# New Approach of Blind Adaptive Equalizer Based on Genetic Algorithms
## 4 Methodology  
To validate the BLE-GA scheme, several simulations were performed, considering 4-QAM, 16-QAM, and 64-QAM digital communication systems, without channel encoding and operating at a rate of 10 Mbps. These systems were modeled as illustrated in Figure 6. The Matlab software [41] was used to implement and simulate the systems, with the GA-based blind equalization algorithm implemented using the Matlab Optimization Toolbox.  
The simulations also included an implementation of the CMA in order to have a reference to evaluate the performance of BLE-GA.  
Each communication system was evaluated for two scenarios of static channels with ISI and AWGN. These channels, here called Channel A and Channel B, are expressed, respectively, by  
\[h_{a}(k)=\delta(k)+0,5\delta(k-3)+0,3\delta(k-6) \tag{30}\]  
and  
\[h_{b}(k)=\delta(k)+0,6\delta(k-1)+0,4\delta(k-2). \tag{31}\]  
The simulations performed generated the bit error rate (BER) curves as a function of \(E_{b}/N_{0}\),  
\[E_{b}=\frac{E[a_{k}^{2}]}{log_{2}(M)}, \tag{32}\]  
where \(E_{b}\) represents the bit energy, and \(\sigma_{r}^{2}\) is the power spectral density of the system noise.  
The parameters of the equalizer were adjusted in a period of \(T_{ad}\) symbols. After this period (\(T_{ad}\) symbols), the error count, i.e., the computation of BER values, started and the equalizer parameters were not updated. The settings employed in the simulations are shown in Tables 2 and 3. Table 2 presents the parameters for the BLE-GA and CMA equalizers for three modulation schemes: 4-QAM, 16-QAM, and 64-QAM. The selection of the equalizer length \(N\), equalization delay \(d_{aq}\), and block length \(K\) for BLE-GA, as well as the adaptation step \(\mu\) for CMA, was based on extensive empirical testing. These values were optimized to balance each modulation scheme's convergence speed, computational cost, and equalization accuracy. For instance, the block length \(K\) in BLE-GA was adjusted to ensure efficient processing for higher modulation levels, such as 64-QAM, where \(K=1000\) symbols were necessary to accommodate increased complexity. Similarly, the adaptation step \(\mu\) in CMA was reduced for 64-QAM to maintain stability, reflecting its sensitivity to modulation order. The adjustment period \(T_{ad}\) of \(10^{7}\) symbols was uniformly set to ensure a fair comparison of the algorithms' steady-state performance under comparable conditions.  
Table 3 details the parameters used for configuring the GA in this study. The parameter \(N\) represents the size of the equalizer, directly influencing the chromosome length, which is calculated as \(2N+2\). The crossover fraction of 0.9 indicates that 90% of the new population is generated through crossover operations, while the elite count ensures that the top 10 individuals from each generation are preserved. The number of generations (100) and the population size (200) were chosen based on extensive empirical testing to balance convergence speed and computational efficiency. These settings were specifically tailored to optimize the stochastic linear programming cost function described in Section 3.4.2, ensuring effective equalization across varying channel conditions and modulation schemes.  
\begin{table}
\begin{tabular}{l r r r} \hline \hline
**Parameter** & **4-QAM** & **16-QAM** & **64-QAM** \\ \hline Equalizer length (\(N\)) & 33 & 33 & 33 \\ Equalization delay (\(d_{eq}\)) & 6 & 6 & 6 \\ BLE-GA Block length (\(K\)) & 400 & 400 & 1000 \\ CMA Adaptation step (\(\mu\)) & \(10^{-3}\) & \(10^{-3}\) & \(10^{-4}\) \\ Adjustment period (\(T_{ad}\)) in symbols & \(1\times 10^{7}\) & \(1\times 10^{7}\) & \(1\times 10^{7}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameter setting of the equalizer and CMA.