Developments in SensorArray Signal ProcessingJohn G McWhirter FRS FREng

Overview of Talk

Sensor array signal processing

historical perspective and overview

Recent developments and current trends

from ABF to BSS

from 2nd order statistics to HOS

convergence with artificial neural networks

Current research and future challenges

Convolutive mixtures

Semi-blind signal separation

Sensor Array Signal Processing

Techniques have recently "come of age"

Enabled by the digital processing revolution

Impressive research results

Wide range of application areas

Key to improving mobile telephone systems

Could revolutionise design of future radars

Medical diagnostic techniques (ECG, EEG)

Adaptive Null Steering

Adaptive Beamforming

Adaptive

algorithmS

w1

w2

wp

w3

w4

…

Output

signal

Array gain Complex weights (representphase and amplitude)

Output signal

Minimise output power subjectto look-direction constraint

µ=)(cwH

)()( tteHxw=

Minimise

subject to

Least squares solution (Gauss normal equations)

where

µ=)(cwH

)(),()( cwM =nn

Least Squares Solution

wMw )()()(1

2ntenE

Hn

t

===

==n

tjiij txtxnM

1

*)()()(

LMS Algorithm

Minimise

where

Stochastic gradient update

Minimal computation

Can be slow to converge

)()()( tytteH

+= xw

)()()()1( *ttett xww µ=+

})(E{2

te

Canonical Problem and GSLC

0=Ac

)()()( tytteH

+= xw

µ=wcH

CANONICAL ADAPTIVE COMBINER

BLOCKING

MATRIX

BEAMFORMER

)(ty)(tx

…...

….

QRD Processor Array

Direct residual extraction

Systolic array implementationx

r11

u4r44

r34r33

r24r23r22

r14r13r12

u3

u2

u1

yx4x3x2x11

Residual

Unstabilised Beam Pattern

)()(

)()()(2/

2/

2

qH

q

Hq dhE

wwZww

cww

=

=+

=+ 2/

2/

)()()( dhHccZ

EknE2)( +

Penalty Function Method

Penalty function

where

Minimise

Closed form solution

Stabilised Beam Pattern

Sonobuoy Array

Application to Sonar(sonobuoy trials data)

Conventional (fixed) Beamformer Adaptive Beamformer (stabilised)

Range Range

Bearing Bearing

Blind Signal Separation

Avoids need for array calibration

Foetal heartbeat monitor

HF communications

Independent component analysis (ICA)

Involves use of higher order statistics (HOS)

Requires signals to be non-Gaussian

Typical of man-made signals

Digital communication signals

Signal model (instantaneous)

Data matrix

Unknown mixture matrix A

Unknown signals S

Input signals are non-Gaussian

and statistically independent

Blind Signal Separation

……

s t1( )

s t2( )

s t3( )

x t1( ) x t2( ) x t3( ) x tp ( )

x As n( ) ( ) ( )t t t= +

X AS N= +

Signal model

Singular value decomposition (SVD)

Signal subspace

Principal Components Analysis (PCA)

NASX +=

[ ]

nnsss

n

ss

ns

VUVDU

V

V

I

DUU

UDVX

+=

=

=

0

0

s

H

ss IVV =

XUDVH

sss

1=

By definition

Now define

Then

Can only conclude that

Hidden Rotation Matrix

sQVS =

s

H

ss IVV =

ss QVV =~

s

HH

ss

H

ss IQVQVVV ==~~

Independent Component Analysis

Higher Order Statistics

0

0

x

00

0

0

0

0

00

0

00

0

0

0

0

xx

0

0

0

0

0

0

0

0

i

k

j

Fourth order cumulant tensor

Statistically independent signals

Separation tensor diagonalisation

Need for novel mathematical research

}E{ lkjiijkl xxxxK =

otherwise0

if

=

==== lkjikK iijkl

}E{}E{}E{}E{}E{}E{ kjliljkilkji xxxxxxxxxxxx

HF Communications Array

BLISS Trials ResultsHF communications data

FSK signal 30dB stronger than SSB voice signal

BLISS algorithm - 16384 samples

TX1 Mode13454kHzSSBTX2 Mode13454kHzFSKAngularOffsetRelativelevelsSampleRateBFOFreq.ReceiveF

BLISS

Voice

Digital

Original

Signal

Foetal Heartbeat Analysis

Input Data

Separated sources

Amplitude(micro volts)

-200 -100 0 100 200 300

-5

0

510

15

20

25

Time (milliseconds)

Triplet 2

Application to triplets

-200 -100 0 100 200 300-6-4-2

02468

1012

Time (milliseconds)

Triplet 314

Time (milliseconds)-200 -100 0 100 200 300

-5

0

5

10

15

20 Triplet 125

Averaged foetal ECG

Triplet 1

Triplet 2

Triplet 3

Fast ICA (real data)

Find unit norm vector to maximise

Nonlinear adaptive filter (stochastic gradient)

Fixed point ( )

Iterative solution (normalise and repeat)

Deflate/project to find next weight vector

t

44 3})E{()kurt( wxwxw =TT

)]()()(3))()()(([)()1(23

ttttttttT

wwwxwxww µ +±=+

23 3})(E{ wwxwxwT

)(3}))((E{)1( 3nnn

Twxwxw =+

Convolutive Mixing

Effects of dispersion, multipath etc

Typical of acoustics in a room

Cocktail party effect

**

*

*

)(2 ts

)(1 ts

)(1 tx )(2 tx

Channel Model

Weighted sum of delayed samples (convolution)

Express in polynomial form (z-transform)

Convolution becomes simple product

)(.........)1()()( 10 pnshnshnshnx p++=

........)(.........)1()0()(

........)(.........)1()0()(

.........)(

1

1

110

+++=

+++=

++=

n

n

pp

znxzxxzx

znszsszs

zhzhhzh

)()()( zszhzx =

Polynomial Matrices

Convolution is product of z-transforms

Two signals and two sensors

Polynomial matrix

Need for new mathematical algorithms

)()()( zszhzx =

=)(

)(

)()(

)()(

)(

)(

2

1

2221

1211

2

1

zs

zs

zhzh

zhzh

zx

zx

)(zH

Second Order Stage (Convolutive)

Strong decorrelation

Whiten or equalise spectra

ij

T

tiji tvtv =

=1

)()()(

=)(0

0)()/1()(

2

1

z

zzz

TVV

ijiji zzvzv )()/1()( =

Paraconjugation

Paraunitary matrix

Apply a decorrelation and whitening filter (2nd order)

Hidden paraunitary matrix

( ) IHHHH == zzzz )(~

)(~)(

Hidden Paraunitary Matrix

( )z

zT 1)(

~HH =

IVV =)(~)( zz

IHVVH =)(~)(

~)()( zzzz

Future Directions

Combine 2nd order and higher order statistics

semi-blind algorithms

Combine PCA and ICA stages

more robust algorithms

Broadband adaptive sensor arrays

broadband subspace identification

Acknowledgements

Colleagues at QinetiQ, Malvern

Dr I J Clarke, Dr C A Speirs, Dr D T Hughes

Dr I K Proudler, Dr T J Shepherd, Mr P Baxter

QinetiQ, Winfrith, Bincleaves, Portsdown

University of Leuven

Dr L De Lathauwer

UK Ministry of Defence

Corporate Research Programme