Exact equality of the MSEs for two types of nonlinear adaptive systems: Saturation and dead-zone types

TL;DR

This paper investigates the steady-state mean square errors of adaptive systems with saturation and dead-zone nonlinearities, revealing conditions under which their MSEs are equal and deriving parameters that maximize MSE.

Contribution

It provides a theoretical analysis showing the equality of MSEs for saturation and dead-zone nonlinearities when their parameters match, and derives formulas for optimal parameter selection.

Findings

MSEs are identical when saturation value equals dead-zone width

Derived equations for maximizing steady-state MSE

MSE depends on nonlinearity parameters, not step size or noise variance

Abstract

Adaptive signal processing systems, commonly utilized in applications such as active noise control and acoustic echo cancellation, often encompass nonlinearities due to hardware components such as loudspeakers, microphones, and amplifiers. Examining the impact of these nonlinearities on the overall performance of adaptive systems is critically important. In this study, we employ a statistical-mechanical method to investigate the behaviors of adaptive systems, each containing an unknown system with a nonlinearity in its output. We specifically address two types of nonlinearity: saturation and dead-zone types. We analyze both the dynamic and steady-state behaviors of these systems under the effect of such nonlinearities. Our findings indicate that when the saturation value is equal to the dead-zone width, the mean square errors (MSEs) in steady states are identical for both nonlinearity types. Furthermore, we derive a self-consistent equation to obtain the saturation value and dead-zone width that maximize the steady-state MSE. We theoretically clarify that these values depend on neither the step size nor the variance of background noise.