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 $R$ with its estimate. Using $K$ $N$-dimensional samples $X_1,X_2,\dots ,X_K$, an unbiased estimate of $R_X$, the $N\times N$ correlation matrix of the array signals, may be obtained by means of a simple averaging scheme:

${\hat{R}}_X=\frac{1}{K}\sum _{k=1}^K{X}_k{X}_k^H,$

where $H$ is the conjugate transpose. The expression of the theoretically optimal weights requires the inverse of $R_X$, 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.