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.
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