Wavelet Neural Networks and Equalization of Nonlinear Satellite Communication Channel

Introduction

Wavelet networks are a class of neural networks which have wavelets as activation function instead of conventional sigmoid function. Wavelets are a special kind of short duration oscillatory functions satisfying certain criteria. In contrast to Fourier transform, wavelet analysis represents a function by the combination of translations and dilations of a single basis function called mother wavelet. Wavelet analysis is especially efficient in decomposition and reconstruction of signals with abrupt changes in components of time or frequency. Although wavelet theory has found application in many fields, the implementations are usually limited to wavelets of small dimensions. Construction and application of wavelets of larger dimensions is complex and prohibitive in cost. Wavelets neural network has been proven to be a powerful tool for handling problems with large number of dimensions (Zhang, Q., 1997).

Artificial neural network are a potent tool for handling large dimensional and ill-defined problems. Neural networks can recover underlying dependencies between the given inputs and outputs by using training data. Training enables neural network to represent high-dimensional nonlinear functions. Neural networks can solve regression and classification problems by changing parameters that control how they learn as they go through training data. These parameters are called weights and influence the quality of classification or regression. Approximation of a function by conventional neural network is not efficient one because at a fundamental level such functions are constructed using weighted sum of sigmoid functions. Sigmoid function has a large support compared to wavelets which are more efficient in representing localized functions. Therefore, because of compact representation provided by wavelets, WNN are expected to be smaller compared to other neural networks.

This chapter explains different architecture of WNN, their specific capabilities and learning algorithm. Static and dynamic modeling of nonlinear system using WNN will be explained in subsequent sections. Similar to multilayer perceptron (MLP), WNN can be trained using backpropagation algorithm. In addition to training network weights, the training algorithm must also adapts the translation and dilation parameters of wavelet. A novel application of WNN, in the form of equalization of nonlinear satellite communication, is given which demonstrates the capability of WNN as nonlinear function approximation tool. Nonlinearity in satellite communication arises because of transmitter power amplifiers operating near saturation region for power efficiency considerations. This nonlinearity in satellite power amplifier causes distortion in the received signal and results in significant increase in bit error rate (BER). Channel with larger BER results in packet loss and delay in case of data transmission, audio distortion and video frame loss in case of multimedia transmission. Compensation of channel and power amplifier nonlinearity requires adaptive equalizer implementation at the front end of the communication receiver. Other techniques of reducing nonlinearity due to on-board amplifier, like cascading another nonlinear device to the amplifier, had been proposed but were not too successful.

Besides countering of the effects of channel nonlinearity, equalizers are also applied for compensating the effects of inter-symbol interference (ISI) and noise over dispersive channel. Conventional linear equalizers employ a linear filter with finite impulse response (FIR) or lattice filter structure and using a least mean square (LMS) or recursive least-squares (RLS) algorithm for training the equalizer which adaptively change the filter coefficients. Once the training is complete, the equalizer characteristic is exactly opposite to the effects of the channel. Thus an adaptive equalizer is an iterative optimization program with mean square error (MSE) as objective function which tries to approximate inverse of channel characteristics (Swayamsiddha, S. et al., 2018).