Introduction

High speed communications channels are often impaired by channel inter-symbol interference, co-channel interference and additive noise. Adaptive equalizers are required in these communication systems to obtain reliable data recovery [1]. The discrete-time model of the digital communications systems considered in this paper is depicted in Figure 1. In this model H₀(z) is the channel transfer function and there are a total of ${\eta}$ interfering co-channels with transfer functions ${H_i(z),{1\leq i\leq\eta}}$. The linear dispersive channel and co-channels are modeled by finite impulse response filters and, therefore, their transfer functions are given by

$$H_i(z)=\sum_{j=0}^{n_s} h_{ij} z^{-j}.\hspace{1 in} 0 \leq i \leq \eta$$ (1)

  

Figure 1: Discrete time model of data transmission system.

The transmitted data d₀(k) and the interfering data $d_i(k), 1 \leq i \leq \eta$ are assumed to be equiprobable and independent sequences. The data $d_i(k), 0 \leq i \leq \eta$, would have zero mean, i.e., E[d_i(k)] = 0 and $E[d_i(k_1)d_j(k_2)]=\delta(i-j) \delta(k_1 - k_2)$, where E[.] denotes the expectation operator. The additive white Gaussian noise e(k) has zero mean and variance $\sigma_e^2$ and is uncorrelated with the data $d_i(k), 0 \leq i \leq \eta$. As defined in [2] the channel observation y(k) = s(k) + u(k) + e(k) contains three terms called the desired signal, the interfering signal and the noise respectively, where $s(k)=\sum_{j=0}^{n_0} h_{0j} d_0(k-j)$ and $u(k)=\sum_{i=1}^{\eta}\sum_{j=0}^{n_i}h_{ij}d_i (k-j)$. Let $E[s^2(k)] = \sigma_s^2$ and $E[u^2(k)] = \sigma_u^2$. The signal to noise ratio is then defined as $SNR = {\sigma_s^2}/{\sigma_e^2}$ and the signal to interference ratio is given by $SIR = {\sigma_s^2}/{\sigma_u^2}$ and finally the signal to interference and noise ratio is given by $SINR = {\sigma_s^2}/({\sigma_e^2}+{\sigma_u^2})$ [4]. The task of the equalizer is to estimate the transmitted data d₀(k) based on the channel observation y(k). There are variety of approaches proposed for which one can refer [3].

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