## 4 Simulation  
In this section, I show via simulation results the efficiency of the truncated expression for the conditional expectation (24) combined with the Lagrange multipliers given in (26). Namely, I show via simulation results the equalization performance from the residual ISI point of view of the new proposed algorithm (with (24) and (26)) compared to the simulation results obtained by the maximum entropy [49] method and [PERSON]'s [53] algorithm. [PERSON]'s [53] algorithm is used for comparison since it is a very efficient algorithm from the equalization performance point of view [28]. In addition,it's computational burden is very low [28]. For [PERSON]'s algorithm [53], the equalizer's taps were updated as follows:  
\[c_{l}\left[n+1\right]=c_{l}\left[n\right]-\mu_{G}\left(\left|z\right|^{2}-\frac{ E\left[\left|x\right|^{4}\right]}{E\left[\left|x\right|^{2}\right]}\right)y^{*} \left[n-l\right] \tag{35}\]  
where \(\mu_{G}\) is a positive step size parameter and \(l\) stands for the l-th tap of the equalizer. The update mechanism of the equalizer's taps associated with the recently obtained maximum entropy algorithm [49] was as  
where  
\[E[x_{1}^{G}]=\tilde{m}_{G}. \tag{44}\]  
In order to get an equalization gain of one, the following gain control was used according to Reference [49]:  
\[c_{l}[n]=\frac{\widetilde{c}_{l}}{\sqrt{\sum_{l}\left|\widetilde{c}_{l}\right|^{ 2}}} \tag{45}\]  
where \(c_{l}[n]\) is the vector of taps after iteration and \(c_{l}[0]\) is some reasonable initial guess. The update mechanism of the equalizer's taps associated with the new proposed blind equalization method involving the truncated version for the conditional expectation and the newly derived Lagrange multipliers was as follows:  
\[\tilde{c}_{l}\left[n+1\right]=c_{l}\left[n\right]-\mu_{BNEW}Wy^{*}[n-l] \tag{46}\]  
with \(W\) given in (37), but with  
\[E\left[x_{1}|z_{  
\(\sigma_{z_{s}}^{2}\) was estimated by  
\[\left\langle z_{s}^{2}\right\rangle=\left(1-\beta_{BNEW}\right)\left\langle z_{s }^{2}\right\rangle_{n-1}+\beta_{BNEW}\left(z_{s}\right)_{n}^{2} \tag{50}\]  
where \(\left\langle z_{s}^{2}\right\rangle_{0}>0\), \(\beta_{BNEW}\) and \(\mu_{BNEW}\) are positive step size parameters. The equalizer's taps in (46) were updated only if \(\hat{N}_{s}>\varepsilon\), where \(\varepsilon\) is a small positive parameter and  
\[\hat{N}_{s}=1+\frac{\delta^{\prime\prime}(z_{s})}{2\delta(z_{s})}\sigma_{p_{s }}^{2}. \tag{51}\]  
I also used here a gain control according to (45). In the following, I denote \"\(MaxEnt_{ANEW}\)\", \"\(Godard\)\", and \"\(MaxEnt_{BNEW}\)\" as the algorithms given in References [49; 53], and (46), respectively. For the \"\(MaxEnt_{ANEW}\)\" and \"\(MaxEnt_{BNEW}\)\" algorithms, I used  
\[E[z_{s}^{2}]=E[x_{s}^{2}] \tag{52}\]  
for initialization. The following channels were considered: \begin{tabular}{l l}
**Channel 1**: (initial ISI = 0.44): Taken according to References [1; 49]: \(h_{n}=\left\{0\quad\text{for}\quad n<0;\quad-0.4\quad\text{for}\quad n=0; \quad 0.84\cdot 0.4^{n-1}\quad\text{for}\quad n>0\right\}\). \\
**Channel 2**: Normalized Channel 1 with \(\sum_{k}|h_{k}|^{2}=1.317\). \\
**Channel 3**: Normalized Channel 1 with \(\sum_{k}|h_{k}|^{2}=0.768\). \\ \end{tabular}   
In my simulation, the equalizer's length was set to 13 taps. For initialization purposes, the center tap of the equalizer was set to one while all others were set to zero. As a source input, I used the 16 QAM constellation. The equalization performance comparison between the new proposed equalization method (\"\(MaxEnt_{BNEW}\)\" (46)), the maximum entropy [49], and [PERSON]'s [53] algorithm is given in Figures 2-4. The equalization performance comparison was carried out for a 16 QAM constellation input sent via Channel 1 with SNR values of 10 dB, 7 dB, and 0 dB, respectively. It should be pointed out that the results in Figures 2 and 3 for \"\(MaxEnt_{ANEW}\)\" and \"\(Godard\)\" were reproduced from Reference [49].