statistical signal processing algorithms for time-varying sensor arrays

ASAP Workshop12-Mar-02

Statistical Signal Processing Algorithms for

Time-Varying Sensor Arrays

David W. RiekenVeridian Systems Division

Ann Arbor R&D FacilityAnn Arbor, MI

[email protected]

Daniel R. FuhrmannDept. of Electrical Engineering

Washington UniversitySt. Louis, MO

[email protected]


Applications of Time-Varying Arrays

• Rigid arrays with predictable motion – E-2C, AWACS– ground-based air surveillance

• Rigid arrays with random but measurable motion – F-15, F/A-18– EW platforms– Ship-mounted radar arrays

• Flexible arrays – Towed hydrophone arrays– Wing-mounted radar arrays

• Multiple-platform arrays – Aircraft formations– UAV or satellite clusters


rad/sec. 2

16M

1-s 32sF

Computer Simulation: Rotating ULA

5N

Isotropic Sensors

2

db30

45,45elaz, :1Jammer 2

db30

0,90elaz, :2Jammer 2

db0NoiseReceiver


Array Signal Processing Algorithms

• Adaptive Beamforming– radar detection – communications arrays

• Spatial Spectrum Estimation– radar/sonar imaging, – threat detection

• Direction-of-Arrival Estimation– threat localization– fire control radar

• Modern algorithms make use of the data covariance matrix to do all of the above.


Outline

• Covariance Estimation for Time-Varying Arrays

• Spatial Spectrum Estimation

• Direction-of-Arrival Estimation


Covariance Estimation for Time-Varying Arrays

Mmm ,,1. estimate :Problem R

. iselement th theofPosition array.element - :Given 3RtnN n p

s

mmmmm FtttCNt

1,~ :dataarray 1 R0zz

Mm ,,1

Assumptions: •Signal sources and strengths does not change. •Only the array geometry changes. •Array manifold is known at each sample time.


Maximum-Likelihood Covariance Estimation

. ˆ is MLE nedUnconstrai mHmmm zzR

.ˆ;; However, 2mmm

Hm dttP RIaaR

Constrained MLE algorithms:

1) Gradient search.

2) Inverse iterations (introduced for time-varying arrays at ASAP 2001).

ii L RD ˆ1

WL ii 1;ˆ DSR


Convergence Comparison


Computer Simulation - MVDR Spectra

-180 -135 -90 -45 0 45 90 135 180-10

0

10

20

30

40m=1

azimuth

MV

DR

(dB

)

-180 -135 -90 -45 0 45 90 135 180-10

0

10

20

30

40m=2

azimuth

MV

DR

(dB

)

-180 -135 -90 -45 0 45 90 135 180-10

0

10

20

30

40m=3

azimuth

MV

DR

(dB

)

-180 -135 -90 -45 0 45 90 135 180-10

0

10

20

30

40m=4

azimuth

MV

DR

(dB

)

Inverse Iteration: Projected: Conjugate Gradient:


Outline





ML Spatial Spectrum Estimation

For K sources, the covariance matrix has the form:

IaaR 2

1;;

K

kmk

Hmkkm ttP

.in points of estimation ML try Could 2SAP

SRRSS ,ln; 1

1

11

M

mmM,APL

Use iterative methods: E-M algorithm. (e.g. Lanterman)


ML Spatial Spectrum Estimation

dB

•Azimuth, elevation each discretized at 5o increments for a total of 1387 estimated parameters.

•Results shown after 200 iterations.

•May be possible to estimate from sequence of covariance matrices.


Solving by Least Squares

k

mkH

mkkm ttp ;; aaR

Relationship between covariance matrices and the spatial spectrum:

or

Wpd

spectrum samplescovariance matrix elements

•The Fourier method of radio astronomy (Swenson, 1968) can be derived from this.

•Requires many covariance matrices for large images.

•Large images formed by inverting large matrices.


Least-Squares Results


Modified MVDR

M

mmmm

H

M

mmmm

H

H

HH

tt

t

1

1

1

1

1

1

;;

;

~

aRa

zRa

aRa

zRazw

Apply MVDR beamformer (Capon, 1969):

MMM t

t

;

; 111

a

a

a

R0

0R

R

z

z

z


Modified MVDR

.E and 0CN~ where;Let 2mnnmmmmm ss,σsts az

.

;;

;;,0CN~~ is response Beamformer 2

1

1

1

21

2

M

mmmm

H

M

mmmm

H

H

tt

tt

aRa

aRazw

waRa

aRaw ~

;;

;; :beamformer New

1

21

1

1

M

mmmm

H

M

mmmm

H

tt

tt

.,0CN~ response have will 2zw H


Modified MVDR

M

mmmm

H

M

mmmm

H

H

tt

tt

1

21

1

1

22

;;

;;

E

aRa

aRa

zw

Therefore,

•Can estimate spectrum from estimated covariance matrices.

•Technique is non-iterative.

•Does not require that the spectrum be estimated over the entire sphere.

•Reduces to the MVDR spectrum for M=1.


Modified MVDR Results


Outline





Previous Work

• MUSIC (time-invariant): Schmidt (1986).

• Single source ML: Friedlander, Zeira (1995).

• Interpolated array, focusing matrices: Zeira, Friedlander (1996).

• Least squares: Sheinvald, Wax, Weiss (1998).

• Arrays of linear arrays: Pesavento, Gershman, Wong (2001).


MUSIC for Matrix Sequences

.,, sequencematrix Covariance :Given ,1 MNM V RRR

There are K sources. We wish to find the direction of each.

: ofposition Eigendecom mR

H

H

V

UVUR

m

m

Nm

m

mmm

,

1,

0

0

Nmmm ,2,1,

.,, NmmKNN

mKN

m CC IUUVU H



ismatrix covarianceth for the spectrum MUSIC The m

1;;

mmNmm ttf aPIaH

.1

Mz

z

z

Define the composite data vector as

HUUP mmm

.col onto from projection theis mNC U

where



.

0

01

MU

U

U

The signal is constrained to lie within the intersection of each of the actual data vectors. The signal subspace of the composite data vector is therefore the column span of

The projection onto the composite signal subspace is

MMM P

P

UU

UU

UUPH

H

H

0

0

0

0 111



The MUSIC spectrum is given by

1 aPIaH NMf

1

1

1

1

1

1

11

1

;;

;

;

;;

M

mm

M

mmmNm

MM

NMM

f

tt

t

t

ttf

aPIa

a

a

P

P

Iaa

H

HH


Computer Simulation - MUSIC Spectra

Projected Sequence Inverse Iterations Sequence

0.0

45.0

90.0

135.0

180.0

225.0

270.0

315.0

0.0 30.0 60.0 90.0 0.0

45.0

90.0

135.0

180.0

225.0

270.0

315.0

0.0 30.0 60.0 90.0


Comparison to CRLB


Summary

• Covariance estimation is important for many array processing applications.

• Time-varying sensor arrays are becoming more common and require different covariance estimation algorithms than do their time-invariant brethren.

• We have developed an algorithm which estimates the covariance matrix sequence which arises from a time-varying array and demonstrated the application of that covariance estimate in estimating the direction-of-arrival and the spatial spectrum.

• The time-varying nature of an array can be advantageous rather than detrimental.

• Performance in real-world situations still not quantified: e.g. imperfections in array manifold calibration, sensor location estimates, etc.


Acknowledgement

• This work supported in part by MIT Lincoln Laboratory and the Boeing Foundation.

statistical signal processing algorithms for time-varying sensor arrays

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