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

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

1Review of the keywords: Compressed sensing, MIMO Radar

3

Brief Review of Compressed Sensing


)dim()dim( sy

y Φ s Goal: Reconstruct s from y.



)dim()dim( sy


jiji

φφ ,max

Incoherence:

is small.



)dim()dim( sy


jiji

φφ ,max

Incoherence:

is small. 0| isiSparsity:

is small.

0| isiSparsity:

is small.



)dim()dim( sy


jiji

φφ ,max

Incoherence:

is small.

Given y and F, s can be perfectly recovered by sparse approximation methods even when dim(y)<dim(s).


0| isiSparsity:

is small.



)dim()dim( sy


jiji

φφ ,max

Incoherence:

is small.

This concept can be applied to sampling and compression.This concept can be applied to sampling and compression.



Review: Compressed Sensing in Radar


[Herman & Strohmer08]

u

ytargets

Range

Doppler




u

ytargets

Range

Doppler

si: target RCS in the i-th Range-Doppler cell.

*

sy Φ

**

*




u

ytargets

Range

Doppler


F is a function of the transmitted waveform u.

*

sy Φ

**

*

*

sy Φ

**

*




u

ytargets

Range

Doppler


Assumption: s is sparse.

Transmitted waveform u can be chosen such that F is incoherent.





Assumption: s is sparse.

Transmitted waveform u can be chosen such that F is incoherent.

Target scene s can be reconstructed by compressed sensing method. High resolution can be achieved. [Herman & Strohmer08]

Target scene s can be reconstructed by compressed sensing method. High resolution can be achieved. [Herman & Strohmer08]


*

sy Φ

**

*

Brief Review of MIMO Radar

u2( )tu1( )t

u0( )t

w2u( )tw1u( )t

w0u( )t

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.

MIMO RadarEach element can transmit an arbitrary waveform.

Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008

2Compressed Sensing in MIMO Radar

15

MIMO Radar Signal Model


…

u0(t) u1(t) uM-1(t)

(p,t, fD)t:delayfD :Dopplerp: direction



…

u0(t) u1(t) uM-1(t)

(p,t, fD)

…

y0(t) y1(t) yN-1(t)


1

0

2)(2

)()(M

m

tfjj

mnD

nmT

eetuty

yxp



…

u0(t) u1(t) uM-1(t)

(p,t, fD)

…

y0(t) y1(t) yN-1(t)


Received signals



…

u0(t) u1(t) uM-1(t)

(p,t, fD)

…

y0(t) y1(t) yN-1(t)


1

0

2)(2

)()(M

m

tfjj

mnD

nmT

eetuty

yxp

Range

1

0

2)(2

)()(M

m

tfjj

mnD

nmT

eetuty

yxp



…

u0(t) u1(t) uM-1(t)

(p,t, fD)

…

y0(t) y1(t) yN-1(t)


xm: location of the m-th transmitteryn: location of the n-th transmitter

Cross range

1

0

2)(2

)()(M

m

tfjj

mnD

nmT

eetuty

yxp



…

u0(t) u1(t) uM-1(t)

(p,t, fD)

…

y0(t) y1(t) yN-1(t)



)(sin2

nm yxje

for linear array

1

0

2)(2

)()(M

m

tfjj

mnD

nmT

eetuty

yxp



…

u0(t) u1(t) uM-1(t)

(p,t, fD)

…

y0(t) y1(t) yN-1(t)



Doppler

1

0

)(2

)1(2

2

)'(

1

M

m

yxNM

j

m

LL

j

Lj

LLL

L

L

n

nm

D

D

e

e

e

u

0

I

0

y

1

0

2)(sin2

)()(M

m

tfjyxj

mnD

nm

eetuty



Discrete Model:



1

0

2)(sin2

)()(M

m

tfjyxj

mnD

nm

eetuty

1

0

)(2

)1(2

2

)'(

1

M

m

yxNM

j

m

LL

j

Lj

LLL

L

L

n

nm

D

D

e

e

e

u

0

I

0

y

Discrete Model:Range

12,1,0 LRange Cell: L: Length of um



1

0

)(2

)1(2

2

)'(

1

M

m

yxNM

j

m

LL

j

Lj

LLL

L

L

n

nm

D

D

e

e

e

u

0

I

0

y

1

0

2)(sin2

)()(M

m

tfjyxj

mnD

nm

eetuty

Discrete Model:

Doppler


12,1,0 LD Doppler Cell:



1

0

)(2

)1(2

2

)'(

1

M

m

yxNM

j

m

LL

j

Lj

LLL

L

L

n

nm

D

D

e

e

e

u

0

I

0

y


M: # of transmitting antennasN: # of receiving antennas

12,1,0 LD 12,1,0 NM

Doppler Cell:

Angle Cell:

1

0

2)(sin2

)()(M

m

tfjyxj

mnD

nm

eetuty

Discrete Model:

Angle



1

0

)(2

)1(2

2

)'(

1

M

m

yxNM

j

m

LL

j

Lj

LLL

L

L

n

nm

D

D

e

e

e

u

0

I

0

y

H

DH

nm

H



1

0

)(2

)1(2

2

)'(

1

M

m

yxNM

j

m

LL

j

Lj

LLL

L

L

n

nm

D

D

e

e

e

u

0

I

0

y

H

DH

nm

H

uHHH

y

y

y

y

D

N

1

1

0

1

1

0

Nu

u

u

u

OverallInput-outputrelation:

uHHH

y

y

y

y

D

N

1

1

0



1

1

0

Nu

u

u

u


αH),,( D

α

uHHH

y

y

y

y

D

N

1

1

0



1

1

0

Nu

u

u

u


αH),,( D

α

D

12,1,0 LRange Cell:12,1,0 LD

12,1,0 NMDoppler Cell:

Angle Cell:

α

ααuHy s

Compressed Sensing MIMO Radar Receiver


),,( Dα

D

α

ααuHy s



),,( Dα

y Received waveforms

D

α

ααuHy s



),,( Dα


u Transmitted waveforms

D

α

ααuHy s



),,( Dα


αHu Transmitted waveforms

Transfer function for the target in the a cell D

α

ααuHy s



),,( Dα

D


αHu

αs

Transmitted waveforms

Transfer function for the target in the a cell

RCS of the target in a cell

α

ααuHy s



),,( Dα

D


αHu

αs

Transmitted waveforms

RCS of the target in a cell

sΦφα

αα s

αφ

Transfer function for the target in the a cell

α

ααuHy s



),,( Dα

D

sΦφα

αα s

s is sparse if the target scene is sparse.

αφ

α

ααuHy s



),,( Dα

D

sΦφα

αα s

s is sparse if the target scene is sparse.

Compressed sensing algorithm can effectively recover s if F is incoherent.

αφ

sΦφα

αα s α

ααuHy s

Waveform Optimization


D

Goal: Design u such that

is small.

uHuH αααα

''

,max

αφ




is small.

uHuH αααα

''

,max

…

α'α

u

…

α'α

uHuH αααα ''ss

TX RX




is small.

uHuH αααα

''

,max

…

α'α

u

…

α'α

uHuH αααα ''ss

Small Correlation

TX RX

Waveform Optimization: Dimension Reduction


uHuH αα ',

uHHHHHHu )()( ''' DD αααH

αααH



uHuH αα ',uHHHHHHu )()( ''' DD ααα

Hααα

H

uHHHHHHu )( ''' DD ααH

αH

ααH

αH




Hααα

H


αH

ααH

αH

uHCHHu )( ''' DD αααααH

αH

1

1

1

KC

k




Hααα

H


αH

ααH

αH


αH

),,',( Dαααα

1

1

1

KC

k


is small.

),,',(max)0,0,'(

),,(D

αααα

ααααD




Hααα

H


αH

ααH

αH


αH

),,',( Dαααα

1

1

1

KC

k

Waveform Optimization: Beamforming


Bα

αα

)0,0,,( 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


Bα

αα

)0,0,,(

Bα Bα

ααB

αα

2

)0,0,,(1

)0,0,,(

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


),,',(max)0,0,'(

),,(D

αααα

ααααD

Bα

αα

)0,0,,(

Incoherent

Stopband

Passband

Bα Bα

ααB

αα

2

)0,0,,(1

)0,0,,(



),,',(max)0,0,'(

),,(D

αααα

ααααD

)1(

+

+

Bα Bα

ααB

αα

2

)0,0,,(1

)0,0,,(

Bα

αα

)0,0,,(



Bα Bα

BαD

αααα

ααB

αα

ααααααD

2

)0,0,'(),,(

)0,0,,(1

)0,0,,()1(

)0,0,,(),,',(max

minu

Incoherent Stopband

Passband

Phase Hopping Waveform


Consider constant-modulus signal:

mljm el 2)( u




mljm el 2)( u

Consider phase on a lattice:

1,2,1,0 , KCK

Cml

mlml




mljm el 2)( u

Consider phase on a lattice:

1,2,1,0 , KCK

Cml

mlml

Bα Bα

BαD

αααα

ααB

αα

ααααααD

2

22

2

)0,0,'(),,(

)0,0,,(1

)0,0,,()1(

)0,0,,(),,',(max

mlCmin

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

Csubject to

…

CC’

…


)(min CCf

C

C

CC

T

fZ

T

f

Z

T

TT

)(exp

)(exp

1)(

Example: Histogram of correlations


Parameters:Uniform linear array# of RX elements N=10# of TX elements M =4Signal length L=31# of phase K=15Angle of interest ALL

0 2 4 6 8 10 12 14 16 18 200

100

200

300

0 2 4 6 8 10 12 14 16 18 200

100

200

300

# of

(a,a

’) pa

irs

Alltop Sequence

Proposed Method',, ' uHuH αα



',, ' uHuH αα

0 2 4 6 8 10 12 14 16 18 200

100

200

300

0 2 4 6 8 10 12 14 16 18 200

100

200

300

# of

(a,a

’) pa

irs

Alltop Sequence

Proposed Method




',, ' uHuH αα

0 2 4 6 8 10 12 14 16 18 200

100

200

300

0 2 4 6 8 10 12 14 16 18 200

100

200

300

# of

(a,a

’) pa

irs

Alltop Sequence

Proposed Method


200 400 600 800 1000 12000

1

2

3

200 400 600 800 1000 12000

20

40

60

200 400 600 800 1000 12000

10

20

30

Cross Range

Ran

ge

10 20 30 40

10

20

30

10 20 30 40

10

20

30

Ran

ge

Cross Range10 20 30 40

10

20

30

Example: Recovering Target Scene


Target Scene

CompressedSensing

Matched Filter

SNR=10dB

200 400 600 800 1000 12000

1

2

3

200 400 600 800 1000 12000

20

40

60

200 400 600 800 1000 12000

10

20

30

Cross Range

Ran

ge

10 20 30 40

10

20

30

10 20 30 40

10

20

30

Ran

ge


10

20

30



Target Scene

CompressedSensing

Matched Filter

SNR=10dB

200 400 600 800 1000 12000

1

2

3

200 400 600 800 1000 12000

20

40

60

200 400 600 800 1000 12000

10

20

30

Cross Range

Ran

ge

10 20 30 40

10

20

30

10 20 30 40

10

20

30

Ran

ge


10

20

30



Target Scene

CompressedSensing

Matched Filter

SNR=10dB

200 400 600 800 1000 12000

1

2

3

200 400 600 800 1000 12000

20

40

60

200 400 600 800 1000 12000

10

20

30

Cross Range

Ran

ge

10 20 30 40

10

20

30

10 20 30 40

10

20

30

Ran

ge


10

20

30



Target Scene

CompressedSensing

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


Q&AThank You!

Any questions?




65

)(min CCf Csubject to

C

C

CC

T

fZ

T

f

Z

T

TT

)(exp

)(exp

1)(

…

CC’

…




– Run the Markov chain Monte Carlo (MCMC)

66

Csubject to

…

CC’

…


)(min CCf

C

C

CC

T

fZ

T

f

Z

T

TT

)(exp

)(exp

1)(

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

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