## 2 System Description  
I consider the system from References [20; 47; 49], illustrated in Figure 1, where I make the following assumptions as were done in References [20; 47; 49]:  
1. The source signal \(x[n]\) is given by \[x[n]=x_{1}[n]+jx_{2}[n]\] (1) where \(x_{1}[n]\) and \(x_{2}[n]\) are the real and imaginary parts of \(x[n]\), respectively. It is assumed that \(x_{1}[n]\) and \(x_{2}[n]\) are independent and that \[E[x[n]]=0\] (2) where \(E[\cdot]\) stands for the expectation operation. 2. The unknown channel \(h[n]\) is possibly a non-minimum phase linear time-invariant filter in which the transfer function has no \"deep zeros\". 3. The filter \(c[n]\) is a tap-delay line. 4. The channel noise \(w[n]\) is an additive Gaussian white noise with variance \(\sigma_{w}^{2}\) where \(\sigma_{w_{r}}^{2}\) and \(\sigma_{w_{i}}^{2}\) are the variances of the real and imaginary parts of \(w[n]\), respectively. 5. The function \(T[\cdot]\) is a memoryless nonlinear function that satisfies the additivity condition: \[T[z_{1}[n]+jz_{2}[n]]=T[z_{1}[n]]+jT[z_{2}[n]]\] (3) where \(z_{1}[n],z_{2}[n]\) are the real and imaginary parts of the equalized output, respectively. As was described in References [20; 47; 49], the source input \(x[n]\) is sent via the channel \(h[n]\) and is corrupted with channel noise \(w[n]\). The ideal equalized output is expressed in Reference [50] as  
\[z[n]=x[n-D]e^{i\theta} \tag{4}\]  
where \(D\) is a constant delay and \(\theta\) is a constant phase shift. Therefore, in the ideal case, we could write  
\[c[n]*h[n]=\delta[n-D]e^{i\theta} \tag{5}\]  
where \"\(*\)\" denotes the convolution operation and \(\delta\) is the Kronecker delta function. In this paper, I assume, as was also done in References [20; 47; 49], that \(D\) and \(\theta\) are equal to zero (please refer to Reference [49] for the explanation). The equalized output is given by  
\[z\left[n\right]=x\left[n\right]+\bar{p}\left[n\right]+\bar{w}\left[n\right] \tag{6}\]  
Figure 1: Block diagram of the system. where \(\tilde{p}[n]\) is the convolutional noise and  
\[\tilde{w}\left[n\right]=w\left[n\right]*c\left[n\right]. \tag{7}\]  
In this paper, the ISI is used to measure the performance of the deconvolution process, defined by  
\[ISI=\frac{\sum_{i\tilde{n}}|\tilde{s}[\tilde{m}]|^{2}-|\tilde{s}|_{max}^{2}}{| \tilde{s}|_{max}^{2}} \tag{8}\]  
where \(|\tilde{s}|_{max}\) is the component of \(\tilde{s}\), given in (9), having the maximal absolute value. \[\tilde{s}[n]=c\left[n\right]*h\left[n\right]=\delta\left[n\right]+\tilde{ \varsigma}\left[n\right] \tag{9}\]  
where \(\tilde{\varsigma}[n]\) stands for some error not having the ideal case. The input sequence \(x[n]\) is estimated with the function \(T[z[n]]\) and is denoted as \(d[n]\). The difference between \(T[z[n]]\) and the equalized output \(z\left[n\right]\) is denoted as \(\tilde{\varepsilon}[n]\):  
\[E[x[n]]=0\]0  
This error plays an important role in updating the equalizer's taps:  
\[E[x[n]]=0\]1  
where \(\left(\cdot\right)^{*}\) is the conjugate operation on \(\left(\cdot\right)\), \(\mu\) is the step size parameter, and \(\underline{c}[n]\) is the equalizer vector, where the input vector is \(\underline{y}[n]=[y[n]...y[n-N+1]]^{T}\). The operator \(\left(\cdot\right)^{T}\) denotes the transpose of the function \(\left(\cdot\right)\), and \(N\) is the equalizer's tap length. Another way to update the equalizer's taps is to use the cost function approach [51]:  
\[E[x[n]]=0\]2  
where \(F[n]\) is a predefined function that characterizes the ISI. In the literature [20; 47; 49; 50], we may find the conditional expectation (\(E[x[n]|z[n]]\)) as a proper option for \(T[z[n]]\).