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 [math]\displaystyle{ R }[/math] with its estimate. Using [math]\displaystyle{ K }[/math] [math]\displaystyle{ N }[/math]-dimensional samples [math]\displaystyle{ X_1, X_2,\dots,X_K }[/math], an unbiased estimate of [math]\displaystyle{ R_{X} }[/math], the [math]\displaystyle{ N \times N }[/math] correlation matrix of the array signals, may be obtained by means of a simple averaging scheme:

[math]\displaystyle{ \hat{R}_{X} = \frac{1}{K} \sum\limits_{k=1}^K X_k X^H_k, }[/math]

where [math]\displaystyle{ H }[/math] is the conjugate transpose. The expression of the theoretically optimal weights requires the inverse of [math]\displaystyle{ R_{X} }[/math], 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". Proceedings of the IEEE 55 (12): 2143–2159. doi:10.1109/proc.1967.6092. http://isl-www.stanford.edu/~widrow/papers/j1967adaptiveantenna.pdf. 
• Haykin, S. (2002). Adaptive Filter Theory. Prentice Hall. pp. 165–168. ISBN 0-13-048434-2. https://archive.org/details/adaptivefilterth00hayk.