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 {\displaystyle R} with its estimate. Using K {\displaystyle K} N {\displaystyle N} -dimensional samples X 1 , X 2 , … , X K {\displaystyle X_{1},X_{2},\dots ,X_{K}} , an unbiased estimate of R X {\displaystyle R_{X}} , the N × N {\displaystyle N\times N} correlation matrix of the array signals, may be obtained by means of a simple averaging scheme: R ^ X = 1 K ∑ k = 1 K X k X k H , {\displaystyle {\hat {R}}_{X}={\frac {1}{K}}\sum \limits _{k=1}^{K}X_{k}X_{k}^{H},} where H {\displaystyle H} is the conjugate transpose. The expression of the theoretically optimal weights requires the inverse of R X {\displaystyle R_{X}} , and the inverse of the estimates matrix is then used for finding estimated optimal weights.
Sample matrix inversion (or direct matrix inversion) is an algorithm that estimates weights of an array (adaptive filter) by replacing the correlation matrix with its estimate. Using -dimensional samples , an unbiased estimate of , the correlation matrix of the array signals, may be obtained by means of a simple averaging scheme:
where is the conjugate transpose. The expression of the theoretically optimal weights requires the inverse of , and the inverse of the estimates matrix is then used for finding estimated optimal weights.
References
[edit]- 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.