Least Mean Squares Filter - Simplifications

Simplifications

For most systems the expectation function must be approximated. This can be done with the following unbiased estimator

$\hat{E}\left\{\mathbf{x}(n) \, e^{*}(n)\right\}=\frac{1}{N}\sum_{i=0}^{N-1}\mathbf{x}(n-i) \, e^{*}(n-i)$

where indicates the number of samples we use for that estimate. The simplest case is

$\hat{E}\left\{\mathbf{x}(n) \, e^{*}(n)\right\}=\mathbf{x}(n) \, e^{*}(n)$

For that simple case the update algorithm follows as

Indeed this constitutes the update algorithm for the LMS filter.

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