DSP Group, EE, Caltech, Pasadena CA1 Beamforming Issues in Modern MIMO Radars with Doppler Chun-Yang...

DSP Group, EE, Caltech, Pasadena CA 1

Beamforming Issues in Modern MIMO Radars with Doppler

Chun-Yang Chen and P. P. Vaidyanathan

California Institute of Technology


Outline Review of the MIMO radar

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

MIMO space-time adaptive processing (STAP) Problem formulation Clutter rank in MIMO STAP Clutter subspace in MIMO STAP

Numerical example


SIMO RadarTransmitter: M elements Receiver: N elements

dT

ej2(ft-x/)

w2 w1 w0

dR

ej2(ft-x/)

Transmitter emits coherent waveforms.

Number of received signals: N


Transmitter: M elements Receiver: N elements

dT

ej2(ft-x/)

dR

ej2(ft-x/)

MF MF…

…

Transmitter emits orthogonal waveforms.

Matched filters extract the M orthogonal waveforms.

Overall number of signals: NM

MIMO Radar


Transmitter: M elements Receiver: N elements

Virtual array: NM elements

dT=NdR

dR

ej2(ft-x/) ej2(ft-x/)

MF MF…

…

MIMO Radar (2)

The spacing dT is chosen as NdR, such that the virtual array is uniformly spaced.


The clutter resolution is the same as a receiving array with NM physical array elements.

A degree-of-freedom NM can be created using only N+M physical array elements.

Receiver: N elements


Transmitter : M elements

+ =

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

MIMO Radar (3)


Space-Time Adaptive Processing (STAP)

vvsini

airborne radar

jammertarget

i-th clutter

vt

ii

Di

vf

c

vf

sin2sin2'

The clutter Doppler frequencies depend on looking directions. The problem is non-separable in space-time.

i

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


Formulation of MIMO STAP

1,,1,01,,1,01,,1,0,,,,,,,,,,

LlMmNn

wvcsy lmnlmnlmnlmnlmn

dT=NdR

ej2(ft-x/)

dR

ej2(ft-x/)

MF MF……

Transmitter : M elements Receiver: N elements

vsin vsin

target

vt

targetvt

target

IRRssR

wvcsy

vcy2

H

clutterjammer

noiseNML

NML x NML


c ii

Ti

RN

i

vTlj

dmj

dnj

ilmn eeec1

sin22sin2sin2

,,

Clutter in MIMO Radar

)(2,,,

, lmnfjlmni

isec

RR

T

iR

is

d

vT

d

d

df

2

sin,

cN

i

Hiiic

HE1

2,][ ccccRc

cN21c cccR span)(range

cN

iii

1

cc size: NML

size: NMLxNML

lmni

N

iic

c

,,,1


Clutter Rank in MIMO STAP: Integer Case

Integer case: and are both integers.

)}.1()1(1,,1,0{ LMNlmn

),),1()1(min()(rank cNNMLLMN cR

This result can be viewed as the MIMO extension of Brennan’s rule.

Theorem: If and are integers,

The set {n+m+l} has at most N+(M-1)+(L-1) distinct elements.

)(2,,,

, lmnfjlmni

isec cN21c cccR span)(range


Clutter Signals and Truncated Sinusoidal Functions

.)(2,,,

, lmnfjlmni

isec ci is NML vector which consists of

)1()1(1,5.0

otherwise.,0

0,),(

,

2

,

,

LMNXW

WfW

Xxexfc

is

xfj

is

is

It can be viewed as a non-uniformly sampled version of truncated sinusoidal signals.

-50 0 50 100 150-1

-0.5

0

0.5

1

x

c(f s,

i),x)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

fs

|C(f s,

i,f s)|

(b)

The “time-and-band limited” signals can be approximated by linear combination of prolate spheroidal wave functions.

X

2W


Prolate Spheroidal Wave Function (PSWF)

Time window Frequency window

X -W W0

Prolate spheroidal wave functions (PSWF) are the solutions to the integral equation [van tree, 2001].

in [0,X]

Only the first 2WX+1 eigenvalues are significant [D. Slepian, 1962].

The “time-and-band limited” signals can be well approximated by the linear combination of the first 2WX+1 basis elements.


PSWF Representation for Clutter Signals

WfW

Xxexfc

is

xfj

is

is

,

2

, otherwise.,0

0,),(

,

-50 0 50 100 150-1

-0.5

0

0.5

1

x

c(f s,

i),x)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

fs

|C(f s,

i,f s)|

(b)

)1()1(12

)1()1(1,5.0

)(),(2

0,,

LMNWX

LMNXW

xxfcWX

kkkiis

The “time-and-band limited” signals can be approximated by 2WX+1 PSWF basis elements.

clutter rank in integer case


PSWF Representation for Clutter Signals (2)

WX

kkkiis xxfc

2

0,, )(),(

WX

kkkilmni lmnc

2

0,,,, )(

Tkkk

Ti LMN ))1()1(1()0(( ccc

1-L1,-M1,-Ni,i,1,0,0i,0,0,0 uc

1

0,

cr

kkkii uc )1()1(12 LMNWXrc

The PSWF (x)can be computed off-line

The vector uk can be obtained by sampling the PSWF.

non-uniformly sample

1211

2, ,

c

c

rH

N

i

Hiiic uuuUUAUccRc

U: NML x rc A: rc x rc


)(2,,,

, lmnfjlmni

isec

xfjis

isexfc ,2, ),(

)(xk

i-th clutter signal

truncated sinusoidal PSWF

Non-uniformlysample

Linearcombination

Non-uniformlysample

)( lmnk Sampled PSWFLinear

combination

Stack

ici-th clutter signal

Stack

kuSampled PSWFLinear

combination

HN

i

Hiiicc

c

R UAUcc

1

0

2 ,

Clutter covariance matrix

U: NML x rc A: rc x rc

)1()1( LMNrc


Numerical Example

0 20 40 60 80 100 120 140 160 180 200

-200

-150

-100

-50

0

50

100

Basis element index

Clu

tter

pow

er (

dB)

Proposed subspace methodEigen decomposition

N+(M-1)+(L-1)

N=10M=5L=16=N=10NML=800N+(M-1)+(L-1)=72.5

Proposed method

Eigenvalues

The figure shows the clutter power in the orthonormalized basis elements.

The proposed method captures almost all the clutter power.

Parameters:

k

qkH Rcqk


Conclusion The clutter subspace in MIMO radar is explored.

Clutter rank for integer/non-integer and Data-independent estimation of the clutter subspace.

Advantages of the proposed subspace estimation method. It is data-independent. It is accurate. It can be computed off-line.


Further and Future Work Further work

The STAP method applying the subspace estimation is submitted to ICASSP 07.

Future work In practice, some effects such as

internal clutter motion (ICM) will change the clutter space.

Estimating the clutter subspace by using a combination of both the geometry and the data will be explored in the future.

-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)

SMI, K=2000LSMI, K=300

PC, K=300

Separate, Kv=20, K

c=300

New method ZF, Kv=20

New method, Kv=20, K

c=300

MVDR, perfect Ry

New method


References

[1] D. W. Bliss and K. W. Forsythe, “Multiple-input multiple-output (MIMO) radar and imaging: degrees of freedom and resolution,” Proc. 37th IEEE Asilomar Conf. on Signals, Systems, and Computers, pp. 54-59, Nov. 2003.

[2] D. Slepian, and H. O. Pollak, "Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty-III: the dimension of the space of essentially time-and-band-limited signals," Bell Syst. Tech. J., pp. 1295-1336, July 1962.

[3] D. J. Rabideau and P. Parker, "Ubiquitous MIMO Multifunction Digital Array Radar," Proc. 37th IEEE Asilomar Conf. on Signals, Systems, and Computers, pp. 1057-1064, Nov. 2003.

[4] N. A. Goodman and J.M. Stiles, "On Clutter Rank Observed by Arbitrary Arrays," accepted to IEEE Trans. on Signal Processing.


Thank you


Comparison of the Clutter Rank in MIMO and SIMO Radar

MIMO SIMO

Clutter rank N+(M-1)+(L-1) N+(L-1)

Total dimension

NML NL

Ratio (=N)

NML

1)-(L

L

1 <

The clutter rank is a smaller portion of the total dimension. The MIMO radar receiver can null out the clutter subspace

without affecting the SINR too much.

>

>

NL

1)-(L

L

1


Formulation of MIMO STAP (2)

)sin(2

2)sinsin(2

,,

vvTlj

dm

dnj

lmn

tt

Tt

R

ees

lfjmnfj tDts ee , , 2)(2

R

T

ttD

tR

ts

d

d

Tvv

f

df

)sin(2

sin

,

,

dT

ej2(ft-x/)

dR

ej2(ft-x/)

MF MF……

Transmitter : M elements Receiver: N elements

vsin vsin

target

vt

targetvt

T: Radar pulse period


Fully Adaptive STAP for MIMO Radar

1 subject to

min

sw

wRw yw

H

H

lfjmnfjlmn

tDts ees , , 2)(2,,

sRs

sRw

y

y

1

1

H

K

k

HkkK 1

1yyRy

^

Difficulty: The size of Ry is NML which is often large. The convergence of the fully adaptive STAP is slow. The complexity is high.

Solution:


Clutter Subspace in MIMO STAP: Non-integer Case Non-integer case: andnot integers.

Basis need for representation of clutter steering vector ci.

Data independent basis is preferred. Less computation Faster convergence of STAP

We study the use of prolate spheroidal wave function (PSWF) for this.


Extension to Arbitrary Array

))1(2)(

)((max(2

1)(rank

',,

',,1,',,',

1

))2((2

,,

,,

LT

ec

mRmR

nRnRT

mmnn

N

i

Tlj

ilmn

cmTnR

Ti

vxx

xxuRu

c

vxxu

This result can be extended to arbitrary array.

XR,n is the location of the n-th receiving antenna.XT,m is the location of the m-th transmitting antenna.ui is the location of the i-th clutter.v is the speed of the radar station.


Review of MIMO radar: Diversity approach

dR

ej2(ft-x/)

MF MF…

…

Receiver: If the transmitting antennas are f

ar enough, the received signals of each orthogonal waveforms becomes independent. [E. Fishler et al. 04]

This diversity can be used to improve target detection.


Prolate Spheroidal Wave Function (PSWF) (2) By the maximum principle, this basis concentrates most of its en

ergy on the band [-W, W] while maintaining the orthogonality.

Only the first 2WX+1 eigenvalues are significant [D. Slepian, 1962].

The “time-and-band limited” signals can be well approximated by the linear combination of the first 2WX+1 basis elements.


Review of MIMO Radar: Degree-of-Freedom Approach

The clutter resolution is the same as a receiving array with NM physical array elements.

A degree-of-freedom NM can be created using only N+M physical array elements.

Receiver: N elements


Transmitter : M elements

dT=NdR

ej2(ft-x/)

dR

ej2(ft-x/)

MF MF……

+ =

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

DSP Group, EE, Caltech, Pasadena CA1 Beamforming Issues in Modern MIMO Radars with Doppler Chun-Yang...

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