I've seen in the literature a few transient analyses for the LMS algorithm. These mostly focus on the mean estimation error for the LMS tap weights as well as the limiting signal estimation error covariance compared to the Weiner filter.

I am interested in simple expressions for the limiting coefficient error covariance.

In other words, if the LMS coefficient update is given by:

$$ \hat{\bf w}_{k+1}=\hat{\bf w}_{k} + \mu {\bf x}_k e_k^* $$

and the true coefficient vector is $\bar{\bf w}$, define the difference as:

$$ {\bf e}_k={\bf \hat{w}}_k-\bar{\bf w}. $$.

What can be said about $\lim_{k\rightarrow \infty} E[{\bf e}_k{\bf e}_k^H]$?


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