Joint MIMO Radar Waveform and Receiving Filter Optimization

Chun-Yang Chen and P. P. Vaidyanathan

California Institute of TechnologyElectrical Engineering/DSP Lab

ICASSP 2009

Outline

Problem Formulation– Extended target and clutter– Detection– MIMO radar

Proposed Algorithm– Iterative algorithm– Receiver– Waveforms

Numerical Examples Conclusions

2Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

1Problem Formulation

3

Extended Target vs. Point Target

4Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

)(tfPoint Target

Extended Target vs. Point Target

5Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

)(tf )( trfPoint Target

r : radar cross section : delay

Extended Target vs. Point Target

6Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

)(tf

)(tf )( ii tfr

)( trfPoint Target

Extended Target vs. Point Target

7Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

dtfr )()(

)(tf

)(tf

)(tf

)( ii tfr

)( trfPoint Target

Extended Target

Extended Target and Clutter

8Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

)(tf

Extended Target Extended Clutter

Extended Target and Clutter

9Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

dtfc

dtfr

)()(

)()(

)(tf

Extended Target Extended Clutter

Extended Target and Clutter

10Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

dtfc

dtfr

)()(

)()(

)(tf

Extended Target

R(s)

C(s)

v(t)f(t)

Extended Clutter

Baseband Equivalent Model

11Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

Modulation R(s)

C(s)

Demodulation

v (t)f(n) D/A A/D r(n)

Baseband Equivalent Model

12Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

Modulation R(s)

C(s)

Demodulation

v (t)f(n) D/A A/D r(n)

R(z)

C(z)

v (n)

f(n)

)()(

)()(

mnfmc

mnfmr

Detection Problem

13Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

H0

H1 Target

Clutter

Detection Problem

14Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

H0

H1 Target

Clutter

R(z)

C(z)

v (n)f(n) H(z) LRT

Receiving filter

H0 or H1

Likelihood ratio test

[Delong & Hofstetter 67] [Pillai et al. 03]

Transmittedwaveform

Detection Problem

15Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

H0

H1 Target

Clutter

R(z)

C(z)

v (n)f(n) H(z) LRT

Receiving filter

H0 or H1

Likelihood ratio test

[Delong & Hofstetter 67] [Pillai et al. 03]

Transmittedwaveform

SINR Maximization

16Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

R(z)

C(z)

v (n)f(n) H(z) LRT

Receiving filter

H0 or H1

Likelihood ratio test

Transmittedwaveform

[Delong & Hofstetter 67] [Pillai et al. 03]

u

)( vCfRfh Hu vhCfhRfh HHH

vhCfhRfh HHH

SINR Maximization

17Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

R(z)

C(z)

v (n)f(n) H(z) LRT

Receiving filter

H0 or H1

Likelihood ratio test

Transmittedwaveform

[Delong & Hofstetter 67] [Pillai et al. 03]

u

)( vCfRfh Hu

1 subject to

max22

2

,

f

vhCfh

Rfhfh

HH

H

EE

vhCfhRfh HHH

SINR Maximization

18Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

R(z)

C(z)

v (n)f(n) H(z) LRT

Receiving filter

H0 or H1

Likelihood ratio test

Transmittedwaveform

[Delong & Hofstetter 67] [Pillai et al. 03]

u

)( vCfRfh HuSignal

1 subject to

max22

2

,

f

vhCfh

Rfhfh

HH

H

EE

vhCfhRfh HHH

SINR Maximization

19Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

R(z)

C(z)

v (n)f(n) H(z) LRT

Receiving filter

H0 or H1

Likelihood ratio test

Transmittedwaveform

[Delong & Hofstetter 67] [Pillai et al. 03]

u

)( vCfRfh HuClutter

1 subject to

max22

2

,

f

vhCfh

Rfhfh

HH

H

EE

vhCfhRfh HHH

SINR Maximization

20Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

R(z)

C(z)

v (n)f(n) H(z) LRT

Receiving filter

H0 or H1

Likelihood ratio test

Transmittedwaveform

[Delong & Hofstetter 67] [Pillai et al. 03]

u

)( vCfRfh HuNoise

1 subject to

max22

2

,

f

vhCfh

Rfhfh

HH

H

EE

vhCfhRfh HHH

SINR Maximization

21Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

R(z)

C(z)

v (n)f(n) H(z) LRT

Receiving filter

H0 or H1

Likelihood ratio test

Transmittedwaveform

[Delong & Hofstetter 67] [Pillai et al. 03]

u

)( vCfRfh Hu

Power constraint1 subject to

max22

2

,

f

vhCfh

Rfhfh

HH

H

EE

The MIMO Case

22Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

[Friedlander 07]

The MIMO Case

23Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

R(z)

C(z)

v (n)f(n) H(z) LRT

Receiving filter

H0 or H1

Likelihood ratio test

Transmittedwaveform

u

)( vCfRfh Hu vhCfhRfh HHH

[Friedlander 07]

Prior Information

24Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

1 subject to

max22

2

f

vhCfh

Rfhfh

HH

H

,EE

Assumptions:

RTarget impulse response is known

Prior Information

25Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

Assumptions:

RTarget impulse response is known

''*jiijCCE2nd order statistics of clutter is known

1 subject to

max22

2

f

vhCfh

Rfhfh

HH

H

,EE

2 Proposed Algorithm

26

Iterative Algorithm

27Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

R(z)

C(z)

v (n)f(n) H(z) LRT

Receiving filter

H0 or H1

Likelihood ratio test

Transmittedwaveform

u

1. Fixed f, solve for h

22

2

maxvhCfh

Rfhh

HH

H

EE

Iterative Algorithm

28Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

R(z)

C(z)

v (n)f(n) H(z) LRT

Receiving filter

H0 or H1

Likelihood ratio test

Transmittedwaveform

u

1. Fixed f, solve for h2. Fixed h, solve for f

1 subject to

max22

2

f

vhCfh

Rfhf

HH

H

EE

Iterative Algorithm

29Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

R(z)

C(z)

v (n)f(n) H(z) LRT

Receiving filter

H0 or H1

Likelihood ratio test

Transmittedwaveform

u

1. Fixed f, solve for h2. Fixed h, solve for f3. Fixed f, solve for h

22

2

maxvhCfh

Rfhh

HH

H

EE

Iterative Algorithm

30Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

R(z)

C(z)

v (n)f(n) H(z) LRT

Receiving filter

H0 or H1

Likelihood ratio test

Transmittedwaveform

u

1. Fixed f, solve for h2. Fixed h, solve for f3. Fixed f, solve for h

22

2

maxvhCfh

Rfhh

HH

H

EE

SINR is guaranteed to be non-decreasing in each iterative step.

Solving for the Receiver

31Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

22

2

maxvhCfh

Rfhh

HH

H

EE

R(z)

C(z)

v (n)f(n) H(z) LRT

Receiving filter

H0 or H1

Likelihood ratio test

Transmittedwaveform

u

Solving for the Receiver

32Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

22

2

maxvhCfh

Rfhh

HH

H

EE

hvvhhCCffh

Rfhh

max

2

HHHHH

H

EE

Solving for the Receiver

33Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

hvvhhCCffh

Rfhh

max

2

HHHHH

H

EE

Solving for the Receiver

34Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

hvvhhCCffh

Rfhh

max

2

HHHHH

H

EE

1 subject to

min

Rfh

hvvCCffhhH

HHHH EE MVDR (Minimum Variance Distortionless)

Solving for the Receiver

35Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

hvvhhCCffh

Rfhh

max

2

HHHHH

H

EE

1 subject to

min

Rfh

hvvCCffhhH

HHHH EE MVDR (Minimum Variance Distortionless)

RfvvCCffh-1 HHH EE

Solving for the Waveforms

36Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

1 subject to

max22

2

f

vhCfh

Rfhf

HH

H

EE

R(z)

C(z)

v (n)f(n) H(z) LRT

Receiving filter

H0 or H1

Likelihood ratio test

Transmittedwaveform

u

Solving for the Waveforms

37Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

1 subject to

max22

2

f

vhCfh

Rfhf

HH

H

EE

1 subject to

max

2

2

f

vhfChhCf

Rfhf

HHHH

H

EE

Solving for the Waveforms

38Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

1 subject to

max22

2

f

vhCfh

Rfhf

HH

H

EE

1 subject to

max

2

2

f

vhfChhCf

Rfhf

HHHH

H

EE

Cannot be solved using MVDR

Solving for the Waveforms

39Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

1 subject to

max22

2

f

vhCfh

Rfhf

HH

H

EE 1 subject to

max

2

2

f

vhfChhCf

Rfhf

HHHH

H

EE

Try Lagrange Method:

Solving for the Waveforms

40Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

1 subject to

max22

2

f

vhCfh

Rfhf

HH

H

EE 1 subject to

max

2

2

f

vhfChhCf

Rfhf

HHHH

H

EE

0

22

22

fvhfChhCf

fChhCRfhvhfChhCfRfhhRλ

EE

EEE

HHHH

HHHHHHHHH

cannot be solved easily

Try Lagrange Method:

Recasting the Waveform Optimization Problem

41Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

1 subject to

max

2

2

f

vhfChhCf

Rfhf

HHHH

H

EE

1 subject to

max

2

2

f

vhfChhCf

Rfhf

HHHH

H

EE

Recasting the Waveform Optimization Problem

42Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

1 f

Recasting the Waveform Optimization Problem

43Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

1 f

1 subject to

max

2

2

f

vhfChhCf

Rfhf

HHHH

H

EE

1 subject to

max

22

2

f

fvhfChhCf

Rfhf

HHHH

H

EE

Recasting the Waveform Optimization Problem

44Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

1 subject to

max

2

2

f

vhfChhCf

Rfhf

HHHH

H

EE

1 f

1 subject to

max

22

2

f

fvhfChhCf

Rfhf

HHHH

H

EE

Recasting the Waveform Optimization Problem

45Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

22

2

max

fvhfChhCf

Rfhf

HHHH

H

EE

Recasting the Waveform Optimization Problem

46Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

22

2

max

fvhfChhCf

Rfhf

HHHH

H

EE

MVDR (Minimum Variance Distortionless)

1 subject to

min2

Rfh

fIvhChhCffH

HHHH EE

Recasting the Waveform Optimization Problem

47Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

22

2

max

fvhfChhCf

Rfhf

HHHH

H

EE

MVDR (Minimum Variance Distortionless)

1 subject to

min2

Rfh

fIvhChhCffH

HHHH EE

hRIvhChhCf H-12

HHH EE

Proposed Algorithm

48Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

R(z)

C(z)

v (n)f(n) H(z)

Receiving filterTransmittedwaveform

HHE CCffR f Compute .1

RfvvRh 1f ])[( .2 HE

Initialize: Choose a start point for f

Proposed Algorithm

49Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

R(z)

C(z)

v (n)f(n) H(z)

Receiving filterTransmittedwaveform

HHE CCffR f Compute .1

RfvvRh 1f ])[( .2 HE

ChhCR HHEh Compute .3

hRIhvvhRf HHHE 1h )][( .4

Initialize: Choose a start point for f

Proposed Algorithm

50Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

HHE CCffR f Compute .1

RfvvRh 1f ])[( .2 HE

ChhCR HHEh Compute .3

hRIhvvhRf HHHE 1h )][( .4

fff .5

RepeatR(z)

C(z)

v (n)f(n) H(z)

Receiving filterTransmittedwaveform

Initialize: Choose a start point for f

Numerical Examples

51Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

0 5 10 15 20 25 30 35 40 45 50

20

22

24

26

28

30

32

34

36

38

40

SIN

R (d

B)

# of iterations

Proposed

Method in [Pillai et al. 03]

LFM (Linear Frequency Modulation)

Matched Filter Bound

Parameters# of transmitters: 2# of receivers: 2Randomly generated impulse response

Numerical Examples

52Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

-10 -5 0 5 10 15 20 25 30 35 40-50

-40

-30

-20

-10

0

10

20

30

CNR (dB)

SN

R (d

B)

Proposed

Method in [Pillai et al. 03]

Matched Filter Bound

Parameters# of transmitters: 2# of receivers: 2Averaging 1000randomly generated examples

LFM (Linear Frequency Modulation)

Conclusions Detection of Extended Target in Clutter

– Prior information• Target impulse response• Clutter statistics

Iterative Algorithm– Recast the problem– MVDR solution

More General Target Impulse Response are considered in the Journal Version– Uncertainty Set (Worst case optimization)– Random

53Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009

[Chen & Vaidyanathan, TSP under review]

Q&AThank You!

Any questions?

54Chun-Yang Chen, Caltech DSP Lab | ICASSP 2009