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 ${\displaystyle R}$ with its estimate. Using ${\displaystyle K}$ ${\displaystyle N}$-dimensional samples ${\displaystyle X_{1},X_{2},\dots ,X_{K}}$, an unbiased estimate of ${\displaystyle R_{X}}$, the ${\displaystyle N\times N}$ correlation matrix of the array signals, may be obtained by means of a simple averaging scheme:

${\displaystyle {\hat {R}}_{X}={\frac {1}{K}}\sum \limits _{k=1}^{K}X_{k}X_{k}^{H},}$

where ${\displaystyle H}$ is the conjugate transpose. The expression of the theoretically optimal weights requires the inverse of ${\displaystyle R_{X}}$, 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" (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.