# New Approach of Blind Adaptive Equalizer Based on Genetic Algorithms
## 5 Results and Discussion  
The results of the simulations concerning channel A, \(h_{a}(k)\), for the 4-QAM, 16-QAM, and 64-QAM systems are presented, respectively, in Figures 7-9. Figure 7 shows a better bit error rate for the 4-QAM system when comparing BLE-GA to the CMA method. In the cases of 16-QAM and 64-QAM systems (Figures 8 and 9), BLE-GA equalized channel A within the period of \(T_{ad}=K\) symbols (16-QAM used \(K=400\) symbols and 64-QAM used \(k=1000\) symbols) in contrast to CMA, which was unable to achieve convergence to a global minimum for the same period. Nevertheless, BLE-GA had a similar performance to CMA when the parameters were adjusted to a much greater period of symbols (4-QAM used \(T_{ad}=1\times 10^{3}\) symbols, 16-QAM used \(T_{ad}=1\times 10^{4}\) symbols and 64-QAM used \(T_{ad}=7\times 10^{4}\) symbols).  
The simulations employing the second scenario with channel B, \(h_{b}(  
Figure 8: BER curve of 16-QAM system for channel A. \(E_{b}/N_{0}\) is expressed in dB.  
Figure 10: BER curve of 4-QAM system for channel B. \(E_{b}/N_{0}\) is expressed in dB.  
Figure 9: BER curve of 64-QAM system for channel A. \(E_{b}/N_{0}\) is expressed in dB.  
The constellation analysis of the equalizer output indicates a substantial performance difference between the BLE-GA equalizer and the CMA equalizer when a signal-to-noise ratio of 30 dB and \(T_{ad}=K\) were used. While the BLE-GA equalizer was able to provide visually recognizable constellations (Figures 13a, 14a, 15a, and 16a), the constellations of the CMA equalizer were completely scattered and unrecognizable (Figures 13b, 14b, 15b, and 16b).  
In all scenarios, both BLE-GA and CMA converged to global minima, i.e., there was no failure in convergence. However, for \(M>4\), CMA required a greater number of iterations to achieve global convergence. For example, in Figure 12 (64-QAM channel B), CMA converged after 100,000 symbols, while BLE-GA required only \(1\times 10^{3}\) symbols to achieve similar performance.  
Another aspect to be taken into account is that BLE-GA is a block channel equalization algorithm. Although BLE-GA is more complex than CMA, it updates the equalizer parameters only after \(K\) symbols, while CMA updates them for each symbol. Even if BLE-GA cannot equalize the channel in a \(K\) symbol block, it still has a much longer time frame than the CMA to converge to the global minimum.  
It is important to note that both methods do not require a training sequence, and they therefore have essentially the same limitation. BLE-GA tries to adjust the constellation of signals within a square with dimensions limited by \(\pm M\tau_{1}\) and \(\pm M\tau_{2}\), while CMA attempts to adjust the constellation to a radius \(\gamma\)[11]. In other words, the challenges are of the same magnitude for both BLE-GA and CMA; therefore, the comparison between these methods is justified. However, BLE-GA does not present the limitations of the CMA of having a cost  
Figure 11: BER curve of 16-QAM system for channel B. \(E_{b}/N_{0}\) is expressed in dB.  
Figure 12: BER curve of 64-QAM system for channel B. \(E_{b}/N_{0}\) is expressed in dB.