# Optimized Accelerated Over-Relaxation Method for Robust Signal Detection: A Metaheuristic Approach
## 3 Proposed Method for Signal Detection
### Derivation of \(\omega^{*}\) and \(\gamma^{*}\) Using Novel PSO  
The optimal \(\omega^{*}\) and \(\gamma^{*}\) can be derived for the OAOR method using our proposed novel PSO approach. The proposed method of derivation is given in Algorithm 3. For the derivation, see Appendix A. ```
0: Population size \(N\), maximum iterations \(t_{\text{max}}\)
0: Optimal \(\omega^{*}\) and \(\gamma^{*}\)
1: Initialize PSO parameters:
2:\(\mathcal{g}_{\text{NO}}\leftarrow\)\(N\) // Number of particles
3:\(w_{\text{max}}\leftarrow\)\(1.2-\left\lfloor\frac{\mathcal{g}_{\text{NO}}-50}{150}\right\rfloor\)
4:for\(i=1\) to \(N\)do
5: Initialize \(\mathbf{x}_{i}=(\omega_{i},\gamma_{i})\) randomly within bounds
6: Initialize \(\mathbf{v}_{i}\) randomly
7: Evaluate fitness \(f(\mathbf{x}_{i})\)
8: Set \(\mathbf{p}_{\text{best},i}\leftarrow\mathbf{x}_{i}\)
9:endfor
10: Set \(\mathbf{g}_{\text{best}}\leftarrow\arg\min_{i}f(\mathbf{p}_{\text{best},i})\)
11:for\(t=1\) to \(t_{\text{max}}\)do
12:\(w(t)\gets w_{\text{max}}\cdot e^{\frac{t^{2}}{2(0.5\text{max})^{2}}}\)
13:\(c_{1}(t)\gets 2+0.5\cos(\frac{\pi t}{2})\)
14:\(c_{2}(t)\gets 2-0.5\cos(\frac{\pi t}{2})\)
15:for each particle \(i=1\) to \(N\)do
16: Update velocity and position:
17:\(\mathbf{v}_{i}(t+1)\gets w(t)\mathbf{v}_{i}(t)+r_{1}c_{1}(t)(\mathbf{p}_ {\text{best},i}-\mathbf{x}_{i}(t))+r_{2}c_{2}(t)(\mathbf{g}_{\text{best}}- \mathbf{x}_{i}(t))\)
18:\(\mathbf{x}_{i}(t+1)\leftarrow\mathbf{x}_{i}(t)+\mathbf{v}_{i}(t+1)\)
19: Evaluate fitness \(f(\mathbf{x}_{i}(t+1))\)
20:if\(f(\mathbf{x}_{i}(t+1))<f(\mathbf{p}_{\text{best},i})\)then
21:\(\mathbf{p}_{\text{best},i}\leftarrow\mathbf{x}_{i}(t+1)\)
22:if\(f(\mathbf{p}_{\text{best},i})<f(\mathbf{g}_{\text{best}})\)then
23:\(\mathbf{g}_{\text{best}}\leftarrow\mathbf{p}_{\text{best},i}\)
24:endif
25:endif
26:endfor
27:if convergence criterion is met then
28:break
29:endif
30:endfor
31:\((\omega^{*},\gamma^{*})\leftarrow\)\(\mathcal{B}_{\text{best}}\)
32:return\((\omega^{*},\gamma^{*})\)
```  
**Algorithm 3** Novel PSO algorithm for \(\omega^{*}\) and \(\gamma^{*}\). Figure 3: Evaluation of novel PSO results using different benchmark functions. By executing the proposed novel PSO algorithm, we obtain the optimal values \(\omega^{*}\) and \(\gamma^{*}\) that minimize the mean square error and help in achieving near-MMSE results by guaranteeing the convergence of the OAOR technique. When the detection process starts, the novel PSO runs and explores the optimal solution by minimizing the error of the objective function. After finding the optimal values based on that reduced error, the OAOR uses these values to update its estimate of the received signal. The appropriate values are crucial for attaining the highest performance in signal detection for large antenna systems. This comprehensive discussion highlights the importance of choosing the optimal parameters to ensure the effectiveness of the OAOR technique in practical situations. After the calculation of \((\omega^{*},\gamma^{*})\) values, we can finally rewrite this whole optimized AOR process as Algorithm 4.