1. A Subspace Method for MIMO Radar Space-Time Adaptive Processing
Chun-Yang Chen and P. P. Vaidyanathan
California Institute of Technology
Electrical Engineering/DSP Lab
ICASSP 2007 student paper contest

2. Outline Review of the background The proposed MIMO-STAP method
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

3. SIMO radar (Traditional)
MIMO Radar
The radar systems which emits orthogonal (or noncoherent) waveforms in each transmitting antennas are called MIMO radar.
MIMO radar
SIMO radar (Traditional)
f2(t)
w2f(t)
f1(t)
w1f(t)
f0(t)
w0f(t)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

4. SIMO radar (Traditional)
MIMO Radar
The radar systems which emits orthogonal (or noncoherent) waveforms in each transmitting antennas are called MIMO radar.
MIMO radar
SIMO radar (Traditional)
f2(t)
w2f(t)
f1(t)
w1f(t)
f0(t)
w0f(t)
[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 | ICASSP 2007 student paper contest

5. SIMO Radar (Traditional)
Transmitter: M antenna elements
Receiver: N antenna elements
ej2p(ft-x/l)
ej2p(ft-x/l) dT dR
w2f(t)
w1f(t)
w0f(t)
Transmitter emits
coherent waveforms.
Number of received signals: N
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

6. MIMO Radar ej2p(ft-x/l) ej2p(ft-x/l) f2(t) f1(t) f0(t)
Transmitter: M antenna elements
Receiver: N antenna elements
ej2p(ft-x/l)
ej2p(ft-x/l) dR dT
f2(t)
f1(t)
f0(t) MF … MF …
Matched filters extract
the M orthogonal waveforms.
Overall number of signals: NM
Transmitter emits
orthogonal waveforms.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

7. MIMO Radar – Virtual Array
ej2p(ft-x/l)
ej2p(ft-x/l) q
f2(t) q dR
dT=NdR … MF MF
f1(t)
f0(t) …
Transmitter: M antenna elements
Receiver: N antenna elements q
Virtual array: NM elements
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

8. MIMO Radar – Virtual Array (2)
[D. W. Bliss and K. W. Forsythe, 03] + =
Virtual array: NM elements
Transmitter: M elements
Receiver: N elements
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

9. Space-Time Adaptive Processing
The adaptive techniques for processing the data from airborne antenna arrays are called space-time adaptive processing (STAP).
airborne radar v
vsinqi qi
jammer
target vt
i-th clutter
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest 9

10. Space-Time Adaptive Processing
The adaptive techniques for processing the data from airborne antenna arrays are called space-time adaptive processing (STAP).
airborne radar v
vsinqi qi
The clutter Doppler frequencies depend on angles. So, the problem is non-separable in space-time.
jammer
target vt
i-th clutter
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest 10

11. Space-Time Adaptive Processing (2)
L: # of radar pulses
Non separable: NL taps
Separable: N+L taps
Angle
processing L
Doppler
processing
Space-time
processing
Jointly process
Doppler frequencies and angles
Independently process
Doppler frequencies and angles
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

12. MIMO Radar STAP + MIMO STAP NML signals MIMO Radar STAP NM signals
NL signals
M waveforms
MIMO
STAP
[D. Bliss and
K. Forsythe 03]
N: # of receiving antennas
M: # of transmitting antennas
L: # of pulses
NML signals
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

13. MIMO Radar STAP (2) MVDR (Capon) beamformer: NML signals
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

14. MIMO Radar STAP (2) MVDR (Capon) beamformer: NMLxNML NML signals Pros
Cons
Very good spatial resolution
High complexity
Slow convergence
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

15. The Proposed Method
clutter
jammer
noise
We first observe each of the matrices Rc and RJ has some special structures.
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

16. Formulation of the Clutter Signals
points …
n-th antenna
m-th matched filter output
l-th radar pulse
Matched filters
Matched filters
Matched filters
Pulse 2
c002
c012
c102
c112
c202
c212
Nc: # of clutter points
ri: ith clutter signal amplitude
Pulse 1
c001
c011
c101
c111
c201
c211
Pulse 0
c000
c010
c100
c110
c200
c210
cnml: clutter signals
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

17. Simplification of the Clutter Expression
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

18. Simplification of the Clutter Expression-2 2 4 6 8 10 12
-1.5 -1
-0.5
0.5 1
1.5 x
Re{c(x;fs,i)}
Re{c(n+gm+bl;fs,i)}
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

19. “Time-and-Band” Limited Signals
The signals are well-localized in a time-frequency region.
Time
domain
[0 X]
To concisely represent these
signals, we can use a basis which
concentrates most of its energy
in this time-frequency region.
Freq.
domain
[ ]
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

20. Prolate Spheroidal Wave Functions (PSWF)
is called PSWF. X
-0.5
0.5
in [0,X]
Time window
Frequency window
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

21. Prolate Spheroidal Wave Functions (PSWF)
is called PSWF. X
-0.5
0.5
in [0,X]
Time window
Frequency window
[D. Slepian, 62]
Only X+1 basis functions are required to well represent the
“time-and-band limited” signal
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

22. Clutter Representation by PSWF
consists of
NML
N+g(M-1)+b(L-1)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

23. Clutter Representation by PSWF
consists of
NML
N+g(M-1)+b(L-1)
can be obtained by sampling from The PSWF
can be computed off-line
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

24. Clutter Representation by PSWF
consists of
NML
N+g(M-1)+b(L-1)
can be obtained by sampling from The PSWF
can be computed off-line
The NMLxNML clutter covariance matrix has only N+g(M-1)+b(L-1) significant eigenvalues. This is the MIMO extension of Brennan’s rule (1994).
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

25. Jammer Covariance Matrix
Matched filters
Matched filters
Matched filters
Pulse 2
j002
j012
j102
j112
j202
j212
Pulse 1
j001
j011
j101
j111
j201
j211
Pulse 0
j000
j010
j100
j110
j200
j210
jnml: jammer signals
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

26. Jammer Covariance Matrix
Jammer signals in different pulses are independent.
Matched filters
Matched filters
Matched filters
Pulse 2
j002
j012
j102
j112
j202
j212
Pulse 1
j001
j011
j101
j111
j201
j211
Pulse 0
j000
j010
j100
j110
j200
j210
jnml: jammer signals
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

27. Jammer Covariance Matrix
Jammer signals in different pulses are independent.
Jammer signals in different matched filter outputs are independent.
Matched filters
Matched filters
Matched filters
Pulse 2
j002
j012
j102
j112
j202
j212
Pulse 1
j001
j011
j101
j111
j201
j211
Pulse 0
j000
j010
j100
j110
j200
j210
jnml: jammer signals
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

28. Jammer Covariance Matrix
Jammer signals in different pulses are independent.
Jammer signals in different matched filter outputs are independent.
Matched filters
Matched filters
Matched filters
Pulse 2
j002
j012
j102
j112
j202
j212
Pulse 1
j001
j011
j101
j111
j201
j211
Pulse 0
j000
j010
j100
j110
j200
j210
jnml: jammer signals
Block diagonal
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

29. The Proposed Method low rank block diagonal
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

30. By Matrix Inversion Lemma
The Proposed Method
low rank
block diagonal
By Matrix Inversion Lemma
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

31. By Matrix Inversion Lemma
The Proposed Method
low rank
block diagonal
By Matrix Inversion Lemma
The proposed method
Compute Y by sampling the prolate spheroidal wave functions.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

32. By Matrix Inversion Lemma
The Proposed Method
low rank
block diagonal
By Matrix Inversion Lemma
The proposed method
Compute Y 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.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

33. By Matrix Inversion Lemma
The Proposed Method
low rank
block diagonal
By Matrix Inversion Lemma
The proposed method
Compute Y 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.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

34. The Proposed Method – Advantages
Inversions are easy to compute
:block diagonal
:small size
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

35. The Proposed Method – Advantages
Low
complexity
Inversions are easy to compute
:block diagonal
:small size
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

36. The Proposed Method – Advantages
Low
complexity
Inversions are easy to compute
:block diagonal
:small size
Fewer parameters need to be estimated
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

37. The Proposed Method – Advantages
Low
complexity
Inversions are easy to compute
:block diagonal
:small size
Fast
convergence
Fewer parameters need to be estimated
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

38. The Proposed Method – Complexity
Direct method
The proposed method
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

39. The Zero-Forcing Method
Typically we can assume that the clutter is very strong and all eigenvalues of Rx are very large.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

40. The Zero-Forcing Method
Typically we can assume that the clutter is very strong and all eigenvalues of Rx are very large.
Zero-forcing method
The entire clutter space is nulled out without estimation
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

41. Simulations K: number of samples Kv: number of clutter free samples
Parameters:
N=10, M=5, L=16
CNR=50dB
2 jammers, JNR=40dB
SINR of a target at angle zero and Doppler frequencies [-0.5, 0.5]
MVDR known R (unrealizable) -2
Sample matrix inversion K=1000 -4
Diagonal loading K=300 -6
Principal component K=300
SINR (dB) -8
Proposed method K=300,Kv=20
-10
Proposed ZF method Kv=20
-12
-14
K: number of samples
Kv: number of clutter free samples
collected in passive mode
-16
-0.5
-0.4
-0.3
-0.2
-0.1
0.1
0.2
0.3
0.4
0.5
Normalized Doppler frequency
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

42. Conclusion and Future Work
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

43. Q&A Thank You! Any questions?
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest