## 4 PSO Techniques for Adaptive Equalization
### Performance Analysis
The performance of least mean squares (LMS) degrades compared to hybrid PSO due to its limited ability to handle nonlinear and non-convex optimization problems. Hybrid PSO incorporates global search capabilities and adaptive techniques, providing better convergence and optimization results. In the conducted experiment, the performance of three different optimization techniques LMS, the PSO VCF, and HPSO was evaluated in the context of a digital communication channel. The experiment involved varying SNR levels to simulate different channel conditions. The SNR represents the ratio of signal power to noise power and serves as a key factor in determining the quality of communication in noisy environments. For each SNR level, the BER was measured using the three optimization techniques: LMS, the PSO VCF, and HPSO. Table 5 shows the comparison of LMS, the PSO variable constriction factor (VCF), and HPSO. A lower BER signifies better channel performance, indicating fewer errors in received bits. The comparison of the BER values among the three techniques at different SNR levels provides insights into their respective abilities to mitigate errors and enhance communication quality. The experiment revolved around evaluating four distinct PSO variants using the sphere function. The sphere function, a common optimization benchmark, calculates the sum of squared differences between the candidate solution and the optimal solution. The aim was to gauge the performance of these PSO variants in terms of mean function values across different dimensions, i.e., 30 and 60. Lower mean function values signify more-proficient optimization, thereby enabling a comparative analysis of the PSO techniques' effectiveness in searching for optimal solutions. The experimental results demonstrated the superior performance of hybrid PSO compared to other variants of PSO in various optimization tasks. In a comparative study, different PSO variants, including standard PSO, adaptive PSO, and HPSO, were evaluated for their convergence speed and solution quality. The results revealed that HPSO outperformed the other variants in terms of both convergence speed and solution quality. HPSO demonstrated faster convergence, \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of Iterations** & **MSE (dB) \(N\) = 10** & **MSE (dB) \(N\) = 20** & **MSE (dB) \(N\) = 40** & **MSE (dB) \(N\) = 60** \\ \hline
0 & 20 & 20 & 20 & 20 \\ \hline
50 & \(-\)12 & \(-\)15 & \(-\)28 & \(-\)29 \\ \hline
100 & \(-\)20 & \(-\)23 & \(-\)29 & \(-\)30 \\ \hline
200 & \(-\)21 & \(-\)25 & \(-\)29.5 & \(-\)30 \\ \hline
300 & \(-\)19.5 & \(-\)24.5 & \(-\)29 & \(-\)30.5 \\ \hline
400 & \(-\)20.5 & \(-\)25 & \(-\)29.5 & \(-\)31 \\ \hline
500 & \(-\)22 & \(-\)26 & \(-\)30 & \(-\)31 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Convergence of hybrid PSO with different values of \(N\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**SNR** & **LMS** & **PSO VCF** & **HPSO** \\ \hline
0 & 0.8922 & 0.8929 & 0.8929 \\ \hline
2 & 0.8017 & 0.8019 & 0.8019 \\ \hline
4 & 0.8402 & 0.8402 & 0.8402 \\ \hline
6 & 0.8051 & 0.805 & 0.805131 \\ \hline
8 & 0.74283 & 0.7428 & 0.7427 \\ \hline
10 & 0.65015 & 0.6505 & 0.6505 \\ \hline
12 & 0.5526 & 0.5527 & 0.5527 \\ \hline
14 & 0.4026 & 0.4026 & 0.4026 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: BER performance of LMS and HPSO. reaching the optimal or near-optimal solution more quickly compared to standard PSO and adaptive PSO. This was attributed to HPSO's ability to balance exploration and exploitation through the combination of particle interactions and adaptive parameters. The solution quality achieved by HPSO was consistently superior to other variants. The algorithm's hybrid nature, incorporating elements of both particle swarm optimization and local search techniques, allowed for better exploration of the search space, leading to improved solutions. HPSO effectively balanced global exploration to escape local optima with local exploitation to refine solutions, resulting in enhanced overall performance.