsdot

minusasymp=

sdot==

sdot+

minus

SCRSCRNP

PP

SCRNSCR

faSCRNfad

aT

T

ln11

1

21 σ1M1E

SCR Signal-to-Clutter Ratio per pulse

N Coherent Integration Gain

Angle Diversity

V ldquotarget extensionrdquoλ carrier wavelength

Different aspects at distance R

dgt λRV (full decorrelation)

Pulsed MIMO Radar Scheme

Pulsed Waveforms

Scheme 2

Underlying Assumption

ldquoNarrow-bandrdquo assumption on the transmitted pulses The target is seen in the same range-cell by all of the antennae

TX 1

TX 2

TX 3

TX s

dmax

RX 1

RX r

hmax

dmax+hmaxltltcB

Ex B=1MHz we have

dmax+hmaxltlt300m

Degrees of Freedom

The number of degrees of freedom is

( ) (tall code matrix)min

(fat code matrix)sr

s r NsN

⎧sdot = ⎨

⎩

Why Because the transmitted signal space has at most dimension N whereby a ldquofatrdquo code matrix means that the codewords are not linearly independent

MIMO Radar The Model

1

0

1 1

i i i

i i

H i rH i r

= + == =

r Aα wr w

helliphellip

receivedat antenna

ldquoirdquoCode-Matrix

(Ntimess)

Coefficientsfrom the TX to ldquoirdquo RX ant

1

0

HH

= +=

R A H WR W

Clutter withCovariance M

Parameters

Designing the test

( )( )

[ ]Hii

r

rrT

H

H

Hf

Hfr

wwM

Mrr

Mααrrαα

E

max

0

1

01

1111

=

ltgt

hellip

helliphelliphellip (GLRT)

GLR Test Statistics

M-12Projector onto

Signal Subspace

||middot||2ri zi

12

1

22 12

1 1

0

r r

i ii i

H

T

H

minusminus

= =

gt=

ltsum sum M Az P M r

GLRT TestStatistics

PV=projector onto the rangespan of V

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Performance

If α is complex Normal with variance σa2 (Swerling-I)

( )

infinrarrsdot

minusasymp=

sdot==

sdot+

minus

SCRSCRNP

PP

SCRNSCR

faSCRNfad

aT

T

ln11

1

21 σ1M1E

SCR Signal-to-Clutter Ratio per pulse

N Coherent Integration Gain

Angle Diversity

V ldquotarget extensionrdquoλ carrier wavelength

Different aspects at distance R

dgt λRV (full decorrelation)

Pulsed MIMO Radar Scheme

Pulsed Waveforms

Scheme 2

Underlying Assumption

ldquoNarrow-bandrdquo assumption on the transmitted pulses The target is seen in the same range-cell by all of the antennae

TX 1

TX 2

TX 3

TX s

dmax

RX 1

RX r

hmax

dmax+hmaxltltcB

Ex B=1MHz we have

dmax+hmaxltlt300m

Degrees of Freedom

The number of degrees of freedom is

( ) (tall code matrix)min

(fat code matrix)sr

s r NsN

⎧sdot = ⎨

⎩

Why Because the transmitted signal space has at most dimension N whereby a ldquofatrdquo code matrix means that the codewords are not linearly independent

MIMO Radar The Model

1

0

1 1

i i i

i i

H i rH i r

= + == =

r Aα wr w

helliphellip

receivedat antenna

ldquoirdquoCode-Matrix

(Ntimess)

Coefficientsfrom the TX to ldquoirdquo RX ant

1

0

HH

= +=

R A H WR W

Clutter withCovariance M

Parameters

Designing the test

( )( )

[ ]Hii

r

rrT

H

H

Hf

Hfr

wwM

Mrr

Mααrrαα

E

max

0

1

01

1111

=

ltgt

hellip

helliphelliphellip (GLRT)

GLR Test Statistics

M-12Projector onto

Signal Subspace

||middot||2ri zi

12

1

22 12

1 1

0

r r

i ii i

H

T

H

minusminus

= =

gt=

ltsum sum M Az P M r

GLRT TestStatistics

PV=projector onto the rangespan of V

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Angle Diversity

V ldquotarget extensionrdquoλ carrier wavelength

Different aspects at distance R

dgt λRV (full decorrelation)

Pulsed MIMO Radar Scheme

Pulsed Waveforms

Scheme 2

Underlying Assumption

ldquoNarrow-bandrdquo assumption on the transmitted pulses The target is seen in the same range-cell by all of the antennae

TX 1

TX 2

TX 3

TX s

dmax

RX 1

RX r

hmax

dmax+hmaxltltcB

Ex B=1MHz we have

dmax+hmaxltlt300m

Degrees of Freedom

The number of degrees of freedom is

( ) (tall code matrix)min

(fat code matrix)sr

s r NsN

⎧sdot = ⎨

⎩

Why Because the transmitted signal space has at most dimension N whereby a ldquofatrdquo code matrix means that the codewords are not linearly independent

MIMO Radar The Model

1

0

1 1

i i i

i i

H i rH i r

= + == =

r Aα wr w

helliphellip

receivedat antenna

ldquoirdquoCode-Matrix

(Ntimess)

Coefficientsfrom the TX to ldquoirdquo RX ant

1

0

HH

= +=

R A H WR W

Clutter withCovariance M

Parameters

Designing the test

( )( )

[ ]Hii

r

rrT

H

H

Hf

Hfr

wwM

Mrr

Mααrrαα

E

max

0

1

01

1111

=

ltgt

hellip

helliphelliphellip (GLRT)

GLR Test Statistics

M-12Projector onto

Signal Subspace

||middot||2ri zi

12

1

22 12

1 1

0

r r

i ii i

H

T

H

minusminus

= =

gt=

ltsum sum M Az P M r

GLRT TestStatistics

PV=projector onto the rangespan of V

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Pulsed MIMO Radar Scheme

Pulsed Waveforms

Scheme 2

Underlying Assumption

ldquoNarrow-bandrdquo assumption on the transmitted pulses The target is seen in the same range-cell by all of the antennae

TX 1

TX 2

TX 3

TX s

dmax

RX 1

RX r

hmax

dmax+hmaxltltcB

Ex B=1MHz we have

dmax+hmaxltlt300m

Degrees of Freedom

The number of degrees of freedom is

( ) (tall code matrix)min

(fat code matrix)sr

s r NsN

⎧sdot = ⎨

⎩

Why Because the transmitted signal space has at most dimension N whereby a ldquofatrdquo code matrix means that the codewords are not linearly independent

MIMO Radar The Model

1

0

1 1

i i i

i i

H i rH i r

= + == =

r Aα wr w

helliphellip

receivedat antenna

ldquoirdquoCode-Matrix

(Ntimess)

Coefficientsfrom the TX to ldquoirdquo RX ant

1

0

HH

= +=

R A H WR W

Clutter withCovariance M

Parameters

Designing the test

( )( )

[ ]Hii

r

rrT

H

H

Hf

Hfr

wwM

Mrr

Mααrrαα

E

max

0

1

01

1111

=

ltgt

hellip

helliphelliphellip (GLRT)

GLR Test Statistics

M-12Projector onto

Signal Subspace

||middot||2ri zi

12

1

22 12

1 1

0

r r

i ii i

H

T

H

minusminus

= =

gt=

ltsum sum M Az P M r

GLRT TestStatistics

PV=projector onto the rangespan of V

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Scheme 2

Underlying Assumption

ldquoNarrow-bandrdquo assumption on the transmitted pulses The target is seen in the same range-cell by all of the antennae

TX 1

TX 2

TX 3

TX s

dmax

RX 1

RX r

hmax

dmax+hmaxltltcB

Ex B=1MHz we have

dmax+hmaxltlt300m

Degrees of Freedom

The number of degrees of freedom is

( ) (tall code matrix)min

(fat code matrix)sr

s r NsN

⎧sdot = ⎨

⎩

Why Because the transmitted signal space has at most dimension N whereby a ldquofatrdquo code matrix means that the codewords are not linearly independent

MIMO Radar The Model

1

0

1 1

i i i

i i

H i rH i r

= + == =

r Aα wr w

helliphellip

receivedat antenna

ldquoirdquoCode-Matrix

(Ntimess)

Coefficientsfrom the TX to ldquoirdquo RX ant

1

0

HH

= +=

R A H WR W

Clutter withCovariance M

Parameters

Designing the test

( )( )

[ ]Hii

r

rrT

H

H

Hf

Hfr

wwM

Mrr

Mααrrαα

E

max

0

1

01

1111

=

ltgt

hellip

helliphelliphellip (GLRT)

GLR Test Statistics

M-12Projector onto

Signal Subspace

||middot||2ri zi

12

1

22 12

1 1

0

r r

i ii i

H

T

H

minusminus

= =

gt=

ltsum sum M Az P M r

GLRT TestStatistics

PV=projector onto the rangespan of V

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Underlying Assumption

ldquoNarrow-bandrdquo assumption on the transmitted pulses The target is seen in the same range-cell by all of the antennae

TX 1

TX 2

TX 3

TX s

dmax

RX 1

RX r

hmax

dmax+hmaxltltcB

Ex B=1MHz we have

dmax+hmaxltlt300m

Degrees of Freedom

The number of degrees of freedom is

( ) (tall code matrix)min

(fat code matrix)sr

s r NsN

⎧sdot = ⎨

⎩

Why Because the transmitted signal space has at most dimension N whereby a ldquofatrdquo code matrix means that the codewords are not linearly independent

MIMO Radar The Model

1

0

1 1

i i i

i i

H i rH i r

= + == =

r Aα wr w

helliphellip

receivedat antenna

ldquoirdquoCode-Matrix

(Ntimess)

Coefficientsfrom the TX to ldquoirdquo RX ant

1

0

HH

= +=

R A H WR W

Clutter withCovariance M

Parameters

Designing the test

( )( )

[ ]Hii

r

rrT

H

H

Hf

Hfr

wwM

Mrr

Mααrrαα

E

max

0

1

01

1111

=

ltgt

hellip

helliphelliphellip (GLRT)

GLR Test Statistics

M-12Projector onto

Signal Subspace

||middot||2ri zi

12

1

22 12

1 1

0

r r

i ii i

H

T

H

minusminus

= =

gt=

ltsum sum M Az P M r

GLRT TestStatistics

PV=projector onto the rangespan of V

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Degrees of Freedom

The number of degrees of freedom is

( ) (tall code matrix)min

(fat code matrix)sr

s r NsN

⎧sdot = ⎨

⎩

Why Because the transmitted signal space has at most dimension N whereby a ldquofatrdquo code matrix means that the codewords are not linearly independent

MIMO Radar The Model

1

0

1 1

i i i

i i

H i rH i r

= + == =

r Aα wr w

helliphellip

receivedat antenna

ldquoirdquoCode-Matrix

(Ntimess)

Coefficientsfrom the TX to ldquoirdquo RX ant

1

0

HH

= +=

R A H WR W

Clutter withCovariance M

Parameters

Designing the test

( )( )

[ ]Hii

r

rrT

H

H

Hf

Hfr

wwM

Mrr

Mααrrαα

E

max

0

1

01

1111

=

ltgt

hellip

helliphelliphellip (GLRT)

GLR Test Statistics

M-12Projector onto

Signal Subspace

||middot||2ri zi

12

1

22 12

1 1

0

r r

i ii i

H

T

H

minusminus

= =

gt=

ltsum sum M Az P M r

GLRT TestStatistics

PV=projector onto the rangespan of V

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

MIMO Radar The Model

1

0

1 1

i i i

i i

H i rH i r

= + == =

r Aα wr w

helliphellip

receivedat antenna

ldquoirdquoCode-Matrix

(Ntimess)

Coefficientsfrom the TX to ldquoirdquo RX ant

1

0

HH

= +=

R A H WR W

Clutter withCovariance M

Parameters

Designing the test

( )( )

[ ]Hii

r

rrT

H

H

Hf

Hfr

wwM

Mrr

Mααrrαα

E

max

0

1

01

1111

=

ltgt

hellip

helliphelliphellip (GLRT)

GLR Test Statistics

M-12Projector onto

Signal Subspace

||middot||2ri zi

12

1

22 12

1 1

0

r r

i ii i

H

T

H

minusminus

= =

gt=

ltsum sum M Az P M r

GLRT TestStatistics

PV=projector onto the rangespan of V

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Parameters

Designing the test

( )( )

[ ]Hii

r

rrT

H

H

Hf

Hfr

wwM

Mrr

Mααrrαα

E

max

0

1

01

1111

=

ltgt

hellip

helliphelliphellip (GLRT)

GLR Test Statistics

M-12Projector onto

Signal Subspace

||middot||2ri zi

12

1

22 12

1 1

0

r r

i ii i

H

T

H

minusminus

= =

gt=

ltsum sum M Az P M r

GLRT TestStatistics

PV=projector onto the rangespan of V

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Designing the test

( )( )

[ ]Hii

r

rrT

H

H

Hf

Hfr

wwM

Mrr

Mααrrαα

E

max

0

1

01

1111

=

ltgt

hellip

helliphelliphellip (GLRT)

GLR Test Statistics

M-12Projector onto

Signal Subspace

||middot||2ri zi

12

1

22 12

1 1

0

r r

i ii i

H

T

H

minusminus

= =

gt=

ltsum sum M Az P M r

GLRT TestStatistics

PV=projector onto the rangespan of V

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

GLR Test Statistics

M-12Projector onto

Signal Subspace

||middot||2ri zi

12

1

22 12

1 1

0

r r

i ii i

H

T

H

minusminus

= =

gt=

ltsum sum M Az P M r

GLRT TestStatistics

PV=projector onto the rangespan of V

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Remark

The maximum number of degrees of freedom (maximum diversity order under full-rank space-time coding) is r min (Ns)= rδ In fact

( )12

112 1 12H H

N

N s

N sminus

minusminus minus minus⎧ ge⎪= ⎨le⎪⎩

M A

M A A M A A MP

I

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Special cases

T

H

Hr

ii

0

12

1 ltgt

sum=

r

White noise N=s and orthogonal codes

White noise N=s and un-coded waveforms (ie rank-1 coding A=11T)

T

H

Hr

ii

T

0

12

1 ltgt

sum=

r1 (Coherent Integrator)

(Incoherent Integrator)

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Performances

1

0

1

1

E 2 2

krT

fak

rH H

d r i ii

TP ek

P Q T

θ

θ

minusminus

=

minus

=

=

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

sum

sumα A M Aα

Under rank-θ coding the performance of the GLRT admits the general expression

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Code Selection

No uniformly optimum (ie for any signal-to-clutter ratio and for any backscattering pdf) strategy exists First Step Assume Gaussian Scattering and study the impact of diverse design strategies aimed at optimizing some figures of merit under some constraint

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Figures of merit (to be maximized)

Lower Chernoff Bound (LCB) to the detection probability

Average received SCR per pulse per receive antenna

Mutual Information (MI)I(r1rrα1hellipαr)

( )20 1

11 min1 1

r

Td

j j a

P eθ

γ

γ γ λ σge=

⎡ ⎤ge minus ⎢ ⎥

+ +⎢ ⎥⎣ ⎦prod

( )2 2

1

1

tr Ha ai

i

SCRN N

θσ σ λminus

=

= = sumA M A

Eigenvaluesof

M-12AAHM-12

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

LCB-Optimal codes

Chose the space-time code A so as to maximize the LCB subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) Rank of the code matrix (θ)

Solution generate as many independent and identically distributed paths as possible

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

MI-Optimal codes

Chose the space-time code A so as to maximize the MI subject to constraints on

1) Average Received Signal-to-Clutter Ratio per pulse and per receive antenna SCR

2) The Maximum Rank of the code matrix (ie rank(A)leθ)

Solution generate as many (ie θr) independent and identically distributed paths as possible Thus with a definite constraint on the rank MI and LCB are equivalent

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

SCR-Optimal codes

Chose the space-time code A so as to maximize the average received SCR subject to a constraint on the transmitted energy

Solution Use rank-1 coding (ie transmit all of the energy along the least interfered direction of the signal space) note the consistency with the results concerning capacity (Boche et alii 2005)

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Other possible strategies

Both LCB and MI could be maximized under a constraint on the transmitted powerThe corresponding solution (ldquowaterfillingrdquo) leads to a performance improvement on the previous strategies but is useless (in that it requires knowledge of the average power of a target whose presence we are trying to ascertain)

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Semi-definite vs Definite rank constraints

With a definite rank constraint we are left free only of allocating the power on a pre-determined number of modesWith a semi-definite constraint we determine the optimum transmit policy ie we can decide between diversity and beam-formingDiversity maximization is not uniformly optimal for either Pd or LCB an indirect evidence that the uncertainty as to the target presence plays a fundamental roleConclusion MI is not a good figure of merit for waveform design in MIMO radar with surveillance functions

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Which Strategy is the best

N=r=s=2 white noise LCB-optimal Un-coded+coherent Un-coded+incoherent

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Diversity order and Energy Integration

( )( )

11

rank

r

dTP

r N SCR

θ

θ θ

θ

⎛ ⎞asymp minus ⎜ ⎟sdot⎝ ⎠

=A

Here the trade-off is apparent increasing the number of diversity paths (ie θ the code-matrix rank) results in a larger number of paths with lower average SCR

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Performance

Optimal rank of the code matrix (ie requiring minimum SCR to achieve the miss probability on the abscissas)Pfa=10-4N=s=4

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Some conclusions

Unlike LCB MI does not appear very suitable a figure of merit (oblivious as to the uncertainty as to the target presence)Mathematically we need a figure of merit which is neither Schur-Concave nor Schur-convex in order to study the trade-offs between diversity and integration and the corresponding interplay with the policy transmissionUnfortunately a semi-definite optimization problem involving the LCB is not very credible nor feasible (the LCB is not tight at low SNRrsquos)

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Alternative Kullback-Leibler Distance

Define the short-hand notation fi(rk)=f(rk|Hi) and recall theKullback-Leibler distance (or divergence)

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

11 0 10 1

0

00 1 01 0

1

ln

ln

fD f f D f d

f

fD f f D f d

f

= =

= =

int

int

rr r r r

r

rr r r r

r

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

A new figure of merit

( ) ( ) ( )( ) ( ) ( ) ( )( )1 0 0 11d D f f D f fλ λ= + minusA r r r r

Design strategy chose λ and design A according to

( ) maxsubject to a constraint on Average receivedTransmitted SCRd =A

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

General Comments

For λ=0 the criterion admits a nice interpretation in the light of Steinrsquos lemma and represents the asymptotically optimum criterion for fixed sample size multi-frame MIMO radarsFor non-zero λ the criterion can be nicely interpreted as an optimality criterion for sequential multi-frame MIMO radars

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Other interpretations

Maximizing D01 distances has a nice interpretation in the light of Steinrsquos lemma for fixed-sample size testsMaximizing D10 and D01 corresponds to some form of optimization (in terms of Average Sample Number) in a sequential MIMO-radar procedure (see Waldrsquos theory)

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Sequential MIMO

Defer detection until the returns from several frames (ie pulse trains) each with different carrier have been received and solve

( ) ( ) ( )( )

m mm

m⎧ +⎪= ⎨⎪⎩

AH WR

WThe decision process proceeds until a given pair (Pd Pfa) is reached by using a random number of samples and two detection thresholds The Average Sample Number corresponds to the expected observation number under the two alternatives

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

An example of performance

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Problem

Everything hinges upon uncorrelated Gaussian scatteringNotice that target scattering correlation is not under the control of the designer since it depends on the antennas spacing the target distance and its extension (the distance is in particular variable)The Gaussian assumption looks less dramatic

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Schur-Convexity

( )( ) ( )

1 1 1 1

Let ) Majorization iff

1 1 with

) is Schur-Convex iff

n n

m m n n

k k k kk k k k

a

x y m n x x

b ϕ

ϕ ϕ

= = = =

isin isin

le = minus =

rArr le

sum sum sum sum

x yx y

x

x y x y

≺

hellip

≺

R R

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Robust Coding (Unpublished HWC)

( )( )

( )

12 1 12

1

Denote

E

Denote

a figure of cost which is Schur-convex Here and in the

following denote the

Hi i

Hi

ni i

f ζ

ζ

minus

=

⎡ ⎤= ⎣ ⎦

α α

αR α α

R A M AR

X eigenvalues of

n nC timesisinX

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Robust Space-Time Coding

Mini-Max Design

( ) ( )12 1 12

1

Assume so we have eigenvalues to play with is chosen as the solution to the constrained problem

subject to constraints ontransmitted energyreceived SCR

maxminsH

i i

N s s

f ζ minus

=

ge

α ααA R

A

R A M AR

Trace of the scattering matrix maximum rank ( )s

αR

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Robust codes

A must be full-rank (ie s)The amount of power to be poured in each mode should ensure equal powers of the modes at the receiver ie we have to equalize all the modes which means that the most interfered modes require more power (dramatically different from water-filling)

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

What cost functions satisfy the conditions stated above

Any decreasing function of the SCR defined as SCR=Trace(ARαAHM-1)Any increasing function of the MMSE of a linear estimator of the scatteringUnder Gaussian scattering any decreasing function of the MI between received signal and target scattering

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Non-Gaussian Target Scattering

Why shouldnrsquot αi be circularly symmetric GaussThe target RCS fluctuation might be more constrained than exponentialThe target scattering might be similar to ldquocomposite surface scatteringrdquo wherein a ldquoslowly varyingrdquo texture is modulated by a rapidly decorrelating speckle

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

How to select the Code under NG

Set AHM-1A=CDCH with C unitary stimess D diagonal with θles positive entries and define

βi=CHαi i=1helliprThe βi are defined exchangeable (a la de Finetti) if

fβ1 hellipβr(x1xr)= fβ1 hellipβr(π(x1hellipxr))If the rvrsquos αi are independent unitarily invariant (ie their

pdf is invariant under unitary transformations) then defineβ=vec(β1 hellipβr)

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

LCB-optimal codes

If the αi are independent unitarily invariant and the βi are exchangeable then

The LCB is strictly concave for given θ=rank(A) and is maximized as we generate θr independent paths (ie as we maximize diversity which can be done through equalization under no power limitation)

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

MI-optimal codes

Under the same invariance condition the solution to the constrained (semi-definite) problem

( )( )( )

1 1

1

max

st Trace

rank

r r

H

I

ν

θ

minus

⎧⎪

le⎨⎪ le⎩

r r α α

A M A

A

hellip hellip

is solved by equalizing θr paths ie by maximizing the diversity order

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

On Unitary Invariance

Is this condition restrictive Yes but a very important family of non-Gaussian processes the Spherically Invariant Random Processes (SIRP) used to model composite scattering are UI Remark that x is an SIRP iff (Yao 1973)

x=sgwith g a complex white circularly symmetric Gaussian vector and

s a non-negative random variable whereby

( )2

2 21 exp ( )

2 2jxzf z dF s

s sπ⎛ ⎞

= minus⎜ ⎟⎝ ⎠

int

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Performance under K-distributed scattering ldquopower unlimitedrdquo

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Performance under K-distributed scattering ldquopower limitedrdquo

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Nexthellip

Optimized STC with constraints on the accuracy (eg in the range-doppler plane)What happens for non-colocated antennas (ie the delays vary for loose synchronization)For highly mobile targets the doppler frequencies vary across the antennas leading to a frequency-dispersive MIMO channel Whatrsquos the model and the coding strategyCan non linear codes help in any of the previous situationsWhat happens to these trade-offs if we use a (random) number of frames to make decisions ie if we merge the concepts of Space-Time Coding and Sequential decision making

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip

Nexthellip

Conditioned to the previous results is it possible to interleave several space-time codes (ie for detection of far-away and nearby objects)Is MIMO radar useful as a diversity generator or we realistically have to use it just as a ldquopower multiplexerrdquo

• Space-Time Coding in Statistical MIMO Radar
• MIMO Detection context
• RADAR Basic Operation
• Pulsed Radars
• Basic Receiver
• Matched Filter
• Performance
• Angle Diversity
• Pulsed MIMO Radar Scheme
• Scheme 2
• Underlying Assumption
• Degrees of Freedom
• MIMO Radar The Model
• Parameters
• Designing the test
• GLR Test Statistics
• Remark
• Special cases
• Performances
• Code Selection
• Figures of merit (to be maximized)
• LCB-Optimal codes
• MI-Optimal codes
• SCR-Optimal codes
• Other possible strategies
• Semi-definite vs Definite rank constraints
• Which Strategy is the best
• Diversity order and Energy Integration
• Performance
• Some conclusions
• Alternative Kullback-Leibler Distance
• A new figure of merit
• General Comments
• Other interpretations
• Sequential MIMO
• An example of performance
• Problem
• Schur-Convexity
• Robust Coding (Unpublished HWC)
• Robust Space-Time Coding
• Robust codes
• What cost functions satisfy the conditions stated above
• Non-Gaussian Target Scattering
• How to select the Code under NG
• LCB-optimal codes
• MI-optimal codes
• On Unitary Invariance
• Performance under K-distributed scattering ldquopower unlimitedrdquo
• Performance under K-distributed scattering ldquopower limitedrdquo
• Nexthellip
• Nexthellip