Signal Processing Algorithms for MIMO Radar Chun-Yang Chen and P. P. Vaidyanathan California...
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Transcript of Signal Processing Algorithms for MIMO Radar Chun-Yang Chen and P. P. Vaidyanathan California...
Signal Processing Algorithms for MIMO Radar
Chun-Yang Chen and P. P. Vaidyanathan
California Institute of TechnologyElectrical Engineering/DSP Lab
Candidacy
Chun-Yang Chen, Caltech DSP Lab | Candidacy
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
Conclusion and future work.
1MIMO Radar and Beamforming
MIMO Radar
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 | Candidacy
MIMO radar SIMO radar (Traditional)
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]
Chun-Yang Chen, Caltech DSP Lab | Candidacy
Radar Systems
Chun-Yang Chen, Caltech DSP Lab | Candidacy
t
Radartarget
R
Received Signal
Matched filter outputthreshold
R=ct/2
Detection
Ranging
Time
Radar was an acronym for Radio Detection and Ranging.Radar was an acronym for Radio Detection and Ranging.
Beampattern of Antennas
Chun-Yang Chen, Caltech DSP Lab | Candidacy
target
Beampattern is the antenna gain as a function of angle of arrival.Beampattern is the antenna gain as a function of angle of arrival.
Beampattern of Antennas
Chun-Yang Chen, Caltech DSP Lab | Candidacy
d/2
-d/2
2/
2/
sin2
0)(d
d
yj
dyeAE
siny
target
Plane wave-front
)(E
Beampattern is the antenna gain as a function of angle of arrival.Beampattern is the antenna gain as a function of angle of arrival.
)sin
sinc(sin
2
2/
2/ 0
ddyeA
y
d
d
j
Beampattern of Antennas
Chun-Yang Chen, Caltech DSP Lab | Candidacy
d/2
-d/2
2/
2/
sin2
0)(d
d
yj
dyeAE
siny
target
Plane wave-front
)(E
Beampattern is the antenna gain as a function of angle of arrival.Beampattern is the antenna gain as a function of angle of arrival.
Beampattern of Antennas
Chun-Yang Chen, Caltech DSP Lab | Candidacy
d/2
-d/2
2/
2/
sin2
0)(d
d
yj
dyeAE
siny
target
Fourier transform
Plane wave-front
)(E
Beampattern is the antenna gain as a function of angle of arrival.Beampattern is the antenna gain as a function of angle of arrival.
)sin
sinc(sin
2
2/
2/ 0
ddyeA
y
d
d
j
Antenna Array
Chun-Yang Chen, Caltech DSP Lab | Candidacy
N-1
I/Q Down-Convert and ADC
w*N-1
1
I/Q Down-Convert and ADC
w*1
0
I/Q Down-Convert and ADC
w*0
+
…
ywH
By linearly combining the output of a group of antennas, we can
control the beampattern digitally.
By linearly combining the output of a group of antennas, we can
control the beampattern digitally.
Antenna Array
Chun-Yang Chen, Caltech DSP Lab | Candidacy
)2( xkTftje
N-1
I/Q Down-Convert and ADC
w*N-1
1
I/Q Down-Convert and ADC
w*1
0
I/Q Down-Convert and ADC
w*0
+
…
Plane wave-front
sindsin)1( dN
ywH
sin2
1
0
*
1
0
sin2
*
)(
d
M
n
jnn
M
n
nd
j
n
ew
ewE
By linearly combining the output of a group of antennas, we can
control the beampattern digitally.
By linearly combining the output of a group of antennas, we can
control the beampattern digitally.
Antenna Array
Chun-Yang Chen, Caltech DSP Lab | Candidacy
)2( xkTftje
N-1
I/Q Down-Convert and ADC
w*N-1
1
I/Q Down-Convert and ADC
w*1
0
I/Q Down-Convert and ADC
w*0
+
…
Plane wave-front
sindsin)1( dN
ywH
sin2
1
0
*
1
0
sin2
*
)(
d
M
n
jnn
M
n
nd
j
n
ew
ewE
Discrete timeFourier transform
By linearly combining the output of a group of antennas, we can
control the beampattern digitally.
By linearly combining the output of a group of antennas, we can
control the beampattern digitally.
Antenna Array (2)
Chun-Yang Chen, Caltech DSP Lab | Candidacy
1
0
*)(M
n
jnnewE
Advantages of antenna array:
…
target
Beampattern can be steered digitally.Beampattern can be steered digitally.
Antenna Array (2)
Chun-Yang Chen, Caltech DSP Lab | Candidacy
1
0
*)(M
n
jnnewE
Advantages of antenna array:
…
target
interferences
Beampattern can be steered digitally.Beampattern can be steered digitally.
Beampattern can be adapted to the interferences.Beampattern can be adapted to the interferences.
Antenna Array (2)
Chun-Yang Chen, Caltech DSP Lab | Candidacy
1
0
*)(M
n
jnnewE
Advantages of antenna array:
…
target
interferences
Beampattern can be steered digitally.Beampattern can be steered digitally.
Beampattern can be adapted to the interferences.Beampattern can be adapted to the interferences.
The signal processing techniques to control the beampattern
is called beamforming.
The signal processing techniques to control the beampattern
is called beamforming.
Phased Array Beamforming
Chun-Yang Chen, Caltech DSP Lab | Candidacy
)2( xkTftje
N-1
I/Q Down-Convert and ADC
w*N-1
1
I/Q Down-Convert and ADC
w*1
0
I/Q Down-Convert and ADC
w*0
+
…
Plane wave-front
sindsin)1( dN
ywH
The response of a desired angle of arrival q can be maximized
by adjust wi.
The response of a desired angle of arrival q can be maximized
by adjust wi.
Phased Array Beamforming
Chun-Yang Chen, Caltech DSP Lab | Candidacy
)2( xkTftje
N-1
I/Q Down-Convert and ADC
w*N-1
1
I/Q Down-Convert and ADC
w*1
0
I/Q Down-Convert and ADC
w*0
+
…
Plane wave-front
sindsin)1( dN
ywH
TNdjdj
ee
)1(sin2
sin2
1
s
1 subject to
max2
w
swwH
The response of a desired angle of arrival q can be maximized
by adjust wi.
The response of a desired angle of arrival q can be maximized
by adjust wi.
Phased Array Beamforming
Chun-Yang Chen, Caltech DSP Lab | Candidacy
)2( xkTftje
N-1
I/Q Down-Convert and ADC
w*N-1
1
I/Q Down-Convert and ADC
w*1
0
I/Q Down-Convert and ADC
w*0
+
…
Plane wave-front
sindsin)1( dN
ywH
TNdjdj
ee
)1(sin2
sin2
1
s
1 subject to
max2
w
swwH
sw
The response of a desired angle of arrival q can be maximized
by adjust wi.
The response of a desired angle of arrival q can be maximized
by adjust wi.
Adaptive Beamforming
Chun-Yang Chen, Caltech DSP Lab | Candidacy
2
2
vw
swvsy
H
H
ESINR
The beamformer can be further designed to maximize the SINR
using second order statistics of received signals.
The beamformer can be further designed to maximize the SINR
using second order statistics of received signals.
Adaptive Beamforming
Chun-Yang Chen, Caltech DSP Lab | Candidacy
2
2
vw
swvsy
H
H
ESINR
H
H
H
E yyR
sw
Rwww
1 subject to
min
The beamformer can be further designed to maximize the SINR
using second order statistics of received signals.
The beamformer can be further designed to maximize the SINR
using second order statistics of received signals.
The SINR can be maximized by minimizing the total variance
while maintaining unity signal response.
The SINR can be maximized by minimizing the total variance
while maintaining unity signal response.
Adaptive Beamforming
Chun-Yang Chen, Caltech DSP Lab | Candidacy
2
2
vw
swvsy
H
H
ESINR
H
H
H
E yyR
sw
Rwww
1 subject to
mins1 Rw [Capon 1969]
MVDR beamformer
(Minimum Variance Distortionless Response)
The beamformer can be further designed to maximize the SINR
using second order statistics of received signals.
The beamformer can be further designed to maximize the SINR
using second order statistics of received signals.
The SINR can be maximized by minimizing the total variance
while maintaining unity signal response.
The SINR can be maximized by minimizing the total variance
while maintaining unity signal response.
An Example of Adaptive Beamforming
Chun-Yang Chen, Caltech DSP Lab | Candidacy
0 10 20 30 40 50 60 70 80 90-60
-50
-40
-30
-20
-10
0
10
20
Angle
Bea
m p
atte
rn (
dB)
Parameters Noise: 0dB Signal: 10dB, 43 degree Jammer1: 40dB, 30 degree Jammer2: 20dB, 75 degree
SINR Phased array: -20.13dB Adaptive: 9.70dB
However, the MVDR beamformer is very sensitive to target DoA (Direction of Arrival) mismatch.However, the MVDR beamformer is very sensitive to target DoA (Direction of Arrival) mismatch.
Adaptive beamforming can be very effective when there exists strong interferences.Adaptive beamforming can be very effective when there exists strong interferences.
Beamforming under Direction-of-Arrival Mismatch
Chun-Yang Chen, Caltech DSP Lab | Candidacy
[2] Chun-Yang Chen and P. P. Vaidyanathan, “Quadratically Constrained Beamforming Robust Against Direction-of-Arrival Mismatch,” IEEE Trans. on Signal Processing, July 2007.
SINR Matched DoA: 9.70dB Mismatched DoA: -8.80dB
Parameters Noise: 0dB Signal: 10dB, 43 degree Jammer1: 40dB, 30 degree Jammer2: 20dB, 75 degree
0 10 20 30 40 50 60 70 80 90-60
-50
-40
-30
-20
-10
0
10
20
Angle
Bea
m p
atte
rn (
dB)
Transmit Beamforming
Chun-Yang Chen, Caltech DSP Lab | Candidacy
N-1
I/Q Down-Convert and ADC
w*N-1
1
I/Q Down-Convert and ADC
w*1
0
I/Q Down-Convert and ADC
w*0
…
By weighting the input of a group of antennas, we can also control
the transmit beampattern digitally.
By weighting the input of a group of antennas, we can also control
the transmit beampattern digitally.
transmitted waveform
Transmit Beamforming
Chun-Yang Chen, Caltech DSP Lab | Candidacy
)2( xkTftje
N-1
I/Q Down-Convert and ADC
w*N-1
1
I/Q Down-Convert and ADC
w*1
0
I/Q Down-Convert and ADC
w*0
…
Plane wave-front
sindsin)1( dN
sin2
1
0
*
1
0
sin2
*
)(
d
M
n
jnn
M
n
nd
j
n
ew
ewE
By weighting the input of a group of antennas, we can also control
the transmit beampattern digitally.
By weighting the input of a group of antennas, we can also control
the transmit beampattern digitally.
transmitted waveform
Transmit Beamforming
Chun-Yang Chen, Caltech DSP Lab | Candidacy
)2( xkTftje
N-1
I/Q Down-Convert and ADC
w*N-1
1
I/Q Down-Convert and ADC
w*1
0
I/Q Down-Convert and ADC
w*0
…
Plane wave-front
sindsin)1( dN
sin2
1
0
*
1
0
sin2
*
)(
d
M
n
jnn
M
n
nd
j
n
ew
ewE
Discrete timeFourier transform
By weighting the input of a group of antennas, we can also control
the transmit beampattern digitally.
By weighting the input of a group of antennas, we can also control
the transmit beampattern digitally.
transmitted waveform
SIMO Radar (Traditional)
Transmitter: M antenna elements
dT
ej2(ft-x/)
w2 w1 w0
Transmitter emits
coherent waveforms.
(transmit beamforming)
Transmitter emits
coherent waveforms.
(transmit beamforming)
Receiver: N antenna elements
dR
ej2(ft-x/)
Number of received signals: N
Number of received signals: N
Chun-Yang Chen, Caltech DSP Lab | Candidacy
MIMO Radar
dT
ej2(ft-x/)
Transmitter emits
orthogonal waveforms.
(No transmit beamforming)
Transmitter emits
orthogonal waveforms.
(No transmit beamforming)
Transmitter: M antenna elements
dR
ej2(ft-x/)
MF MF…
…
Matched filters extract the M orthogonal waveforms.Overall number of signals:
NM
Matched filters extract the M orthogonal waveforms.Overall number of signals:
NM
Receiver: N antenna elements
Chun-Yang Chen, Caltech DSP Lab | Candidacy
MIMO Radar – Virtual Array
Transmitter: M antenna elements
Virtual array: NM elements
dT=NdR
ej2(ft-x/)
Receiver: N antenna elements
dR
ej2(ft-x/)
MF MF…
…
Chun-Yang Chen, Caltech DSP Lab | Candidacy
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 | Candidacy
However, a processing gain of M is lost because of the broad transmitting beam.
MIMO Transmitter vs. SIMO Transmitter
Chun-Yang Chen, Caltech DSP Lab | Candidacy
dT
w2 w1 w0 dT=NdR
…
In the application of scanning or imaging, global illumination is required. In this case the SIMO system needs to steer the transmit beam. This cancels the processing gain obtained by the focused beam in SIMO system.
In the application of scanning or imaging, global illumination is required. In this case the SIMO system needs to steer the transmit beam. This cancels the processing gain obtained by the focused beam in SIMO system.
2Space-Time Adaptive Processing
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).
Chun-Yang Chen, Caltech DSP Lab | Candidacy
The goal in STAP is to detect the moving target on the ground and estimate its
position and velocity.
The goal in STAP is to detect the moving target on the ground and estimate its
position and velocity.
Doppler Processing
Radartarget
v
ftje 2
Chun-Yang Chen, Caltech DSP Lab | Candidacy
Doppler Processing
fc
vfd
2 Doppler
effect:
Radartarget
v
ftje 2
tffj de )(2
Radartarget
v
The phenomenon can be used to estimate velocity.
The phenomenon can be used to estimate velocity.
Chun-Yang Chen, Caltech DSP Lab | Candidacy
Adaptive Temporal Processing
Chun-Yang Chen, Caltech DSP Lab | Candidacy
tfj de 2
I/Q Down-Convert and ADC
w*0 w*
1 w*L-1
T T
…
+
ywH
The same idea in adaptive beamforming can be applied
in Doppler processing.
The same idea in adaptive beamforming can be applied
in Doppler processing.
Adaptive Temporal Processing
Chun-Yang Chen, Caltech DSP Lab | Candidacy
tfj de 2
I/Q Down-Convert and ADC
w*0 w*
1 w*L-1
T T
…
+
ywH
s1 Rw
The same idea in adaptive beamforming can be applied
in Doppler processing.
The same idea in adaptive beamforming can be applied
in Doppler processing.
H
H
H
E yyR
sw
Rwww
1 subject to
min
Separable Space-Time Processing
Chun-Yang Chen, Caltech DSP Lab | Candidacy
N-1
I/Q Down-Convert and ADC
w*N-1
1
I/Q Down-Convert and ADC
w*1
0
I/Q Down-Convert and ADC
+
…
w*0 w*
1 w*L-1
T T
…
+
w*0
Filtered outthe unwanted angles
Filtered outthe unwanted frequencies
When the Doppler frequencies
and looking-directions are independent,
the spatial and temporal filtering
can be implemented separately.
When the Doppler frequencies
and looking-directions are independent,
the spatial and temporal filtering
can be implemented separately.
Example of Separable Space-Time Processing
Chun-Yang Chen, Caltech DSP Lab | Candidacy
Normalized Spatial Frequency
Nor
mal
ized
Dop
pler
Fre
quen
cy
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-70
-60
-50
-40
-30
-20
-10
Parameters Noise: 0dB Signal: 10dB, (0.11, 0.15) Jammer1: 40dB, (-0.22, x ) Jammer2: 20dB, (0.33, x ) Clutter: 40dB, (x , 0 )
However, the beampattern is not always separable.However, the beampattern is not always separable.
Space-time beampattern is the antenna gain as a function of angle of arrival and Doppler frequency.
Space-time beampattern is the antenna gain as a function of angle of arrival and Doppler frequency.
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 i
Di
sin2
Chun-Yang Chen, Caltech DSP Lab | Candidacy
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 i
Di
sin2
Chun-Yang Chen, Caltech DSP Lab | Candidacy
Example of a Non-Separable Beampattern
Chun-Yang Chen, Caltech DSP Lab | Candidacy
Normalized Spatial Frequency
Nor
mal
ized
Dop
pler
Fre
quen
cy
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Normalized Spatial Frequency
Nor
mal
ized
Dop
pler
Fre
quen
cy
-0.5 0 0.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-70
-60
-50
-40
-30
-20
-10
Non-Separable Separable
In a stationary radar, clutter Doppler frequency is zero for all angle of arrival.
In a stationary radar, clutter Doppler frequency is zero for all angle of arrival.
In airborne radar, clutter Doppler frequency is proportional to the angle of arrival. Consequently,
the beampattern becomes non-separable.
In airborne radar, clutter Doppler frequency is proportional to the angle of arrival. Consequently,
the beampattern becomes non-separable.
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 N: # of antennas
L
Chun-Yang Chen, Caltech DSP Lab | Candidacy
NL signals
As in beamforming and Doppler
processing, the maximum SINR can be
obtained by minimizing the
total variance while maintaining
unity signal response.
As in beamforming and Doppler
processing, the maximum SINR can be
obtained by minimizing the
total variance while maintaining
unity signal response.
Chun-Yang Chen, Caltech DSP Lab | Candidacy
Optimal Space-Time Adaptive Processing
Optimal Space-Time Adaptive Processing
NL signals
As in beamforming and Doppler
processing, the maximum SINR can be
obtained by minimizing the
total variance while maintaining
unity signal response.
As in beamforming and Doppler
processing, the maximum SINR can be
obtained by minimizing the
total variance while maintaining
unity signal response.
Chun-Yang Chen, Caltech DSP Lab | Candidacy
H
H
H
E yyR
sw
Rwww
1 subject to
min
s1 Rw
NL signals
As in beamforming and Doppler
processing, the maximum SINR can be
obtained by minimizing the
total variance while maintaining
unity signal response.
As in beamforming and Doppler
processing, the maximum SINR can be
obtained by minimizing the
total variance while maintaining
unity signal response.
Chun-Yang Chen, Caltech DSP Lab | Candidacy
H
H
H
E yyR
sw
Rwww
1 subject to
min
Optimal Space-Time Adaptive Processing
3An Efficient Space-Time Adaptive Processing Algorithm for MIMO Radar
MIMO Radar STAPSTAP MIMO Radar
NL signals
MIMOSTAP
M waveforms
NML signals
N: # of receiving antennas
M: # of transmitting antennas
L: # of pulses
[D. Bliss and K. Forsythe 03]
+
NM signals
Chun-Yang Chen, Caltech DSP Lab | Candidacy
MIMO Radar STAP (2)
1),( subject to
min
DH
H
fsw
Rwww
NML signals
MVDR (Capon) beamformer:
Chun-Yang Chen, Caltech DSP Lab | Candidacy
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 | Candidacy
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 | Candidacy
[Chun-Yang Chen and
P. P. Vaidyanathan,
ICASSP 07]
The MIMO STAP Signals
Chun-Yang Chen, Caltech DSP Lab | Candidacy
Received signal: yn,m,l n: receiving antenna index m: transmitting antenna index l: pulse trains index
The signals contain four components:
Target Noise Jammer Clutter
vvsinqi
airborne radar
jammer
vt
target
i
i-th clutter
Target Noise Jammer Clutter
Formulation of the Clutter Signals
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
Chun-Yang Chen, Caltech DSP Lab | Candidacy
n-th antennam-th matched filter outputl-th radar pulse
c ii
Ti
RN
i
vTlj
dmj
dnj
ilmn eeec1
sin22sin2sin2
,,
Formulation of the Clutter Signals
Matchedfilters
Pulse 2
Pulse 1
Pulse 0
Matchedfilters
Matchedfilters
…
Clutter points
n-th antennam-th matched filter outputl-th radar pulse
Nc: # of clutter points ri: i-th clutter signal amplitude Receiving antenna Transmitting antenna Doppler effect
c002 c012 c102
c001 c011 c101
c000 c010 c100
c112 c202 c212
c111 c201 c211
c110 c200 c210
cnml: clutter signals
Chun-Yang Chen, Caltech DSP Lab | Candidacy
Simplification of the Clutter Expression
Chun-Yang Chen, Caltech DSP Lab | Candidacy
c ii
Ti
RN
i
vTlj
dmj
dnj
ilmn eeec1
sin22sin2sin2
,,
R
T
d
d
Rd
vT2
, sinRs i i
df
Simplification of the Clutter Expression
Chun-Yang Chen, Caltech DSP Lab | Candidacy
5.05.0
)1()1(1
,
isf
LMNX
otherwise,0
0),2exp();( ,
,
Xxxffxc is
is
c ii
Ti
RN
i
vTlj
dmj
dnj
ilmn eeec1
sin22sin2sin2
,,
R
T
d
d
Rd
vT2
, sinRs i i
df
Simplification of the Clutter Expression
Chun-Yang Chen, Caltech DSP Lab | Candidacy
5.05.0
)1()1(1
,
isf
LMNX
otherwise,0
0),2exp();( ,
,
Xxxffxc is
is
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, );(
Simplification of the Clutter Expression
Chun-Yang Chen, Caltech DSP Lab | Candidacy
5.05.0
)1()1(1
,
isf
LMNX
otherwise,0
0),2exp();( ,
,
Xxxffxc is
is
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, );(
Trick: We can view the three dimensional signal as non-uniformly sampled one
dimensional signal.
Simplification of the Clutter Expression (2)
Chun-Yang Chen, Caltech DSP Lab | Candidacy
c ii
Ti
RN
i
vTlj
dmj
dnj
ilmn eeec1
sin22sin2sin2
,,
cN
ilmnisi fxc
1, );(
-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)}
otherwise,0
0),2exp();( ,
,
Xxxffxc is
is
-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 is
is
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 | Candidacy
is called PSWF. is called PSWF.
Prolate Spheroidal Wave Functions (PSWF)
dx k
X
kk )())-sinc((x)(0 ( )k x
in [0,X]
Frequency window
-0.5 0.5
Time window
X0( )k x ( )k xk
Chun-Yang Chen, Caltech DSP Lab | Candidacy
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
Frequency window
-0.5 0.5
Time window
X0( )k x ( )k xk
( )k x
Chun-Yang Chen, Caltech DSP Lab | Candidacy
X
kkkiis xfxc
0,, )();(
cN
ilmnisi fxc
1,lm,n, );(c
[D. Slepian, 62]
Concise Representation of the Clutter Signals
Chun-Yang Chen, Caltech DSP Lab | Candidacy
X
kkkiis xfxc
0,, )();(
cN
ilmnisi fxc
1,lm,n, );(c
X
kkki
N
ii lmn
c
0,
1
)(
X
kkk lmn
0
)( )1()1(1 LMNX
[D. Slepian, 62]
Concise Representation of the Clutter Signals
Chun-Yang Chen, Caltech DSP Lab | Candidacy
Chun-Yang Chen, Caltech DSP Lab | Candidacy
X
kkkiis xfxc
0,, )();(
cN
ilmnisi fxc
1,lm,n, );(c
X
kkki
N
ii lmn
c
0,
1
)(
X
kkk lmn
0
)( )1()1(1 LMNX
Hc ΨΨRR
Ψ )( lmnk consists ofc Ψξ
NML X+1
[D. Slepian, 62]
Concise Representation of the Clutter Signals
)1()1(1,,1,0 LMNk
Concise Representation of the Clutter Signals (2)
Hc ΨΨRR Ψ )( lmnk consists of
NMLN+(M-1)+(L-1)
Chun-Yang Chen, Caltech DSP Lab | Candidacy
)1()1(1,,1,0 LMNk
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 | Candidacy
Concise Representation of the Clutter Signals (2)
)1()1(1,,1,0 LMNk
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 | Candidacy
Concise Representation of the Clutter Signals (2)
[Chun-Yang Chen and P. P. Vaidyanathan, IEEE Trans SP, to appear]
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 | Candidacy
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 | Candidacy
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 | Candidacy
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 | Candidacy
The Proposed Methodlow rank
block diagonalNMLJc IRRR 2 H
v ΨR Ψ R
Chun-Yang Chen, Caltech DSP Lab | Candidacy
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 | Candidacy
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 | Candidacy
The proposed method
– Compute by sampling the prolate spheroidal wave functions.
– Instead of estimating R, we estimate Rv and Rx. The matrix Rv can
be estimated using a small number of clutter free samples.k
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 | Candidacy
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 Rx. 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 | Candidacy
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 | Candidacy
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 | Candidacy
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 | Candidacy
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 | Candidacy
The Proposed Method – Complexity
1111111 )( vH
vH
vv RΨΨRΨRΨRRR
Complexity:1 3: (( ( 1) ( 1)) )O N M L R
)(: 31 NOvR
Chun-Yang Chen, Caltech DSP Lab | Candidacy
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
1R )( 333 LMNO
Chun-Yang Chen, Caltech DSP Lab | Candidacy
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 | Candidacy
The Zero-Forcing Method
Typically we can assume that the clutter is very strong and all eigenvalues of Rx are very large.
1111111 )( vH
vH
vv RΨΨRΨRΨRRR
1 0 R
Chun-Yang Chen, Caltech DSP Lab | Candidacy
The Zero-Forcing Method
Typically we can assume that the clutter is very strong and all eigenvalues of Rx are 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 | Candidacy
Proposed method K=300,Kv=20
Simulations – SINR
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 | Candidacy
Parameters:N=10, M=5, L=16, CNR=50dB2 jammers, JNR=40dB
Target: (0,0.25)
Target: (0,0.25)
Chun-Yang Chen, Caltech DSP Lab | Candidacy
Simulations – Beampattern
Proposed ZF MethodProposed ZF Method
Normalized Spatial Frequency
Nor
mal
ized
Dop
pler
Fre
quen
cy
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Target
Jammer
Clutter
Jammer
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 clutter subspace
mismatch.– Develop clutter subspace estimation methods using a combination
of both the geometry and the received data.
Chun-Yang Chen, Caltech DSP Lab | Candidacy
4Research Topics
Chun-Yang Chen, Caltech DSP Lab | Candidacy
Research Topics
Robust Beamforming Algorithm against DoA Mismatch [2]
An Efficient STAPAlgorithm for
MIMO Radar [3]
Precoded V-BLAST Transceiver for MIMO
Communication [1]
Precoded V-BLAST Transceiver for MIMO
Communication [1]
Beamforming techniques for Radar systems Beamforming techniques for Radar systems
An Efficient STAPAlgorithm for
MIMO Radar [3]
Chun-Yang Chen, Caltech DSP Lab | Candidacy
Publications
Chun-Yang Chen, Caltech DSP Lab | Candidacy
[1] Chun-Yang Chen and P. P. Vaidyanathan, “Precoded FIR and Redundant V-BLAST Systems for Frequency-Selective MIMO Channels,” IEEE Trans. on Signal Processing, July, 2007.
[2] Chun-Yang Chen and P. P. Vaidyanathan, “Quadratically Constrained Beamforming Robust Against Direction-of-Arrival Mismatch,” IEEE Trans. on Signal Processing, Aug., 2007.
[3] Chun-Yang Chen and P. P. Vaidyanathan, “MIMO Radar Space-Time Adaptive Processing Using Prolate Spheroidal Wave Functions,” accepted to IEEE Trans. on Signal Processing.
[4] Chun-Yang Chen and P. P. Vaidyanathan, “MIMO Radar Space-Time Adaptive Processing and Signal Design,” invited chapter in MIMO Radar Signal Processing, J. Li and P. Stoica, Wiley, to be published.
Journal Papers
Book Chapter
Publications
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
[5] Chun-Yang Chen and P. P. Vaidyanathan, “A Subspace Method for MIMO Radar Space-Time Processing,” IEEE International Conference on Acoustics, Speech, and Signal Processing Honolulu, Hi, April 2007.
[6] Chun-Yang Chen and P. P. Vaidyanathan, “Beamforming issues in modern MIMO Radars with Doppler,” Proc. 40th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, Nov. 2006.
[7] Chun-Yang Chen and P. P. Vaidyanathan, “A Novel Beamformer Robust to Steering Vector Mismatch,” Proc. 40th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, Nov. 2006.
[8] Chun-Yang Chen and P. P. Vaidyanathan, “Precoded V-BLAST for ISI MIMO channels,” IEEE International Symposium on Circuit and System Kos, Greece, May 2006,
[9] Chun-Yang Chen and P. P. Vaidyanathan, “IIR Ultra-Wideband Pulse Shaper Design,” Proc. 39th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, Nov. 2005.
Conference Papers
Future Topic – Waveform Design in MIMO Radar
Chun-Yang Chen, Caltech DSP Lab | Candidacy
In SIMO radar, chirp waveform is often used in the transmitter to increase the range resolution. This technique is called pulse compression.
Radartarget
R
Received Signal
Matched filter outputTime
Range resolution
Range resolution
Future Topic – Waveform Design in MIMO Radar
Chun-Yang Chen, Caltech DSP Lab | Candidacy
In MIMO radar, multiple orthogonal waveforms are transmitted.
These waveforms affects not only the range resolution but also angle and Doppler resolution.
It is not clear how to design multiple waveforms which provide good range, angle and Doppler resolution.
Range resolution
Angleresolution
Doppler
Q&AThank You!
Any questions?
Chun-Yang Chen, Caltech DSP Lab | Candidacy
Parameters:N=10, M=5, L=16, CNR=50dB2 jammers, JNR=40dB
Normalized Spatial Frequency
Nor
mal
ized
Dop
pler
Fre
quen
cy
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Jammer 1
Clutter
Target
Jammer 2
Target: (0,0.25)
Target: (0,0.25)
Chun-Yang Chen, Caltech DSP Lab | Candidacy
Simulations – Beampattern
Proposed ZF MethodProposed ZF Method
Space-Time Beam Pattern
Normalized Spatial Freq.
Normalized Doppler
Freq.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Space-Time Beam Pattern
Normalized Spatial Freq.
Normalized Doppler
Freq.
Velocity mismatchVelocity mismatch
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Space-Time Beam Pattern
Normalized Spatial Freq.
Normalized Doppler
Freq.
Velocity misalignmentVelocity misalignment
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Space-Time Beam Pattern
Normalized Spatial Freq.
Normalized Doppler
Freq.
Internal clutter motion (ICM)Internal clutter motion (ICM)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
MIMO vs. SIMO
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Simulations
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Clutter Power in PSWF Vector Basis
0 50 100 150 200
-200
-150
-100
-50
0
50
100
Basis element index
Clu
tter
po
we
r (d
B)
Proposed subspace methodEigen decomposition
N+(M-1)+(L-1)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Proposed method K=300,Kv=20
Simulations
-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)
MVDR perfect R
Proposed ZF method Kv=20
Sample matrix inversion K=2000
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
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
MIMO Radar – Virtual Array (2)
Receiver: N elementsVirtual 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.
A processing gain of M is lost because of the broad transmitting beam.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Efficient Representation for the Clutter
X
kkkiis xfxc
0,, )();(
cN
ilmnisi fxc
1,lm,n, );(c
X
kkki
N
ii lmn
c
0,
1
)(
X
kkk lmn
0
)( )1()1(1 LMNX
[D. Slepian, 62]
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Efficient Representation for the Clutter
X
kkkiis xfxc
0,, )();(
cN
ilmnisi fxc
1,lm,n, );(c
X
kkki
N
ii lmn
c
0,
1
)(
X
kkk lmn
0
)( )1()1(1 LMNX
Hc ΨΨRR
Ψ )( lmnk consists ofc Ψξ
NML X+1
[D. Slepian, 62]
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
Simplification of the Clutter Expression
cN
iisi lmnfj
1, ))(2exp(
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 is
is
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)}
Receiver Transmitter Doppler
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
T
T
…
T
T
T
…
T
T
T
…T
…
…
T
T
T
…
Time window Frequency window
X -W W0 in [0,X]
( )k x ( )k k x