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.

  • Widrow, B.; Mantey, P. E.; Griffiths, L. J.; Goode, B. B. (1967). (PDF). Proceedings of the IEEE. 55 (12): 2143–2159. doi:.
  • Haykin, S. (2002). . Prentice Hall. pp. –168. ISBN 0-13-048434-2.