April 2008
Intermediate
832 pages
26h 2m
English
Content preview from Adaptive Filters
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CHAPTER 25 Data-Normalized Filters
We extend the transient analysis of the earlier chapters to more general filter recursions, starting with the normalized LMS algorithm.
25.1 NLMS FILTER
We thus consider the 
-NLMS recursion
for which the data normalization in (22.2) is given by
(25.2) 
In this case, relations(22.26)–(22.27) and (22.29) become
and we see that we need to evaluate the moments
Unfortunately, closed form expressions for these moments are not available in general, even for Gaussian regressors. Still, we will be able to show that the filter is convergent in the mean and is also mean-square stable for step-sizes satisfying μ < 2, and regardless of the input distribution (Gaussian or otherwise) — see App. 25.B. We therefore treat the general case directly. Since the arguments are similar to those in Chapter 24 for LMS, we shall be brief.
Thus, introduce the M2 × 1 vectors
as well as the M2 × M2 matrices
and the M × M matrix
The matrix ...
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