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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

“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 [-0.5 0.5] Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The Proposed Method – Complexity Direct method The proposed method 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 Rx are very large. 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 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

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

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

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