Compressed Sensing in MIMO Radar Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP Lab Asilomar 2008

Outline  Review of the background –Compressed sensing [Donoho 06, Candes&Tao 06…] Compressed sensing in radar [Herman & Strohmer 08] –MIMO radar [Bliss & Forsythe 03, Robey et al. 04, Fishler et al. 04….]  Compressed sensing in MIMO radar –Compressed sensing receiver –Waveform optimization –Examples  Conclusion 2Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008

1 Review of the keywords: Compressed sensing, MIMO Radar 3

Brief Review of Compressed Sensing 4Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Goal: Reconstruct s from y.

Brief Review of Compressed Sensing 5Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Goal: Reconstruct s from y. Incoherence: is small.

Brief Review of Compressed Sensing 6Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Goal: Reconstruct s from y. Incoherence: is small. Sparsity: is small.

Sparsity: is small. Brief Review of Compressed Sensing 7Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Goal: Reconstruct s from y. Incoherence: is small. Given y and , s can be perfectly recovered by sparse approximation methods even when dim(y)<dim(s). Given y and , s can be perfectly recovered by sparse approximation methods even when dim(y)<dim(s).

Sparsity: is small. Brief Review of Compressed Sensing 8Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Goal: Reconstruct s from y. Incoherence: is small. This concept can be applied to sampling and compression. Given y and , s can be perfectly recovered by sparse approximation methods even when dim(y)<dim(s). Given y and , s can be perfectly recovered by sparse approximation methods even when dim(y)<dim(s).

Review: Compressed Sensing in Radar 9Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 [Herman & Strohmer08] u y targets Range Doppler

Review: Compressed Sensing in Radar 10Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 [Herman & Strohmer08] u y targets Range Doppler s i : target RCS in the i-th Range-Doppler cell. * * * *

Review: Compressed Sensing in Radar 11Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 [Herman & Strohmer08] u y targets Range Doppler s i : target RCS in the i-th Range-Doppler cell.  is a function of the transmitted waveform u. * * * *

* * * * Review: Compressed Sensing in Radar 12Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 [Herman & Strohmer08] u y targets Range Doppler s i : target RCS in the i-th Range-Doppler cell. Assumption: s is sparse. Transmitted waveform u can be chosen such that  is incoherent.  is a function of the transmitted waveform u.

Review: Compressed Sensing in Radar 13Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 s i : target RCS in the i-th Range-Doppler cell. Assumption: s is sparse. Transmitted waveform u can be chosen such that  is incoherent. Target scene s can be reconstructed by compressed sensing method. High resolution can be achieved. [Herman & Strohmer08]  is a function of the transmitted waveform u. * * * *

Brief Review of MIMO Radar u   u   u   w 2 u  w 1 u  w 0 u   Advantages –Better spatial resolution [Bliss & Forsythe 03] –Flexible transmit beampattern design [Fuhrmann & San Antonio 04] –Improved parameter identifiability [Li et al. 07] Phased array radar (Traditional) Each element transmits a scaled version of a single waveform. Phased array radar (Traditional) Each element transmits a scaled version of a single waveform. MIMO Radar Each element can transmit an arbitrary waveform. MIMO Radar Each element can transmit an arbitrary waveform. Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008

2 Compressed Sensing in MIMO Radar 15

MIMO Radar Signal Model 16Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 … u 0 (t)u 1 (t)u M-1 (t) (p,,fD)(p,,fD)  delay f D  Doppler p  direction

MIMO Radar Signal Model 17Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 … u 0 (t)u 1 (t)u M-1 (t) (p,,fD)(p,,fD) … y 0 (t)y 1 (t)y N-1 (t) (p,,fD)(p,,fD)  delay f D  Doppler p  direction

MIMO Radar Signal Model 18Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 … u 0 (t)u 1 (t)u M-1 (t) (p,,fD)(p,,fD) … y 0 (t)y 1 (t)y N-1 (t) (p,,fD)(p,,fD)  delay f D  Doppler p  direction Received signals

MIMO Radar Signal Model 19Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 … u 0 (t)u 1 (t)u M-1 (t) (p,,fD)(p,,fD) … y 0 (t)y 1 (t)y N-1 (t) (p,,fD)(p,,fD)  delay f D  Doppler p  direction Range

MIMO Radar Signal Model 20Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 … u 0 (t)u 1 (t)u M-1 (t) (p,,fD)(p,,fD) … y 0 (t)y 1 (t)y N-1 (t) (p,,fD)(p,,fD)  delay f D  Doppler p  direction x m : location of the m-th transmitter y n : location of the n-th transmitter Cross range

MIMO Radar Signal Model 21Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 … u 0 (t)u 1 (t)u M-1 (t) (p,,fD)(p,,fD) … y 0 (t)y 1 (t)y N-1 (t) (p,,fD)(p,,fD)  delay f D  Doppler p  direction x m : location of the m-th transmitter y n : location of the n-th transmitter for linear array

MIMO Radar Signal Model 22Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 … u 0 (t)u 1 (t)u M-1 (t) (p,,fD)(p,,fD) … y 0 (t)y 1 (t)y N-1 (t) (p,,fD)(p,,fD)  delay f D  Doppler p  direction x m : location of the m-th transmitter y n : location of the n-th transmitter Doppler

MIMO Radar Signal Model 23Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Discrete Model:

MIMO Radar Signal Model 24Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Discrete Model: Range Range Cell:L: Length of u m

MIMO Radar Signal Model 25Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Discrete Model: Doppler Range Cell:L: Length of u m Doppler Cell:

MIMO Radar Signal Model 26Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Range Cell:L: Length of u m M: # of transmitting antennas N: # of receiving antennas Doppler Cell: Angle Cell: Discrete Model: Angle

MIMO Radar Signal Model 27Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008

MIMO Radar Signal Model 28Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Overall Input-output relation: Overall Input-output relation:

MIMO Radar Signal Model 29Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Overall Input-output relation: Overall Input-output relation:

MIMO Radar Signal Model 30Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Overall Input-output relation: Overall Input-output relation: Range Cell: Doppler Cell: Angle Cell:

Compressed Sensing MIMO Radar Receiver 31Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008

Compressed Sensing MIMO Radar Receiver 32Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Received waveforms

Compressed Sensing MIMO Radar Receiver 33Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Received waveforms Transmitted waveforms

Compressed Sensing MIMO Radar Receiver 34Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Received waveforms Transmitted waveforms Transfer function for the target in the  cell

Compressed Sensing MIMO Radar Receiver 35Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Received waveforms Transmitted waveforms Transfer function for the target in the  cell RCS of the target in  cell

Compressed Sensing MIMO Radar Receiver 36Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Received waveforms Transmitted waveforms RCS of the target in  cell Transfer function for the target in the  cell

Compressed Sensing MIMO Radar Receiver 37Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 s is sparse if the target scene is sparse.

Compressed Sensing MIMO Radar Receiver 38Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 s is sparse if the target scene is sparse. Compressed sensing algorithm can effectively recover s if  is incoherent.

Waveform Optimization 39Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Goal: Design u such that is small. Goal: Design u such that is small.

Waveform Optimization 40Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Goal: Design u such that is small. Goal: Design u such that is small. … … TX RX

Waveform Optimization 41Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Goal: Design u such that is small. Goal: Design u such that is small. … … Small Correlation TX RX

Waveform Optimization: Dimension Reduction 42Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008

Waveform Optimization: Dimension Reduction 43Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008

Waveform Optimization: Dimension Reduction 44Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008

Waveform Optimization: Dimension Reduction 45Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008

Goal: Design u such that is small. Goal: Design u such that is small. Waveform Optimization: Dimension Reduction 46Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008

Waveform Optimization: Beamforming 47Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 B: the set consisting of angles of interest.  To concentrate the transmit energy on the angles of interest, we want the following term to be small

Waveform Optimization: Beamforming 48Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008  To uniformly illuminate the angles of interest, we want the following term to be small  To concentrate the transmit energy on the angles of interest, we want the following term to be small B: the set consisting of angles of interest.

Waveform Optimization: Cost function 49Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Incoherent Stopband Passband

Waveform Optimization: Cost function 50Chun-Yang Chen, Caltech DSP Lab | Asilomar

Waveform Optimization: Cost function 51Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 IncoherentStopband Passband

Phase Hopping Waveform 52Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Consider constant-modulus signal:

Phase Hopping Waveform 53Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Consider constant-modulus signal: Consider phase on a lattice:

Phase Hopping Waveform 54Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Consider constant-modulus signal: Consider phase on a lattice:

Simulated Annealing Algorithm  Simulated annealing –Create a Markov chain on the set A with the equilibrium distribution –Run the Markov chain Monte Carlo (MCMC) –Decrease the temperature T from time to time 55 subject to … C C’ … Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008

Example: Histogram of correlations 56Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Parameters: Uniform linear array # of RX elements N=10 # of TX elements M =4 Signal length L=31 # of phase K=15 Angle of interest ALL # of (  ’) pairs Alltop Sequence Proposed Method

Example: Histogram of correlations 57Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 # of (  ’) pairs Alltop Sequence Proposed Method Parameters: Uniform linear array # of RX elements N=10 # of TX elements M =4 Signal length L=31 # of phase K=15 Angle of interest ALL

Example: Histogram of correlations 58Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 # of (  ’) pairs Alltop Sequence Proposed Method Parameters: Uniform linear array # of RX elements N=10 # of TX elements M =4 Signal length L=31 # of phase K=15 Angle of interest ALL

Example: Recovering Target Scene 59Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Target Scene Compressed Sensing Matched Filter SNR=10dB

Example: Recovering Target Scene 60Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Target Scene Compressed Sensing Matched Filter SNR=10dB

Example: Recovering Target Scene 61Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Target Scene Compressed Sensing Matched Filter SNR=10dB

Example: Recovering Target Scene 62Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 Target Scene Compressed Sensing Matched Filter SNR=10dB

Conclusion  Compressed sensing based receiver –Applicable when the target scene is sparse –Better resolution than the matched filter receiver  Waveform design –Incoherent –Beamforming –Simulated annealing 63Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008

Q&A Thank You! Any questions? 64Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008

Simulated Annealing Algorithm  Simulated annealing –Create a Markov chain on the set A with the equilibrium distribution 65 subject to … C C’ … Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008

Simulated Annealing Algorithm  Simulated annealing –Create a Markov chain on the set A with the equilibrium distribution –Run the Markov chain Monte Carlo (MCMC) 66 subject to … C C’ … Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008