Digital adaptive filters: Conditions for convergence, rates of convergence, effects of noise and errors arising from the implementation

Abstract

A variety of theoretical results are derived for a well-known class of discrete-time adaptive filters. First the following idealized identification problem is considered: a discrete-time system has vector input <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> and 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 time-invariant coefficient vector. The filter considered adjusts an estimate vector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{h}(t)</tex> in a control loop according to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{h}(t + \Delta t) = \hat{h}(t) + K[z(t) - \hat{z} (t)]x(t)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{z}( t)= \hat{h}( t) \' x( t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> is the control loop gain. 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(t) = h - \hat{h}(t)</tex> . It is shown that a certain nondegeneracy "mixing" condition on the Input { x(t)} is necessary and sufficient for the exponential convergence of the misalignment. Qualitatively identical upper and lower bounds are derived for the rate of convergence. Situations where noise is present in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z(t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> and 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 are analyzed. Nonmixing inputs are also considered, and it is shown that in the idealized model the above stability results apply with only minor modifications. However, nonmixing input in conjunction with certain types of noise lead to bounded input - unbounded output, i.e., instability.