# New Approach of Blind Adaptive Equalizer Based on Genetic Algorithms
## 3 Methods
### Adaptive Blind Equalization
#### 3.3.3 Adaptive Blind Equalization Using Linear Programming  
The works [8; 9; 10] proposed a globally convergent convex cost function based on linear programming. This function does not attempt to directly identify complex channel gain, and instead focuses on eliminating ISI. Once the ISI is removed, the equalizer output becomes a scaled version of the channel input, and it is possible to estimate the scalar gain by simpler methods, such as comparing the input power of the channel with the equalizer  
Figure 5: Basic scheme of a blind linear adaptive equalizer. 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 (\(a(k-d_{eq})\)). output. Thus, the convex cost function for the Blind Linear Equalizer proposed by the above literature can be expressed by  
\[J_{PL}(\mathbf{w})=J_{PL}\Big{(}\mathbf{w}^{I},\mathbf{w}^{Q}\Big{)}\equiv \max\Big{|}\tilde{a}^{I}(k)\Big{|}+\max\Big{|}\tilde{a}^{Q}(k)\Big{|}. \tag{15}\]  
In this model, it is assumed that the transmitted signal \(a(k)\) is modulated in the scheme \(M\)-QAM, with \(M\) representing the size of the QAM constellation, and its real and imaginary components are independent and identical, with  
\[M\equiv\max\Big{|}a^{I}(k)\Big{|}=\max\Big{|}a^{Q}(k)\Big{|}\text{for any }k, \tag{16}\]  
or they can be transformed to such by rotational operations. The above model does not take into account the effects of noise, which is impractical in many real-life systems. The work [11] extends the above bibliographies [8; 9; 10], proving that when considering the effects of AWGN (Additive White Gaussian Noise) in the modeling, the function obtained from Equation (15) remains convex. This proof can be performed by expanding the expression of \(\tilde{a}(k)\) from Equations (7)-(9) and then isolating the terms of \(\tilde{a}(k)\) according to Equation (4), obtaining the expression for \(\tilde{a}^{I}(k)\), given by  
\[\tilde{a}^{I}(k) = \sum_{l=0}^{N-1}\sum_{i=0}^{L-1}\Big{(}\rho_{i}^{I}a^{I}(k-\tau_{ i}-l)\] \[-\rho_{i}^{Q}a^{Q}(k-\tau_{i}-l)\Big{)}w_{l}^{I}(k)+\] \[\sum_{l=0}^{N-1}\sum_{i=0}^{L-1}\Big{(}-\rho_{i}^{I}a^{Q}(k-\tau_ {i}-l)\] \[-\rho_{i}^{Q}a^{I}(k-\tau_{i}-l)\Big{)}w_{l}^{Q}(k)+\] \[\sum_{l=0}^{N-1}r^{I}(k-l)w_{l}^{I}(k)+\sum_{l=0}^{N-1}-r^{Q}(k-l )w_{l}^{Q}(k).\]  
The term \(\tilde{a}^{Q}(k)\) can be obtained in the same way. The cost function proposed in [8] seeks to minimize the sum of the maximums of the terms of \(\tilde{a}(k)\) (see Equation (15)), which are formed by a sum of linear expressions. Thus, by expanding the terms in evidence of Equation (17), we obtain  
\[\max\Big{|}\tilde{a}^{I}(k)\Big{|}=\] \[\max\Bigg{|}\sum_{l=0}^{N-1}\sum_{i=1}^{L-1}\rho_{i}^{I}a^{I}(k- \tau_{i}-l)w_{l}^{I}(k)\Bigg{|}\] \[J_{PL}(\mathbf{w})=J_{PL}\Big{(}\mathbf{w}^{I},\mathbf{w}^{Q}\Big{)}\equiv \max\Big{|}\tilde{a}^{I}(k)\Big{|}+\max\Big{|}\tilde{a}^{Q}(k)\Big{|}. \tag{15}\]0 \[J_{PL}(\mathbf{w})=J_{PL}\Big{(}\mathbf{w}^{I},\mathbf{w}^{Q}\Big{)}\equiv \max\Big{|}\tilde{a}^{I}(k)\Big{|}+\max\Big{|}\tilde{a}^{Q}(k)\Big{|}. \tag{15}\]1 \[J_{PL}(\mathbf{w})=J_{PL}\Big{(}\mathbf{w}^{I},\mathbf{w}^{Q}\Big{)}\equiv \max\Big{|}\tilde{a}^{I}(k)\Big{|}+\max\Big{|}\tilde{a}^{Q}(k)\Big{|}. \tag{15}\]2 \[J_{PL}(\mathbf{w})=J_{PL}\Big{(}\mathbf{w}^{I},\mathbf{w}^{Q}\Big{)}\equiv \max\Big{|}\tilde{a}^{I}(k)\Big{|}+\max\Big{|}\tilde{a}^{Q}(k)\Big{|}. \tag{15}\]3Once again, the terms of \(\max\|\tilde{a}^{Q}(k)\big{|}\) can be obtained in the same way. It is important to note that Equation (18) is only valid considering that the finite values of the QAM alphabet are limited between \(-M\) and \(+M\), as defined in Equivalence (16).