A Subspace Method for MIMO Radar Space-Time Adaptive Processing Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP.

A Subspace Method for MIMO Radar Space-Time Adaptive Processing

Chun-Yang Chen and P. P. Vaidyanathan

California Institute of TechnologyElectrical Engineering/DSP Lab

ICASSP 2007 student paper contest

Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest

Outline

Review of the background– 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


MIMO Radar

MIMO radar

SIMO radar (Traditional)

The radar systems which emits orthogonal (or noncoherent) waveforms in each transmitting antennas are called MIMO radar.

w2w1

w0


MIMO Radar

MIMO radar

SIMO radar (Traditional)

w2w1

w0

The radar systems which emits orthogonal (or noncoherent) waveforms in each transmitting antennas are called MIMO radar.

[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]

SIMO Radar (Traditional)

Transmitter: M antenna elements Receiver: N antenna elements

dT

ej2(ft-x/)

w2 w1 w0dR

ej2(ft-x/)

Transmitter emits

coherent waveforms.

Transmitter emits

coherent waveforms.Number of received signals:

N

Number of received signals: N


MIMO Radar

dT

ej2(ft-x/)

dR

ej2(ft-x/)

MF MF…

…

Transmitter emits

orthogonal waveforms.

Transmitter emits

orthogonal waveforms.

Matched filters extract the M orthogonal waveforms.Overall number of signals:

NM

Matched filters extract the M orthogonal waveforms.Overall number of signals:

NM



MIMO Radar – Virtual Array


Virtual array: NM elements

dT=NdR

ej2(ft-x/)

dR

ej2(ft-x/)

MF MF…

…


MIMO Radar – Virtual Array (2)

Receiver: N elements

Virtual array: NM elements

Transmitter: M elements

+ =

[D. W. Bliss and K. W. Forsythe, 03]

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.


Space-Time Adaptive Processing

vvsini

airborne radar

jammertarget

i-th clutter

vt

i

The adaptive techniques for processing the data from airborne antenna arrays are called space-time adaptive processing (STAP).

fc

vf iDi

sin2


Space-Time Adaptive Processing

vvsini

airborne radar

jammertarget

i-th clutter

vt

iThe clutter Doppler frequencies

depend on angles. So, the problem is non-separable in

space-time.

The clutter Doppler frequencies depend on angles. So, the

problem is non-separable in space-time.

The adaptive techniques for processing the data from airborne antenna arrays are called space-time adaptive processing (STAP).

fc

vf iDi

sin2


Space-Time Adaptive Processing (2)

Separable: N+L tapsNon separable: NL taps

Jointly processDoppler frequencies and angles

Jointly processDoppler frequencies and angles

Independently process Doppler frequencies and angles

Independently process Doppler frequencies and angles

Angle processing

Doppler processingSpace-time

processing

L: # of radar pulses

L


MIMO Radar STAPSTAP MIMO Radar

NL signals

MIMOSTAP

M waveforms

NML signals

N: # of receiving antennasM: # of transmitting

antennasL: # of pulses

[D. Bliss and K. Forsythe 03]

+

NM signals


MIMO Radar STAP (2)

1),( subject to

min

DH

H

fsw

Rwww

NML signals

MVDR (Capon) beamformer:


MIMO Radar STAP (2)

1),( subject to

min

DH

H

fsw

Rwww

NML signals

MVDR (Capon) beamformer:

),(1DfsRw

Very good spatial resolutionVery good spatial resolution

Pros ConsCons

High complexityHigh complexity

Slow convergenceSlow convergence

NMLxNML


The Proposed Method

),(1DfsRw

NMLJc IRRR 2

We first observe each of the matrices Rc and RJ has

some special structures.

clutter jammer noise

We show how to exploit the structures of these

matrices to compute R-1 more accurately and

efficiently.


c ii

Ti

RN

i

vTlj

dmj

dnj

ilmn eeec1

sin22sin2sin2

,,

Formulation of the Clutter Signals

Nc: # of clutter points ith clutter signal amplitude

Matchedfilters

Pulse 2

Pulse 1

Pulse 0

Matchedfilters

Matchedfilters

c002 c012 c102

c001 c011 c101

c000 c010 c100

c112 c202 c212

c111 c201 c211

c110 c200 c210

cnml: clutter signals

…

Clutter points

n-th antennam-th matched filter outputl-th radar pulse


Simplification of the Clutter Expression

c ii

Ti

RN

i

vTlj

dmj

dnj

ilmn eeec1

sin22sin2sin2

,,

R

T

d

d

Rd

vT2

, sinRs i i

df

cN

ilmnisi fxc

1, );(

otherwise,0

0),2exp();( ,

,

Xxxffxc isis

5.05.0

)1()1(1

,

isf

LMNX


Simplification of the Clutter Expression

c ii

Ti

RN

i

vTlj

dmj

dnj

ilmn eeec1

sin22sin2sin2

,,

R

T

d

d

Rd

vT2

, sinRs i i

df

cN

ilmnisi fxc

1, );(

otherwise,0

0),2exp();( ,

,

Xxxffxc isis

5.05.0

)1()1(1

,

isf

LMNX

-2 0 2 4 6 8 10 12-1.5

-1

-0.5

0

0.5

1

1.5

x

Re{c(x;fs,i)}Re{c(n+m+l;fs,i)}


-50 0 50 100 150-1

0

1

-1 -0.5 0 0.5 10

20

40

60

80

100

“Time-and-Band” Limited Signals

otherwise,0

0),2exp();( ,

,

Xxxffxc isis

5.05.0

)1()1(1

,

isf

LMNX

[0 X]

[-0.5 0.5]

Timedomain

Freq.domain

The signals are well-localized in a time-frequency region.

The signals are well-localized in a time-frequency region.

To concisely represent these signals, we can use a basis which concentrates most of its energy in this time-frequency region.

To concisely represent these signals, we can use a basis which concentrates most of its energy in this time-frequency region.


is called PSWF. is called PSWF.

Prolate Spheroidal Wave Functions (PSWF)

dx k

X

kk )())-sinc((x)(0 ( )k x

Time window Frequency window

X -0.5 0.50 in [0,X]

( )k x ( )k xk


is called PSWF. is called PSWF.

Prolate Spheroidal Wave Functions (PSWF)

, ,0

( ; )X

s i i kk

c x f

dx k

X

kk )())-sinc((x)(0

[D. Slepian, 62]

( )k x

in [0,X]

Only X+1 basis functions are required to well represent the “time-and-band limited” signal

Only X+1 basis functions are required to well represent the “time-and-band limited” signal

Time window Frequency window

X -0.5 0.50( )k x ( )k xk

( )k x


)1()1(1,,1,0 LMNk

Clutter Representation by PSWF

Hc ΨΨRR Ψ )( lmnk consists of

NMLN+(M-1)+(L-1)


)1()1(1,,1,0 LMNk



can be obtained by sampling from . The PSWF can be computed off-line can be obtained by sampling from . The PSWF can be computed off-lineΨ k

NMLN+(M-1)+(L-1)

k


)1()1(1,,1,0 LMNk



can be obtained by sampling from . The PSWF can be computed off-line can be obtained by sampling from . The PSWF can be computed off-lineΨ k

NMLN+(M-1)+(L-1)

k

The NMLxNML clutter covariance matrix has only N+(M-1)+(L-1) significant eigenvalues. This is the MIMO extension of Brennan’s rule (1994).

The NMLxNML clutter covariance matrix has only N+(M-1)+(L-1) significant eigenvalues. This is the MIMO extension of Brennan’s rule (1994).

cR


Jammer Covariance Matrix

Matchedfilters

jammer

Pulse 2

Pulse 1

Pulse 0

Matchedfilters

Matchedfilters

j002 j012 j102

j001 j011 j101

j000 j010 j100

j112 j202 j212

j111 j201 j211

j110 j200 j210

jnml: jammer signals



Matchedfilters

jammer

Pulse 2

Pulse 1

Pulse 0

Jammer signals in different pulses are independent.


Matchedfilters

Matchedfilters

j002 j012 j102

j001 j011 j101

j000 j010 j100

j112 j202 j212

j111 j201 j211

j110 j200 j210




Matchedfilters

jammer

Pulse 2

Pulse 1

Pulse 0



Jammer signals in different matched filter outputs are independent.

Jammer signals in different matched filter outputs are independent.Matched

filtersMatched

filters

j002 j012 j102

j001 j011 j101

j000 j010 j100

j112 j202 j212

j111 j201 j211

j110 j200 j210




Matchedfilters

jammer

Pulse 2

Pulse 1

Pulse 0





Js

Js

Js

J

R00

0

R0

00R

R

Matchedfilters

Matchedfilters

Block diagonalBlock diagonal

j002 j012 j102

j001 j011 j101

j000 j010 j100

j112 j202 j212

j111 j201 j211

j110 j200 j210



The Proposed Methodlow rank

block diagonalNMLJc IRRR 2 H

v ΨR Ψ R




v ΨR Ψ R

1111111 )( vH

vH

vv RΨΨRΨRΨRRR

By Matrix Inversion Lemma




v ΨR Ψ R

1111111 )( vH

vH

vv RΨΨRΨRΨRRR

The proposed method– Compute by sampling the prolate spheroidal wave

functions.





v ΨR Ψ R

1111111 )( vH

vH

vv RΨΨRΨRΨRRR


functions.

– Instead of estimating R, we estimate Rv and R. The

matrix Rv can be estimated using a small number of

clutter free samples.





v ΨR Ψ R


functions.

– Instead of estimating R, we estimate Rv and R. The

matrix Rv can be estimated using a small number of

clutter free samples.

– Use the above equation to compute R-1.

1111111 )( vH

vH

vv RΨΨRΨRΨRRR



The Proposed Method – Advantages

vR

R

:block diagonal

:small size

Inversions are easy to compute


1111111 )( vH

vH

vv RΨΨRΨRΨRRR



vR

R

:block diagonal

:small size



1111111 )( vH

vH

vv RΨΨRΨRΨRRR

Low complexityLow complexity



vR

R

:block diagonal

:small size



Fewer parameters need to be estimated


1111111 )( vH

vH

vv RΨΨRΨRΨRRR




vR

R

:block diagonal

:small size





1111111 )( vH

vH

vv RΨΨRΨRΨRRR


FastconvergenceFastconvergence


The Proposed Method – Complexity

1111111 )( vH

vH

vv RΨΨRΨRΨRRR

Complexity:1 3: (( ( 1) ( 1)) )O N M L R

)(: 31 NOvR

Direct method

The proposed method

),(1DfsR )( 333 LMNO )))1()1((( 3 LMNO

1R )))1()1((( 222 LMNLMNO )( 333 LMNO


The Zero-Forcing Method

Typically we can assume that the clutter is very strong and all eigenvalues of Rare

very large.

1111111 )( vH

vH

vv RΨΨRΨRΨRRR

1 0 R


The Zero-Forcing Method

Typically we can assume that the clutter is very strong and all eigenvalues of Rare

very large.

1 1 1 1 1 1( )H Hv v v v

R R R Ψ Ψ R Ψ Ψ R

Zero-forcing method

– The entire clutter space is nulled out without estimation

1111111 )( vH

vH

vv RΨΨRΨRΨRRR

1 0 R


Proposed method K=300,Kv=20

Simulations

MVDR known R (unrealizable)

Proposed ZF method Kv=20

Sample matrix inversion K=1000

Diagonal loading K=300

Principal component K=300

SINR of a target at angle zero and Doppler frequencies [-0.5, 0.5]

SINR of a target at angle zero and Doppler frequencies [-0.5, 0.5]

Parameters:N=10, M=5, L=16CNR=50dB2 jammers, JNR=40dB

K: number of samplesKv: number of clutter free samples collected in passive mode

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-16

-14

-12

-10

-8

-6

-4

-2

0

Normalized Doppler frequency

SIN

R (

dB

)


Conclusion and Future Work

Conclusion– 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.


Q&AThank You!

Any questions?


A Subspace Method for MIMO Radar Space-Time Adaptive Processing Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP.

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