Sample matrix inversion

Sample matrix inversion (or direct matrix inversion) is an algorithm that estimates weights of an array (adaptive filter) by replacing the correlation matrix R_u with its estimate. Using K samples $x(k),k=1,2,\dots ,K-1$ , an unbiased estimate of R_u, the correlation matrix of the array signals, may be obtained by means of a simple averaging scheme:

{\hat {R}}_{u}(k)=(1/k)\sum (x(k)x^{H}(k)).

The expression of the theoretically optimal weights requires the inverse of R_u, and the inverse of the estimates matrix is then used for finding estimated optimal weights.

References

Widrow, B.; Mantey, P. E.; Griffiths, L. J.; Goode, B. B. (1967). "Adaptive antenna systems" (PDF). Proceedings of the IEEE. 55 (12): 2143–2159. doi:10.1109/proc.1967.6092.
Haykin, S. (2002). Adaptive Filter Theory. Prentice Hall. pp. 165–168. ISBN 0-13-048434-2.

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