# New Approach of Blind Adaptive Equalizer Based on Genetic Algorithms
## 3 Methods
### Blind Linear Equalizer Based on Genetic Algorithms
#### 3.4.2 Cost Function  
According to the Equation (15), the gains of the \(K\)-block equalizer should be adjusted by using the following linear programming strategy:  
\[\min\quad\left\{\begin{array}{c}\max\big{|}\tilde{a}^{I}(k-d_{eq})\big{|}\\ +\max\big{|}\tilde{a}^{Q}(k-d_{eq})\big{|}\\ \vdots\\ \max\big{|}\tilde{a}^{I}(k-K+1-d_{eq})\big{|}\\ +\max\big{|}\tilde{a}^{Q}(k-K+1-d_{eq})\big{|},\end{array}\right. \tag{21}\]  
subject to  
\[w_{d_{eq}}^{I}(k)=1. \tag{22}\]  
This constraint is based on one of the initial conditions of CMA and ensures the equalization of the channel by avoiding the null output, which is the cost function trivial global minimum when \(w^{I}=w^{Q}=0\). Due to the linear nature of this constraint, neither the convexity nor the global convergence of the cost function are affected.  
Substituting the values of \(\tilde{a}^{I}(k-d_{eq})\) and \(\tilde{a}^{Q}(k-d_{eq})\) for Equation (4), and using the strategy of introducing two auxiliary variables \(\tau_{1}\) and \(\tau_{2}\), presented earlier in [8; 9; 10; 11], the  
Figure 6: Structure of the Blind Linear Adaptive Equalizer based on GA. The arrows indicate the transmitted signal (\(a(k)\)), the output of the channel (\(x(k)\)), the noise (\(r(k)\)), the equalizer input \(u(k)\), the parameters of the equalizer (\(w(k)\)), and the delayed equalizer output (\(a(k-d_{eq})\)).  
linear programming strategy presented in Equations (16), (21), and (22) can be expressed by minimizing the following equation:  
\[\text{min}\ \ M\tau_{1}+M\tau_{2} \tag{23}\]  
subject to the following constraints:  
\[\left\{\begin{array}{c}\sum_{l=0}^{N-1}w_{l}^{l}(k)u^{l}(k-l)\\ -w_{l}^{Q}(k)u^{Q}(k-l)\leq M\tau_{1}\\ \vdots\\ \sum_{l=0}^{N-1}w_{l}^{l}(k)u^{l}(k-l-K+1)\\ -w_{l}^{Q}(k)u^{Q}(k-l-K+1)\leq M\tau_{1}\end{array}\right. \tag{24}\]  
\[\left\{\begin{array}{c}\sum_{l=0}^{N-1}w_{l}^{l}(k)u^{l}(k-l)\\ -w_{l}^{Q}(k)u^{Q}(k-l)\geq-M\tau_{1}\\ \vdots\\ \sum_{l=0}^{N-1}w_{l}^{l}(k)u^{l}(k-l-K+1)\\ -w_{l}^{Q}(k)u^{Q}(k-l-K+1)\geq-M\tau_{1}\end{array}\right. \tag{25}\]  
\[\left\{\begin{array}{c}\sum_{l=0}^{N-1}w_{l}^{Q}(k)u^{l}(k-l)\\ -w_{l}^{l}(k)u^{Q}(k-l)\leq M\tau_{2}\\ \vdots\\ \sum_{l=0}^{N-1}w_{l}^{Q}(k)u^{l}(k-l-K+1)\\ -w_{l}^{l}(k)u^{Q}(k-l-K+1)\leq M\tau_{2}\end{array}\right. \tag{26}\]  
\[\left\{\begin{array}{c}\sum_{l=0}^{N-1}w_{l}^{Q}(k)u^{l}(k-l)\\ -w_{l}^{l}(k)u^{Q}(k-l)\geq-M\tau_{2}\\ \vdots\\ \sum_{l=0}^{N-1}w_{l}^{Q}(k)u^{l}(k-l-K+1)\\ -w_{l}^{l}(k)u^{Q}(k-l-K+1)\geq-M\tau_{2}\end{array}\right. \tag{27}\]  
and  
\[w_{d_{eq}}^{l}(k)=1. \tag{28}\]  
where \(K\) represents the number of times the output signal must be stored to construct the optimization curve and \(d_{eq}\) is an integer value that represents the equalization delay, so that \(0\leq d_{eq}\leq N-1\). Under the optimum condition, \(M\tau_{1}=max|\overline{a}^{l}(k-d_{eq})|\) and \(M\tau_{2}=max|\overline{a}^{Q}(k-d_{eq})|\). It is important to note that the signal \(u(k)\) is stochastic (see Equation (7)), converting the linear programming problem to a stochastic linear programming problem.  
The boundary conditions for Equations (24)-(27) are defined as follows to ensure the validity of the linear constraints under stochastic conditions. For the auxiliary variables \(\tau_{1}\) and \(\tau_{2}\), the boundaries are set such that \(0\leq M\tau_{1}\), \(M\tau_{2}\leq\max|a(k)|\), where \(a(k)\) represents the transmitted signal bounded by the modulation alphabet. The equalizer weights \(\mathbf{w}_{l}\) and \(\mathbf{w}_{Q}\) are constrained to values ensuring that the output signal adheres to the range \([-M,M]\) for each QAM level. Additionally, the condition \(w_{d_{eq}}^{l}(k)=1\) prevents trivial solutions by ensuring that the equalizer output does not collapse to zero. These conditionsare critical to preserving the convexity of the cost function and ensuring robust convergence during optimization.