Transcript Unsupervised Pattern Recognition for the Classification of

Eigenstructure Methods for Noise
Covariance Estimation
Olawoye Oyeyele
AICIP Group Presentation
April 29th, 2003
Outline
 Background
 Adaptive Antenna Arrays
 Array Signal Processing
 Discussion
 Next Steps
Objective
 Discuss Antenna Arrays and similarities to sensor
arrays
 Investigate methods used for covariance estimation
in adaptive antenna arrays with a focus on applicable
eigenstructure methods
Background
 Antenna Arrays are a group of antenna elements with
signal processing capability which enables the
dynamic update of the beam pattern
 Various elemental configurations possible:
– Linear
– Circular
– Planar
 Major objective is to cancel interference
 Sensor Arrays are similar to antenna arrays
Uniformly spaced Linear Array
Signals arriving at the (K-1)th element lag those at the (k-2)th
element but lead in time
…
. . . … ..
d
0
1
3
…
k-2 k-1
Adaptive Antenna Array
Generally, complex weights are used.
Basic Antenna Array Parameters
Array Propagation Vector: contains the information on
angle of arrival of the signal


vT  1 e jkd sin 0 ... e jk ( K 1) d sin 0
k  2 /  and   wavelength
Array Factor: the radiation pattern of the array
consisting of isotropic elements
N 1
AF   An e j ( nkd sin  k )
n 0
Steering Vector
 Contains the responses of all the vectors of an array.
 Used to accomplish electronic Beam Steering – each
element of vector performs phase delay with respect
to the next.
 In electronic steering no physical movement of the
array is done.
 Mechanical beam steering involves physically moving
the elements of the array.
 Multiple steering vectors constitute an Array Manifold
– Array manifold is an array of steering vectors
Comparison between Sensor and
Antenna Arrays
No.
Sensor Arrays
Antenna Arrays
1.
Multiple sensors’
readouts used to
make final decision
Reception of multiple
elements are
combined to estimate
signal
2.
Different sensors
provide different
“views”
Different elements
receive multipath*
components
3.
Not necessarily all
sensor readouts are
combined
Not necessarily all
element receptions
are combined
*Multipath components are signal waves arriving at different
times because each sample traveled varying distances as a
result of reflections.
Array Signal Processing
 Techniques employed in adaptive antenna arrays
 They include:
– Beamforming(Adaptive & Partially Adaptive)
– Direction of Arrival Estimation(DOA)
 These techniques require the estimation of
covariance matrices
Beamforming
 Adjusting signal amplitudes and phases to form a
desired beam
 Estimation of signal arriving from a desired direction
in the presence of noise by exploiting the spatial
separation of the source of the signals.
 Applicable to radiation and reception of energy.
 May be classified as:
–
–
–
–
Data Independent
Statistically optimum
Adaptive
Partially Adaptive
Adaptive Beamforming
 Can be performed in both frequency and time




domains
Sample Matrix Inversion
Least Mean Squares(LMS)
Recursive Least Squares(RLS)
Neural Network
Two-Element Example
S (t )  Ae
j 2f 0t
 / 6I (t )  Ne j 2f0t
Desired Array Output:
yd (t )  Ae j 2f0t (1  2 )
Interference arrives at angle of pi/6
w1
0 / 2

y
w2
Received Interference signals:
yI (t )  Ne j 2f0t1  Ne j ( 2f0t  / 2)2
To completely cancel interference
(yd=y) the following weights must be
used:
w1=1/2-j/2;
w2=1/2+j/2
Wiener (Optimal) Solution
 2 (t )  [d * (t )  w H x(t )]2

 

E  2 (t )  E d *2 (t )  2 w H r  w H Rw
where r  E{d * (t)x(t)}and R  E{x(t)xH (t)}
R  covariancematrix
ComputingtheMinimumsquared error
 w ( E{ 2 (t )})  2r  2 Rw  0
which gives
w opt  R 1r  WienerOptimumSolution
Eigenstructure Technique
 For L x L matrix
 Largest M eigenvalues correspond to M directional
sources
 L-M smallest eigenvalues represent the background
noise power
 Eigenvectors are orthogonal – may be thought of as
spanning L-dimensional space
Eigenstructure Technique
 The space spanned by eigenvectors may be
partitioned into two subspaces
– Signal subspace
– Noise subspace
 The steering vectors corresponding to the directional
sources are orthogonal to the noise subspace
– noise subspace is orthogonal to signal subspace thus
steering vectors are contained in the signal subspace
 When explicit correlation matrix is required it may be
estimated from the samples.
Sample Matrix Inversion(SMI)
 Operates directly on the snapshot of data to estimate
covariance matrix
^
R

N2
H
x
(
i
)
x
(i )

i  N1
N2
^
r   d * (i ) x(i )
i  N1
Weight Vector can be estimated as:
^
^ 1 ^
w R r
SMI Disadvantages
 Increased computational complexity
 Inversion of large matrices and numerical instability
due to roundoff errors
Recursive Least Squares(RLS)
N
~
n i
H
(
n
)


x
(
i
)
x
(i )

R
i 1
N
~
n i
(
n
)


d * (i ) x(i )

r
n 1
where 0    1 is the forgetting factor – ensures that data
in the previous data are forgotten
~
R ( n)  
~
~
H
(
n

1
)

x
(
n
)
x
( n)
R
~
r (n)   r (n  1)  d * (n) x(n)
Thus, the matrix is found recursively
Recursive Least Squares(RLS)
 Fast convergence even with large eigenvalue spread.
 Recursively updates estimates
Beam Pattern
Direction of Arrival Estimation(DOA)
 DOA involves computing the spatial spectrum and
determining the maximas.
– Maximas correspond to DOAs
 Typical DOA algorithms include:
– Multiple SIgnal Classification(MUSIC)
– Estimation of Signal Parameters via Rotational Invariance
Techniques(ESPRIT)
– Spectral Estimation
– Minimum Variance Distortionless Response(MVDR)
– Linear Prediction
– Maximum Likelihood Method(MLM)
 MUSIC is explored in this presentation
MUSIC Algorithm
 Useful for estimating
–
–
–
–
Number of sources
Strength of cross-correlation between source signals
Directions of Arrival
Strength of noise
 Assumes number of sources < Number of antenna
elements.
– else signals may be poorly resolved
 Estimates noise subspace from available samples
MUSIC algorithm-contd
U (t )  As(t )  n(t )
T akingtheExpectation of both sides
Ruu  E[u (t )u (t   ) H ]  E[ As(t )  n(t ))(As(t )  n(t ))H ]
where H denotesHermitianMatrix
Thus,
Ruu  AE[ ss H ] A H  E[nnH ]
Ruu  ARss A H   n I
2
assumes thatnoise at each arrayelementis additive white and gaussian(A
Assumes
that
at each array
element
is additive
whitesignal
and is assu
and have
thenoise
same variance
, also the
mean of
each arriving
gaussian(AWGN) uncorrelated between elements with the same
variance and that arriving signals have a mean of zero.
MUSIC Algorithm-contd
 After computing the eigenvalues of Ruu,the
eigenvalues of ARssAH can be computing by
subtracting the variances as follows:
i  i   n
2
 If number of incident signals D, is less than number
of number of antenna elements M, then M-D
eigenvalues are zero.
Spatial Spectrum
2 signals, 8 array elements
Discussion
 Signal should lie mostly in subspace spanned by
eigenvectors associated with large eigenvalues noise is weak in this subspace.
 Idea of communicating where noise is weak similar to
other spectrum optimization problems – e.g. waterfilling solution to communication spectrum allocation
problem
 Signal strength is maximum in subspace where noise
is weak
Next steps
 Apply to the Restricted Matched Filter problem- select a fixed
subset of sensors in a cluster
 Obtain results that demonstrate the optimality of the Receiver
operating characteristic
References
 Lal C. Godara, "Application of Antenna Arrays to Mobile
Communications, Part I: Performance Improvements, Feasibility
and System Considerations," Proc. of the IEEE, Vol. 85, No 7,
pp. 1031- 1060, July 1997.
 Lal C. Godara, "Application of Antenna Arrays to Mobile
Communications, Part II: Beamforming and Direction of Arrival
Considerations," Proc. of the IEEE, Vol. 85, No 8, pp. 11951245, July 1997.
 B.D. Van Veen and K. M. Buckley "Beamforming: A versatile
Approach to Spatial Filtering" IEEE ASSP Magazine, pp. 4-24,
April 1988.
 John Litva and Titus Kwok-Yeung Lo, Digital Beamforming in
wireless communications, Artech House Publishers, 1996.