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Publication# A Unified Approach to the Steady-state and Tracking Analyses of Adaptive Filtering Algorithms

Abstract

Most adaptive filters are inherently nonlinear and time variant systems. The nonlinearities in the update equations of these filters usually lead to significant difficulties in the study of their performance. This paper develops a new feedback approach to the steady-state and tracking analyses of adaptive algorithms that bypasses many of the difficulties encountered in traditional approaches. In this new formulation, we not only re-derive several earlier results in the literature, but we often do so under weaker assumptions, in a considerably more compact way, and we also obtain new results.

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Basic signal processing concepts, Fourier analysis and filters. This module can
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Digital Signal Processing III

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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 algorithms, almost all adaptive filters are digital filters. Adaptive filters are required for some applications because some parameters of the desired processing operation (for instance, the locations of reflective surfaces in a reverberant space) are not known in advance or are changing.

Compact space

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact.

Locally compact space

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. In mathematical analysis locally compact spaces that are Hausdorff are of particular interest; they are abbreviated as LCH spaces. Let X be a topological space. Most commonly X is called locally compact if every point x of X has a compact neighbourhood, i.

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