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:

: $\hat{R}_{X} = \frac{1}{K} \sum\limits_{k=1}^K X_k X^H_k,$

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

{{cite journal |first1=B. |last1=Widrow |first2=P. E. |last2=Mantey |first3=L. J. |last3=Griffiths |first4=B. B. |last4=Goode |year=1967 |title=Adaptive antenna systems |journal=Proceedings of the IEEE |volume=55 |number=12 |pages=2143–2159 |url=http://isl-www.stanford.edu/~widrow/papers/j1967adaptiveantenna.pdf |doi=10.1109/proc.1967.6092}}
{{cite book |first1=S. |last1=Haykin |year=2002 |title=Adaptive Filter Theory |url=https://archive.org/details/adaptivefilterth00hayk |url-access=limited |publisher=Prentice Hall |pages=[https://archive.org/details/adaptivefilterth00hayk/page/n164 165]–168 |isbn=0-13-048434-2}}

Category:Covariance and correlation

Category:Filter theory