Magnitude response peak detection and control using balanced model reduction and leakage to a target

Abstract

Abstract-Adaptive filters are often used in systems that need to adjust to unknown environments. Communication channels with frequency nulls, signals that lack energy in a frequency band, and transducers with a finite bandwidth present special problems since adaptive filters can develop a large gain at frequencies where excitation is lacking. Such magnitude response peaks can cause problems if unchecked. This paper suggests a procedure for detecting and controlling magnitude response peaks that uses a balanced model reduction technique to form a low-order IIR filter that approximates the performance of the filter. The poles are then studied to determine if magnitude response peaks are present. If a peak is detected, then "leakage to a target" is used to gradually reduce the peak with minimal effect on the equalizer's response at other frequencies. Several useful bounds on the equalizer frequency response magnitude are derived, and the frequency domain behavior of the leakage to a target algorithm is analyzed. A case study is provided.

Unbiased and stable leakage-based adaptive filters

“…Consider an adaptive update of the general form with some initial condition (11) where is the error vector we want to minimize. The function is stochastic, i.e., for every and , is a random vector.…”

“…Theorem 1 (Averaging Result): Consider the error equation (11) and its averaged forms (12) and 13, where the sequence is uniform-mixing (see [21, p. 357]). Assume the following.…”