A Subspace Method for MIMO Radar Space-Time Adaptive Processing Chun-Yang Chen and P. P....
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Transcript of A Subspace Method for MIMO Radar Space-Time Adaptive Processing Chun-Yang Chen and P. P....
A Subspace Method for MIMO Radar Space-Time Adaptive Processing
Chun-Yang Chen and P. P. Vaidyanathan
California Institute of TechnologyElectrical Engineering/DSP Lab
ICASSP 2007 student paper contest
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Outline
Review of the background– MIMO radar– Space-Time Adaptive Processing (STAP)
The proposed MIMO-STAP method– Formulation of the MIMO-STAP– Prolate spheroidal representation of the clutter signals– Deriving the proposed method
Simulations
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
MIMO Radar
MIMO radar
SIMO radar (Traditional)
The radar systems which emits orthogonal (or noncoherent) waveforms in each transmitting antennas are called MIMO radar.
w2w1
w0
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
MIMO Radar
MIMO radar
SIMO radar (Traditional)
w2w1
w0
The radar systems which emits orthogonal (or noncoherent) waveforms in each transmitting antennas are called MIMO radar.
[D. J. Rabideau and P. Parker, 03] [D. Bliss and K. Forsythe, 03][E. Fishler et al. 04]
[F. C. Robey, 04][D. R. Fuhrmann and G. S. Antonio, 05]
SIMO Radar (Traditional)
Transmitter: M antenna elements Receiver: N antenna elements
dT
ej2(ft-x/)
w2 w1 w0dR
ej2(ft-x/)
Transmitter emits
coherent waveforms.
Transmitter emits
coherent waveforms.Number of received signals:
N
Number of received signals: N
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
MIMO Radar
dT
ej2(ft-x/)
dR
ej2(ft-x/)
MF MF…
…
Transmitter emits
orthogonal waveforms.
Transmitter emits
orthogonal waveforms.
Matched filters extract the M orthogonal waveforms.Overall number of signals:
NM
Matched filters extract the M orthogonal waveforms.Overall number of signals:
NM
Transmitter: M antenna elements Receiver: N antenna elements
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
MIMO Radar – Virtual Array
Transmitter: M antenna elements Receiver: N antenna elements
Virtual array: NM elements
dT=NdR
ej2(ft-x/)
dR
ej2(ft-x/)
MF MF…
…
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
MIMO Radar – Virtual Array (2)
Receiver: N elements
Virtual array: NM elements
Transmitter: M elements
+ =
[D. W. Bliss and K. W. Forsythe, 03]
The spatial resolution for clutter is the same as a receiving array with NM physical array elements.
NM degrees of freedom can be created using only N+M physical array elements.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Space-Time Adaptive Processing
vvsini
airborne radar
jammertarget
i-th clutter
vt
i
The adaptive techniques for processing the data from airborne antenna arrays are called space-time adaptive processing (STAP).
fc
vf iDi
sin2
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Space-Time Adaptive Processing
vvsini
airborne radar
jammertarget
i-th clutter
vt
iThe clutter Doppler frequencies
depend on angles. So, the problem is non-separable in
space-time.
The clutter Doppler frequencies depend on angles. So, the
problem is non-separable in space-time.
The adaptive techniques for processing the data from airborne antenna arrays are called space-time adaptive processing (STAP).
fc
vf iDi
sin2
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Space-Time Adaptive Processing (2)
Separable: N+L tapsNon separable: NL taps
Jointly processDoppler frequencies and angles
Jointly processDoppler frequencies and angles
Independently process Doppler frequencies and angles
Independently process Doppler frequencies and angles
Angle processing
Doppler processingSpace-time
processing
L: # of radar pulses
L
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
MIMO Radar STAPSTAP MIMO Radar
NL signals
MIMOSTAP
M waveforms
NML signals
N: # of receiving antennasM: # of transmitting
antennasL: # of pulses
[D. Bliss and K. Forsythe 03]
+
NM signals
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
MIMO Radar STAP (2)
1),( subject to
min
DH
H
fsw
Rwww
NML signals
MVDR (Capon) beamformer:
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
MIMO Radar STAP (2)
1),( subject to
min
DH
H
fsw
Rwww
NML signals
MVDR (Capon) beamformer:
),(1DfsRw
Very good spatial resolutionVery good spatial resolution
Pros ConsCons
High complexityHigh complexity
Slow convergenceSlow convergence
NMLxNML
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
The Proposed Method
),(1DfsRw
NMLJc IRRR 2
We first observe each of the matrices Rc and RJ has
some special structures.
clutter jammer noise
We show how to exploit the structures of these
matrices to compute R-1 more accurately and
efficiently.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
c ii
Ti
RN
i
vTlj
dmj
dnj
ilmn eeec1
sin22sin2sin2
,,
Formulation of the Clutter Signals
Nc: # of clutter points ith clutter signal amplitude
Matchedfilters
Pulse 2
Pulse 1
Pulse 0
Matchedfilters
Matchedfilters
c002 c012 c102
c001 c011 c101
c000 c010 c100
c112 c202 c212
c111 c201 c211
c110 c200 c210
cnml: clutter signals
…
Clutter points
n-th antennam-th matched filter outputl-th radar pulse
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Simplification of the Clutter Expression
c ii
Ti
RN
i
vTlj
dmj
dnj
ilmn eeec1
sin22sin2sin2
,,
R
T
d
d
Rd
vT2
, sinRs i i
df
cN
ilmnisi fxc
1, );(
otherwise,0
0),2exp();( ,
,
Xxxffxc isis
5.05.0
)1()1(1
,
isf
LMNX
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Simplification of the Clutter Expression
c ii
Ti
RN
i
vTlj
dmj
dnj
ilmn eeec1
sin22sin2sin2
,,
R
T
d
d
Rd
vT2
, sinRs i i
df
cN
ilmnisi fxc
1, );(
otherwise,0
0),2exp();( ,
,
Xxxffxc isis
5.05.0
)1()1(1
,
isf
LMNX
-2 0 2 4 6 8 10 12-1.5
-1
-0.5
0
0.5
1
1.5
x
Re{c(x;fs,i)}Re{c(n+m+l;fs,i)}
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
-50 0 50 100 150-1
0
1
-1 -0.5 0 0.5 10
20
40
60
80
100
“Time-and-Band” Limited Signals
otherwise,0
0),2exp();( ,
,
Xxxffxc isis
5.05.0
)1()1(1
,
isf
LMNX
[0 X]
[-0.5 0.5]
Timedomain
Freq.domain
The signals are well-localized in a time-frequency region.
The signals are well-localized in a time-frequency region.
To concisely represent these signals, we can use a basis which concentrates most of its energy in this time-frequency region.
To concisely represent these signals, we can use a basis which concentrates most of its energy in this time-frequency region.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
is called PSWF. is called PSWF.
Prolate Spheroidal Wave Functions (PSWF)
dx k
X
kk )())-sinc((x)(0 ( )k x
Time window Frequency window
X -0.5 0.50 in [0,X]
( )k x ( )k xk
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
is called PSWF. is called PSWF.
Prolate Spheroidal Wave Functions (PSWF)
, ,0
( ; )X
s i i kk
c x f
dx k
X
kk )())-sinc((x)(0
[D. Slepian, 62]
( )k x
in [0,X]
Only X+1 basis functions are required to well represent the “time-and-band limited” signal
Only X+1 basis functions are required to well represent the “time-and-band limited” signal
Time window Frequency window
X -0.5 0.50( )k x ( )k xk
( )k x
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
)1()1(1,,1,0 LMNk
Clutter Representation by PSWF
Hc ΨΨRR Ψ )( lmnk consists of
NMLN+(M-1)+(L-1)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
)1()1(1,,1,0 LMNk
Clutter Representation by PSWF
Hc ΨΨRR Ψ )( lmnk consists of
can be obtained by sampling from . The PSWF can be computed off-line can be obtained by sampling from . The PSWF can be computed off-lineΨ k
NMLN+(M-1)+(L-1)
k
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
)1()1(1,,1,0 LMNk
Clutter Representation by PSWF
Hc ΨΨRR Ψ )( lmnk consists of
can be obtained by sampling from . The PSWF can be computed off-line can be obtained by sampling from . The PSWF can be computed off-lineΨ k
NMLN+(M-1)+(L-1)
k
The NMLxNML clutter covariance matrix has only N+(M-1)+(L-1) significant eigenvalues. This is the MIMO extension of Brennan’s rule (1994).
The NMLxNML clutter covariance matrix has only N+(M-1)+(L-1) significant eigenvalues. This is the MIMO extension of Brennan’s rule (1994).
cR
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Jammer Covariance Matrix
Matchedfilters
jammer
Pulse 2
Pulse 1
Pulse 0
Matchedfilters
Matchedfilters
j002 j012 j102
j001 j011 j101
j000 j010 j100
j112 j202 j212
j111 j201 j211
j110 j200 j210
jnml: jammer signals
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Jammer Covariance Matrix
Matchedfilters
jammer
Pulse 2
Pulse 1
Pulse 0
Jammer signals in different pulses are independent.
Jammer signals in different pulses are independent.
Matchedfilters
Matchedfilters
j002 j012 j102
j001 j011 j101
j000 j010 j100
j112 j202 j212
j111 j201 j211
j110 j200 j210
jnml: jammer signals
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Jammer Covariance Matrix
Matchedfilters
jammer
Pulse 2
Pulse 1
Pulse 0
Jammer signals in different pulses are independent.
Jammer signals in different pulses are independent.
Jammer signals in different matched filter outputs are independent.
Jammer signals in different matched filter outputs are independent.Matched
filtersMatched
filters
j002 j012 j102
j001 j011 j101
j000 j010 j100
j112 j202 j212
j111 j201 j211
j110 j200 j210
jnml: jammer signals
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Jammer Covariance Matrix
Matchedfilters
jammer
Pulse 2
Pulse 1
Pulse 0
Jammer signals in different pulses are independent.
Jammer signals in different pulses are independent.
Jammer signals in different matched filter outputs are independent.
Jammer signals in different matched filter outputs are independent.
Js
Js
Js
J
R00
0
R0
00R
R
Matchedfilters
Matchedfilters
Block diagonalBlock diagonal
j002 j012 j102
j001 j011 j101
j000 j010 j100
j112 j202 j212
j111 j201 j211
j110 j200 j210
jnml: jammer signals
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
The Proposed Methodlow rank
block diagonalNMLJc IRRR 2 H
v ΨR Ψ R
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
The Proposed Methodlow rank
block diagonalNMLJc IRRR 2 H
v ΨR Ψ R
1111111 )( vH
vH
vv RΨΨRΨRΨRRR
By Matrix Inversion Lemma
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
The Proposed Methodlow rank
block diagonalNMLJc IRRR 2 H
v ΨR Ψ R
1111111 )( vH
vH
vv RΨΨRΨRΨRRR
The proposed method– Compute by sampling the prolate spheroidal wave
functions.
By Matrix Inversion Lemma
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
The Proposed Methodlow rank
block diagonalNMLJc IRRR 2 H
v ΨR Ψ R
1111111 )( vH
vH
vv RΨΨRΨRΨRRR
The proposed method– Compute by sampling the prolate spheroidal wave
functions.
– Instead of estimating R, we estimate Rv and R. The
matrix Rv can be estimated using a small number of
clutter free samples.
By Matrix Inversion Lemma
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
The Proposed Methodlow rank
block diagonalNMLJc IRRR 2 H
v ΨR Ψ R
The proposed method– Compute by sampling the prolate spheroidal wave
functions.
– Instead of estimating R, we estimate Rv and R. The
matrix Rv can be estimated using a small number of
clutter free samples.
– Use the above equation to compute R-1.
1111111 )( vH
vH
vv RΨΨRΨRΨRRR
By Matrix Inversion Lemma
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
The Proposed Method – Advantages
vR
R
:block diagonal
:small size
Inversions are easy to compute
Inversions are easy to compute
1111111 )( vH
vH
vv RΨΨRΨRΨRRR
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
The Proposed Method – Advantages
vR
R
:block diagonal
:small size
Inversions are easy to compute
Inversions are easy to compute
1111111 )( vH
vH
vv RΨΨRΨRΨRRR
Low complexityLow complexity
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
The Proposed Method – Advantages
vR
R
:block diagonal
:small size
Inversions are easy to compute
Inversions are easy to compute
Fewer parameters need to be estimated
Fewer parameters need to be estimated
1111111 )( vH
vH
vv RΨΨRΨRΨRRR
Low complexityLow complexity
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
The Proposed Method – Advantages
vR
R
:block diagonal
:small size
Inversions are easy to compute
Inversions are easy to compute
Fewer parameters need to be estimated
Fewer parameters need to be estimated
1111111 )( vH
vH
vv RΨΨRΨRΨRRR
Low complexityLow complexity
FastconvergenceFastconvergence
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
The Proposed Method – Complexity
1111111 )( vH
vH
vv RΨΨRΨRΨRRR
Complexity:1 3: (( ( 1) ( 1)) )O N M L R
)(: 31 NOvR
Direct method
The proposed method
),(1DfsR )( 333 LMNO )))1()1((( 3 LMNO
1R )))1()1((( 222 LMNLMNO )( 333 LMNO
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
The Zero-Forcing Method
Typically we can assume that the clutter is very strong and all eigenvalues of Rare
very large.
1111111 )( vH
vH
vv RΨΨRΨRΨRRR
1 0 R
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
The Zero-Forcing Method
Typically we can assume that the clutter is very strong and all eigenvalues of Rare
very large.
1 1 1 1 1 1( )H Hv v v v
R R R Ψ Ψ R Ψ Ψ R
Zero-forcing method
– The entire clutter space is nulled out without estimation
1111111 )( vH
vH
vv RΨΨRΨRΨRRR
1 0 R
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Proposed method K=300,Kv=20
Simulations
MVDR known R (unrealizable)
Proposed ZF method Kv=20
Sample matrix inversion K=1000
Diagonal loading K=300
Principal component K=300
SINR of a target at angle zero and Doppler frequencies [-0.5, 0.5]
SINR of a target at angle zero and Doppler frequencies [-0.5, 0.5]
Parameters:N=10, M=5, L=16CNR=50dB2 jammers, JNR=40dB
K: number of samplesKv: number of clutter free samples collected in passive mode
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-16
-14
-12
-10
-8
-6
-4
-2
0
Normalized Doppler frequency
SIN
R (
dB
)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Conclusion and Future Work
Conclusion– The clutter subspace is derived using the geometry of the
problem.(data independent)
– A new STAP method for MIMO radar is developed.– The new method is both efficient and accurate.
Future work– This method is entirely based on the ideal model.– Find algorithms which are robust against model
mismatch.– Develop clutter subspace estimation methods using a
combination of both the geometry and the received data.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Q&AThank You!
Any questions?
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest