Qualitative behavior of nonlinear differential equations describing adaptive filters using nonideal multipliers

Abstract

We analyze the behavior of a well-known class of analog adaptive filters in which all the multipliers are replaced by nonideal multipliers for which the linearity requirement with respect to one of the inputs is relaxed. Applications of these filters have been proposed in many different engineering contexts which have in common the following idealized problem: a system has a vector input x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> and a scalar output <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_{t} = h' x_{t}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</tex> is an unknown lime-invariant coefficient vector. From a knowledge of x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> and z <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> it is required to estimate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</tex> . The estimate vector is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(\hat{h_{t}})</tex> where h <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> is generated in the filter as the solution of a differential equation and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(\cdot)</tex> is a nonlinear map defined by the characteristics of the nonideal multipliers. The effectiveness of the filter is determined by the convergence properties of the misalignment vector, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r=h-f(\hat{h})</tex> . With a weak nondegeneracy requirement on x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> , the "mixing condition," we prove the exponential convergence of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\parallel r_{t} \parallel</tex> . Considerable emphasis is placed on the analysis of the effects of various important departures from the idealized problem as when noise is present, the coefficient vector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</tex> is time varying, etc. The results show that in every important aspect the qualitative behavior is similar to that of the conventional filter in which ideal multipliers are used. However, new techniques are required for the analysis of this inherently nonlinear system.