April 2008
Intermediate
832 pages
26h 2m
English
Content preview from Adaptive Filters
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CHAPTER 44 Indefinite Least-Squares
We end our treatment of adaptive filtering in this book by studying the robustness of adaptive filters in the presence of disturbances and uncertainties in the data. A study of this kind requires that we first define what we mean by robustness. For our purposes, and in loose terms, a robust filter will be one for which small disturbances in the data do not degrade the performance of the filter appreciably. The measure of smallness and largeness of a signal will be chosen as its energy, so that a robust filter will be one such that disturbances with small energy cannot lead to estimation errors with large energy and, more generally, the estimation error energy will remain bounded as long as the disturbance energy is bounded.
There are of course other characterizations of robustness. The one described above lends itself to analysis and mathematical manipulations. In particular, its characterization will involve studying quadratic cost functions that share many of the characteristics of regularized least-squares, except for the appearance of indefinite weighting matrices (as opposed to positive-definite weighting matrices). For this reason, many of the features of least-squares solutions will manifest themselves again in this part of the book; albeit in modified forms that result from the presence of indefinite weights.
44.1 INDEFINITE LEAST-SQUARES FORMULATION
The notion of robust adaptive filters will be defined in mathematical terms in Sees. ...
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