Theoretical derivation of minimum mean square error of RBF based equalizer

Abstract

In this paper, the minimum mean square error (MSE) convergence of the RBF equalizer is evaluated and compared with the linear equalizer based on the theoretical minimum MSE. The basic idea of comparing these two equalizers comes from the fact that the relationship between the hidden and output layers in the RBF equalizer is also linear. For the purpose of theoretically evaluating exact minimum MSE for both RBF and linear equalizer, a linear time dispersive channel whose order is one is selected. As extensive studies of this research, various channel models are selected, which include linearly separable channel, slightly distorted channel, and severely distorted channel models. In this work, the theoretical minimum MSE for both RBF and linear equalizers were computed, compared and the sensitivity of minimum MSE due to RBF center spreads was analyzed. It was found that RBF based equalizer always produced lower minimum MSE than linear equalizer, and that the minimum MSE value of RBF equalizer was obtained with the center spread parameter which is relatively higher (approximately 2–10 times more) than variance of AWGN. As a result of that, it leads to the better bit error rate. This work provides an analytical framework for the practical training of RBF equalizer system.

Introduction

In digital communication systems, time dispersion caused by the non-ideal channel frequency response characteristics or multi-path transmission creates overlapping of the received symbols due to intersymbol interference (ISI). The conventional way to combat ISI is to include an equalizer in the receiver. The most widely known equalizer is an adaptive linear transversal equalizer, in which the output signal is compared to the expected signal and the tap coefficients are updated in accordance with the error between the desired signal and actual filter output. For more than a decade, there has been much attention given to applying neural networks to the digital communication areas, including channel equalization problems [1]. Multi-layer perceptrons (MLP) equalizer [2], [3], [4] is able to equalize non-minimum phase channels without the introduction of any time delay; and it is less susceptible than a linear equalizer to the effects of high levels of additive noise. However, the network architecture and training algorithm of the MLP equalizer is much more complex than the linear equalizer. Also, the RBF network has received a great deal of attention by many researchers because of its structural simplicity and more efficient learning [5], [6], [7], [8], [9], [10], [11], [12]. In [5], Chen et al. applied RBF network to channel equalization problem to get the optimal Bayesian solution. Also, the application of RBF network to the modeling of Bayesian equalizer was discussed [13]. In this paper, the effect of delay order on making decision boundary and optimum delay for minimum bit error rate performance was studied. Although the RBF based equalizers provide great advantages over MLP based equalizer, they still require considerable number of RBF centers when channel order goes high, which finally leads to computational burden in training. To solve this problem, some researchers developed techniques for either reducing the number of centers or selecting them wisely for training [6], [8], [10], [12]. In a paper by Kumar et al. [12], the minimal RBF network [11] was applied to channel equalization. It says that channel equalizer with minimal RBF network provide efficient solution without requiring much complexities, such as channel order estimation and fixing of the number of channel states.

Although many of studies, mentioned above, claims that RBF based equalizers are superior to conventional linear equalizer due to both RBF network's structural linearity (or simplicity) and efficient training, none of them tried to compare those two from the theoretical minimum mean square error (MSE) point of view. The basic idea of comparing these two equalizers comes from the fact that once input domain of RBF network is transformed to another domain through Gaussian basis functions, the relationship between the hidden and output layers in the RBF equalizer is also linear. In this paper, the theoretical minimum MSE for various channels for both RBF and the linear equalizers were evaluated and compared. Also, the sensitivity of minimum MSE to equalizer order, channel delay, and RBF center spreads was analyzed.