Eigenstructure Methods for Noise Covariance Estimation

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

. . . … ..

…

d0 1 3 … k-2 k-1

Signals arriving at the (K-1)th element lag those at the (k-2)th element but lead in time

Adaptive Antenna Array

Generally, complex weights are used.

Basic Antenna Array Parameters

1

0

) sin(N

n

knkdjneAAF

Array Factor: the radiation pattern of the array consisting of isotropic elements

Array Propagation Vector: contains the information on angle of arrival of the signal

wavelengthk

eev dKjkjkdT

and /2

... 1 00 sin)1(sin

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

w1 w2

tfjNetI 02)( tfjAetS 02)( 6/

Desired Array Output:

2/0

)()( 212 0 tfj

d Aety

Interference arrives at angle of pi/6

2)2/2(

12 00)( tfjtfj

I NeNety

Received Interference signals:

To completely cancel interference (yd=y) the following weights must be used:

w1=1/2-j/2; w2=1/2+j/2

y

Wiener (Optimal) Solution

Solution Optimum Wiener w

giveswhich

022)})({(

error squared Minimum theComputing

matrix covarianceR

(t)}E{x(t)xR and (t)x(t)}*E{d r where

2)(*)(

)]()(*[)(

1opt

2w

H

22

22

rR

RwrtE

RwwrwtdEtE

txwtdtHH

H

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

2

1

2

1

)()(*

)()(

^

^

N

Ni

N

Ni

H

ixid

ixix

r

R

rRw^1^^

Weight Vector can be estimated as:

SMI Disadvantages

Increased computational complexity Inversion of large matrices and numerical instability

due to roundoff errors

Recursive Least Squares(RLS)

N

n

in

N

i

Hin

ixidn

ixixn

r

R

1

~

1

~

)()(*)(

)()()(

where 10 is the forgetting factor – ensures that data in the previous data are forgotten

)()(*)1()(

)()()1()(~~

~~

nxndnn

nxnxnn

rr

RR H

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

)()()( tntAstU

MatrixHermitian denotes H where

]))()())(()([])()([

sidesboth ofn Expectatio theTakingHH

uu tntAstntAsEtutuER

Thus,

Assumes that noise at each array element is additive white and 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:

2nii

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. water-filling 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. 1195- 1245, 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.