A note on the mathematical modeling of power amplifier/loudspeaker nonlinearity in acoustic echo cancellers

In this paper a new simple, three-parameter nonlinear mathematical model is proposed to represent the nonlinear characteristics of the power amplifier/loudspeaker nonlinearity in acoustic echo cancellers. The three parameters of the model are directly related to three distinct different sections of the nonlinear characteristic. The model and its first derivatives are continuous and stretch over the full range of input amplitudes. Using this model in conjunction with a conventional linear adaptive filter, the design of computationally efficient acoustic echo cancellers would be feasible.

I. Introduction

It is well known that users of many hand-free telecommunication systems; for example hands-free telephones and teleconferencing systems, suffer from the annoying effects of acoustic echo. Acoustic Echo Cancellers (AECs) are, therefore, developed to improve the audio quality by reducing acoustic echoes which result in disturbing comfortable communications. The general setup of a communication system with AEC comprises a power amplifier, a loudspeaker, an enclosure, a microphone, adaptive filters and a nonlinear compensator. Acoustic echoes are generated in the loudspeaker-enclosure-microphone system (LEMS) due to the coupling between the loudspeaker and the microphone. If the echo path is linear, then adaptive filters alone can be used in AECs. However, due to the nonlinear characteristics of the power amplifier and the loudspeaker, the echo path is not linear. This is specially the case in low cost portable communication systems where the power supply voltage is reduced, the power amplifiers are pushed into their saturation regions and small size loudspeakers are used. Thus, it is essential to model the echo path nonlinearity in order to improve the performance of AECs. In the literature several mathematical models have been proposed to represent the echo path nonlinearity. For example, hard clipping nonlinearity [1], [2], soft clipping nonlinearity [2], polynomials of different orders [3]–[5], hyperbolic tangent function [6], Sigmoid function [7], [8] and the piecewise defined raised-cosine function [9]. While all these functions are monotonically increasing and exhibit saturation behaviour for high input amplitudes, they suffer from one or more of the following disadvantages: 1.

It cannot represent the linear part of the nonlinear characteristics of the power amplifier/loudspeaker especially under small input amplitudes [7], [8].

2.

It has only one adaptive parameter available for adjustment to fit the saturation curve. Thus, it cannot model various saturation characteristics [1], [2], [6]–[8].

3.

It is based on a piecewise definition. Thus, it is not a continuous function stretching over the full range of input amplitudes. This would complicate any theoretical analysis to investigate convergence of its parameters [9].

4.

It cannot represent the saturation characteristics of the power amplifier/loudspeaker characteristics unless a high-order polynomial is used [3]–[5].