Compressed Sensing in MIMO Radar ChunYang Chen and

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 Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 2

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

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

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

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

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

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

Review: Compressed Sensing in Radar [Herman & Strohmer 08] Range u y Doppler targets Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 9

Review: Compressed Sensing in Radar [Herman & Strohmer 08] Range u Doppler y targets * * si: target RCS in the i-th Range-Doppler cell. Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 10

Review: Compressed Sensing in Radar [Herman & Strohmer 08] Range u Doppler y targets * * si: target RCS in the i-th Range-Doppler cell. F is a function of the transmitted waveform u. Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 11

Review: Compressed Sensing in Radar [Herman & Strohmer 08] Range u Doppler y targets * * F is a function of the transmitted waveform u. si: target RCS in the i-th Range-Doppler cell. Assumption: s is sparse. Transmitted waveform u can be chosen such that F is incoherent. Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 12

Review: Compressed Sensing in Radar Target scene s can be reconstructed by compressed sensing method. High resolution can be achieved. [Herman & Strohmer 08] * * F is a function of the transmitted waveform u. si: target RCS in the i-th Range-Doppler cell. Assumption: s is sparse. Transmitted waveform u can be chosen such that F is incoherent. Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 13

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

2 Compressed Sensing in MIMO Radar 15

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Waveform Optimization: Beamforming § 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. Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 47

Waveform Optimization: Beamforming § 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. § To uniformly illuminate the angles of interest, we want the following term to be small Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 48

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

Waveform Optimization: Cost function + + Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 50

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

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

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

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

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

Example: Histogram of correlations # of (a, a’) pairs Alltop Sequence 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 Proposed Method Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 56

Example: Histogram of correlations # of (a, a’) pairs Alltop Sequence 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 Proposed Method Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 57

Example: Histogram of correlations # of (a, a’) pairs Alltop Sequence 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 Proposed Method Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 58

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

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

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

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

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 Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008 63

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

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

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