Sample matrix inversion (or direct matrix inversion) is an

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...

that estimates weights of an array (

adaptive filter An adaptive filter is a system with a linear filter that has a transfer function controlled by variable parameters and a means to adjust those parameters according to an optimization algorithm. Because of the complexity of the optimization algorit ...

) by replacing the

correlation matrix In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistic ...

R

with its estimate. Using

K

N

-dimensional samples

X_1, X_2,\dots,X_K

, an unbiased estimate of

R_

, the

N \times N

correlation matrix of the array signals, may be obtained by means of a simple

averaging In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...

scheme: :

\hat_ = \frac \sum\limits_^K X_k X^H_k,

where

H

is the

conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...

. The expression of the theoretically optimal weights requires the inverse of

R_

, and the inverse of the estimates matrix is then used for finding estimated optimal weights.

References

* * {{cite book , first1=S. , last1=Haykin , year=2002 , title=Adaptive Filter Theory , url=https://archive.org/details/adaptivefilterth00hayk , url-access=limited , publisher=Prentice Hall , page
165
��168 , isbn=0-13-048434-2 Covariance and correlation Filter theory