Eigenstructure Methods for Noise Covariance Estimation

1. Eigenstructure Methods for Noise Covariance Estimation Olawoye Oyeyele AICIP Group Presentation April 29th, 2003

2. Outline • Background • Adaptive Antenna Arrays • Array Signal Processing • Discussion • Next Steps

3. 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

4. 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

5. … d 0 1 3 … k-2 k-1 Uniformly spaced Linear Array Signals arriving at the (K-1)th element lag those at the (k-2)th element but lead in time . . . … ..

6. Adaptive Antenna Array Generally, complex weights are used.

7. Basic Antenna Array Parameters Array Propagation Vector: contains the information on angle of arrival of the signal Array Factor: the radiation pattern of the array consisting of isotropic elements

8. 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

9. Comparison between Sensor and Antenna Arrays *Multipath components are signal waves arriving at different times because each sample traveled varying distances as a result of reflections.

10. 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

11. 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

12. Adaptive Beamforming • Can be performed in both frequency and time domains • Sample Matrix Inversion • Least Mean Squares(LMS) • Recursive Least Squares(RLS) • Neural Network

13. Two-Element Example Desired Array Output: Interference arrives at angle of pi/6 w1 w2 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

14. Wiener (Optimal) Solution

15. 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

16. 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.

17. Sample Matrix Inversion(SMI) • Operates directly on the snapshot of data to estimate covariance matrix Weight Vector can be estimated as:

18. SMI Disadvantages • Increased computational complexity • Inversion of large matrices and numerical instability due to roundoff errors

19. Recursive Least Squares(RLS) where is the forgetting factor – ensures that data in the previous data are forgotten Thus, the matrix is found recursively

20. Recursive Least Squares(RLS) • Fast convergence even with large eigenvalue spread. • Recursively updates estimates

21. Beam Pattern

22. 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

23. 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

24. MUSIC algorithm-contd 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.

25. MUSIC Algorithm-contd • After computing the eigenvalues of Ruu,the eigenvalues of ARssAH can be computing by subtracting the variances as follows: • If number of incident signals D, is less than number of number of antenna elements M, then M-D eigenvalues are zero.

26. Spatial Spectrum 2 signals, 8 array elements

27. 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

28. 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

29. 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.