# Optimized Accelerated Over-Relaxation Method for Robust Signal Detection: A Metaheuristic Approach
## 3 Proposed Method for Signal Detection
### Proposed Novel PSO
#### 3.2.4 Convergence Behavior of Proposed Novel PSO  
The convergence of the novel PSO algorithm depends on the behavior of the inertia weight \(w(t)\) and acceleration coefficients \(c_{1}(t)\) and \(c_{2}(t)\):  
1. The inertia weight \(w(t)\) decreases exponentially over time, allowing for initial exploration and later exploitation.  
2. The acceleration coefficients \(c_{1}(t)\) and \(c_{2}(t)\) oscillate between 1.5 and 2.5, balancing the influence of the personal and global best solutions.  
As \(t\to t_{\max}\),  
\[\lim_{t\to t_{\max}}w(t) =w_{\max}\cdot e^{-\frac{t_{\max}^{2}}{2(0.5\max)^{2}}}\approx 0 \tag{24}\] \[\lim_{t\to t_{\max}}c_{1}(t) =2+0.5\cos(\frac{\pi t_{\max}}{2})=1.5\] (25) \[\lim_{t\to t_{\max}}c_{2}(t) =2-0.5\cos(\frac{\pi t_{\max}}{2})=2.5 \tag{26}\]This behavior ensures that the algorithm converges to a solution as the number of iterations approaches \(t_{\text{max}}\). We examine the convergence of our novel PSO over the following:  
(a) Sphere Benchmark Function  
\[f(\mathbf{x})=\sum_{i=1}^{n}x_{i}^{2} \tag{27}\]  
(b) Michalewicz Benchmark Function  
\[f(\mathbf{x})=-\sum_{i=1}^{n}\sin(x_{i})\left(\sin\left(\frac{ix_{i}^{2}}{\pi} \right)\right)^{2m} \tag{28}\]  
Figure 3 represents the curve of the novel PSO for both benchmark functions [52] in 2D space. The sphere function is a very simple uni-modal function to test the optimization problems because of its simplicity. In the 3D shape, it looks like a perfect bowl or parabola. For the sphere function, the goal is to find the global minimum, which is at the bottom center of the bowl (the darkest point) and it can be seen in Figure 3a that our proposed algorithm achieves the optimal solution for the sphere function perfectly, which is (0,0) in the x-y plane. The Michalewicz function is a multimodel complex function and due to its irregular shape, it poses a great challenge for optimization algorithms. In Figure 3b, we are looking for the minimum value which is the nature of the Michalewicz function and it should correspond to the deepest \"valley\" or darkest area on the plot. This shows that the novel PSO has converged to a very good solution, as the red dot is located in one of the darkest regions of the plot and the algorithm is effectively navigating the complex, multimodal landscape of the Michalewicz function. It shows that our proposed algorithm has achieved convergence to find the best optimal solution in both simple and complex test functions. We carry out this test on a system of Intel Core i5-6500U CPU with 12 GB of RAM. The set parameters with the obtained results are given in Table 2. Meanwhile, for the specific use case of novel PSO in signal detection, we put early stopping criteria with a reduced iteration count \(t_{\text{max}}\) during simulation to avoid the high computational complexity, which is evident from our results and discussion section.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Aspect** & **Michalewicz Function** & **Sphere Function** \\ \hline
**Parameters** & & \\ Dimension & 2 & 2 \\ Number of Particles & 150 & 150 \\ Maximum Iterations & 1000 & 1000 \\ Search Space & [0, \(\pi\)] for each dim & [\(-5.12\), \(5.12\)] for each dim \\ \hline
**Expected Results** & & \\ Global Optimum (\(f_{min}\)) & \(-1.80130\) & 0.00000 \\ Optimal Solution (\(x_{1}\), \(x_{2}\)) & (2.20291, \(1.57080\)) & (0.00000, 0.00000) \\ \hline
**Actual Results** & & \\ Global Optimum (\(f_{min}\)) & \(-1.80129\) & 0.00014 \\ Optimal Solution (\(x_{1}\), \(x_{2}\)) & (2.20275, \(1.57052\)) & (0.00029, 0.00043) \\ Iterations to Converge & 76 & 51 \\ Time Taken (ms) & 143 & 118 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Novel PSO results for benchmark functions.