# Uni-Cycle Genetic Algorithm as an Adaptation Engine for Wireless Channel Equalizers
## 3 Uni-Cycle Genetic Algorithm
The equalizer starts to adjust the equalizer taps immediately after receiving the first sample from the channel. It is the typical approach for the initialization of adaptive channel equalizers. * Fitness evaluation  
In every \(n\)th iteration (signaling interval), the UCGA population, \(\Omega(n)\), consists of \(P\) individuals. As mentioned in Section 2, the optimization criterion for the proposed scheme  
Figure 2: Uni-Cycle Genetic Algorithm framework. The GA operators are the base for the transition from one iteration to the next. assumes minimization of the mean-square value of the error signal, given by Equation (2) (the MSE criterion). Thus, to evaluate the fitness of a given individual (represented by chromosome \(\overset{\wedge}{\mathbf{c}}\)) in iteration \(n\), we use the following cost function:  
\[J_{\mathbf{c}}(n)=\left|\rho(n)-\sum_{l=1}^{L}\hat{c}_{l}(n)\cdot r(n+l-\lfloor L /2\rfloor)\right|^{2}, \tag{4}\]  
where \(\hat{c}_{l}(n)=\hat{c}_{l}^{\text{R}}+j\hat{c}_{l}^{\text{J}}\). Let us denote the chromosome exhibiting the best fitness in iteration \(n\) as  
\[\overset{\wedge}{\mathbf{c}}(n)=\arg\min_{\mathbf{c}}J_{\mathbf{c}}(n) \tag{5}\]  
In every \(n\)th iteration, the UCGA searches for the minimum estimation error  
\[J^{\min}(n)=\min_{\mathbf{a}\in\Omega(n)}\left(J_{\mathbf{a}}(n)\right) \tag{6}\]  
across the entire population \(\Omega(n)\) to obtain a reference fitness value for comparisons between individual chromosomes; starting from the 2 nd iteration, the worst chromosomes are replaced with newborns, being the result of genetic operations like selection, crossover, and mutation. * Selection  
In the current work, selection routine utilizes a method called roulette wheel, i.e., the probability that a given individual becomes a parent is proportional to its fitness [14]. Accordingly, the selection probability in the \(n\)th iteration for an individual with chromosome \(\overset{\wedge}{\mathbf{c}}\) can be defined as  
\[S_{\mathbf{c}}(n)=\frac{J_{\mathbf{c}}(n)}{\sum_{\mathbf{a}\in\Omega(n)}J_{ \mathbf{a}}(n)} \tag{7}\]  
To avoid the loss of the best genetic material, the elite strategy [15] is employed. It means that some high-ranked individuals (in terms of the fitness value) co-exist with the offspring generation. The number of such elite individuals is called an elite count. * Crossover and Mutation  
The intermediate crossover is employed for the UCGA model to merge genes from two parents' chromosomes in the current iteration and produce a new child. The mutation operation consists of adding a random complex zero-mean Gaussian number \(A\) to the chromosome value. The standard deviation of \(A\) has been set to an appropriate value, specified in Table 1, after some preliminary experiments. Thanks to the mutation, which brings some random changes to the chromosomes, the UCGA is unlikely to stick at one of the local optima. Neither elite individuals nor crossover children undergo mutation procedure. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Population Size & 200 \\ Number of generations & 1 \\ Elite count & 72 \\ Selection method & Roulette wheel \\ Crossover type & Intermediate \\ Crossover probability & 0.85 \\ Mutation method & Gaussian \\ Std. deviation of mutation probability & 0.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: GA settings.