# Optimized Accelerated Over-Relaxation Method for Robust Signal Detection: A Metaheuristic Approach
## 3 Proposed Method for Signal Detection
### Proposed Novel PSO
#### 3.2.3 Modified Acceleration Coefficients  
In the velocity update Equation (19), \(p_{best}-x(t)\) represents the direction towards the best position so far. Thus, \(c_{1}\) governs the cognitive capacity of the particle, namely, its ability to locate its optimal location. Similarly, \(g_{best}-x(t)\) shows the optimal location of the population, whereas \(c_{2}\) determines the particle's cognitive capacity to locate this place. The fundamental aspect is the updating of both the velocity and location. Velocity updating refers to the process in which the algorithm follows a certain rule to randomly update the location according to a specified rule. To enhance the capabilities of PSO, the acceleration coefficients might be made variable, especially when we have to use signal detection to achieve minimum error rate and low convergence. So a better trade-off between exploration and exploitation needs to be achieved by the algorithm. To encourage this approach more aggressively in searching a wider range of potential \(\omega*\) and \(\gamma*\) values at the early stages, the modified acceleration coefficients are given below:  
\[ \begin{split} c_{1}&=2+0.5\cos(\frac{\pi t}{2})\\ c_{2}&=2-0.5\cos(\frac{\pi t}{2})\end{split}  \tag{22}\]  
Here, \(t\) denotes the evaluation step of the current particle. This helps the algorithm quickly get closer to the best solution as the number of iterations progresses. The algorithm finds the optimal combination of \(\omega\) and \(\gamma\) for maximum performance with the aid of this dynamic balance between global and local search. So, the modified velocity update and position update formula are given by  
\[ \begin{split} v(t+1)&=\left(1.2-\left\lfloor\frac{ g_{NO.}-50}{150}\right\rfloor\right)\cdot e^{-\frac{t^{2}}{2(0.5t_{max})^{2}}} \cdot v(t)\\ &+r_{1}\bigg{(}2+0.5\cos(\frac{\pi t}{2})\bigg{)}(p_{best}-x(t)) \\ &+r_{2}\bigg{(}2-0.5\cos(\frac{\pi t}{2})\bigg{)}(g_{best}-x(t)) \\ x(t+1)&=x(t)+v(t+1)\end{split}  \tag{23}\]  
The complete set of parameter values for the different PSO algorithms is provided in Table 1 for comparison and to highlight the differences between them. The optimization accuracy and speed of the PSO algorithm are enhanced by optimizing the inertia weight and acceleration coefficients, as discussed in [44]. Additionally, we further improve the algorithm by employing cosine functions for the acceleration coefficients. The use of cosine functions for \(c_{1}\) and \(c_{2}\) potentially offers several advantages for optimizing OAOR parameters in massive MIMO systems. The cosine approach provides a more aggressive initial exploration phase due to a higher starting value for \(c_{1}\), followed by a steeper transition to exploitation. This dynamic allows the algorithm to quickly identify promising regions in the \(\omega\) and \(\gamma\) parameters space while ensuring thorough fine-tuning in the later stages. The pronounced oscillation between exploration and exploitation phases helps in escaping local optima and navigating the complex optimization landscape typical of massive MIMO systems. Additionally, the end-phase behavior of cosine, with a higher \(c_{2}\) value, leads to better convergence on the global optimum. Figures 1 and 2 illustrate the curves of the proposed algorithm for both the inertia weight coefficient and the acceleration coefficients. The detailed algorithm is outlined in Algorithm 2.