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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
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