Complex-Valued Neural Networks for Equalization of Communication Channels

Introduction

The complex-valued neural networks have attracted a great deal of interest in recent years. The growing interest could be attributed to superior learning ability of complex-valued neural networks in comparison with the real valued counterparts. A number of new low complexity and fast learning algorithms have also been proposed by the researchers for complex-valued neural networks, facilitating their use in various applications (Nitta, 1997; Leung & Haykin, 1991). The applications of complex-valued neural networks have emerged in the areas of seismic, sonar and radar signal processing, speech and image processing, environmental science and wireless communications where signals are typically complex valued (Haykin, 1994a; Mandic & Chambers, 2001).

Digital communication through multipath channels is subject to intersymbol interference and its cancellation using adaptive equalizers has been studied for several years by the signal processing community. Also, in order to maximize efficiency of digital radio links, the transmitter high-power amplifiers are often required to operate near saturation, introducing nonlinearities which in turn cause degradation of the received signal. The nonlinearity may affect both amplitude and phase of the signal. It is well known that the signals like QAM are very sensitive to nonlinear distortion which causes spectral spreading, inter-symbol interference (ISI) and constellation warping. The classical approaches to equalization rely on the existence of a training sequence in the transmitted signal to identify the channel (Proakis, 1995).

In adaptive equalization, when the channel is varying, even slowly, the training sequence has to be sent periodically to update the equalizer. Since the inclusion of reference training signals in conventional equalizers sacrifices valuable channel capacity, adaptation without using the training signals i.e. blind equalization is preferred (Godard, 1980; Haykin, 1994b).

With the development of complex-valued versions of training algorithms, the nonlinear adaptive filters can be realized as neural networks for both conventional as well as blind equalization as they are well suited to deal with the complex-valued communication signals. Although, the real valued and complex-valued networks have been both extensively studied for equalization of communication channels, the advantages of using complex-valued neural networks instead of a real valued counterpart fed with a pair of real values is well established. Among neural network equalizers, the application of radial basis function networks (RBFN) has been extensively covered in the literature. The studies of Chen, Mulgrew and, Grant (1993), Cha and Kassam (1995), Jianping, Sundarrajan, and, Saratchandran (2002) and Li, Huang, Saratchandran and, Sundarrajan (2006) are some of the examples. The technique proposed by Chen et al. (1993) is based on the clustering of data and uses a real valued network. Cha and Kassam (1995) have proposed an adaptive complex-valued RBFN for equalization. In their approach, the inputs and outputs of the network are both complex-valued while the radial basis functions are real valued. Jianping et al. (2002) and Li et al. (2006) have considered RBFN- based equalization of channels, where growing and pruning strategy is used for selection of nodes. All these approaches show a remarkable improvement in the performance of the equalizers in comparison with conventional equalizers.

Uncini, Vecci, Campolucci, and Piazza (1999) have presented a study on the use of complex-valued neural networks with adaptive activation functions. An intelligent use of activation function has been shown to reduce the number of synaptic interconnections while preserving the universal approximation and regularization properties of the network, resulting in efficient implementation of nonlinear filters.