# Optimized Accelerated Over-Relaxation Method for Robust Signal Detection: A Metaheuristic Approach
## 3 Proposed Method for Signal Detection
### Proposed OAOR Based Method
To derive Equation (6), we start with the MMSE equation from Equation (4):  
\[(\mathbf{G}^{H}\mathbf{G}+\beta\mathbf{I})\mathbf{\hat{x}}=\mathbf{G}^{H} \mathbf{r} \tag{7}\]  
Substituting the matrix decomposition, we obtain  
\[(\mathbf{D}-\mathbf{L}-\mathbf{U}+\beta\mathbf{I})\mathbf{\hat{x}}=\mathbf{G} ^{H}\mathbf{r} \tag{8}\]  
Rearranging the terms,  
\[\mathbf{D}\mathbf{\hat{x}}=(\mathbf{L}+\mathbf{U}-\beta\mathbf{I})\mathbf{ \hat{x}}+\mathbf{G}^{H}\mathbf{r} \tag{9}\]  
Multiplying both sides by \(\mathbf{D}^{-1}\),  
\[\mathbf{\hat{x}}=\mathbf{D}^{-1}(\mathbf{L}+\mathbf{U}-\beta\mathbf{I}) \mathbf{\hat{x}}+\mathbf{D}^{-1}\mathbf{G}^{H}\mathbf{r} \tag{10}\]  
Now, we introduce the iteration index \(k\) by setting up the initialization \(\mathbf{x}^{(0)}=\mathbf{D}^{-1}\mathbf{G}^{H}\mathbf{r}\) and the first relaxation parameter \(\omega^{*}\)  
\[\mathbf{x}^{(k+1)}=\mathbf{x}^{(k)}+\omega^{*}[\mathbf{D}^{-1}\mathbf{G}^{H} \mathbf{r}-\mathbf{x}^{(k)}+\mathbf{D}^{-1}(\mathbf{L}+\mathbf{U}-\beta \mathbf{I})\mathbf{x}^{(k)}] \tag{11}\]  
To further improve the convergence properties, we introduce the second relaxation parameter \(\gamma^{*}\):  
\[\mathbf{x}^{(k+1)}=\mathbf{x}^{(k)}+\omega^{*}[\mathbf{D}^{-1}\mathbf{G}^{H} \mathbf{r}-\mathbf{x}^{(k)}+\gamma^{*}\mathbf{D}^{-1}(\mathbf{L}+\mathbf{U}) \mathbf{x}^{(k)}-\mathbf{D}^{-1}\beta\mathbf{I}\mathbf{x}^{(k)}] \tag{12}\]  
Simplifying this expression, we obtain  
\[\mathbf{x}^{(k+1)}=\mathbf{x}^{(k)}+\omega^{*}[\mathbf{D}^{-1}\mathbf{G}^{H} \mathbf{r}-\mathbf{D}^{-1}(\mathbf{D}-\gamma^{*}(\mathbf{L}+\mathbf{U})+\beta \mathbf{I})\mathbf{x}^{(k)}] \tag{13}\]Finally, substituting back \(\mathbf{G}^{H}\mathbf{G}=\mathbf{D}-\mathbf{L}-\mathbf{U}\), we arrive at the OAOR iteration formula:  
\[\mathbf{G}^{H}\mathbf{G}=\mathbf{D}-\mathbf{L}-\mathbf{U} \tag{5}\]0  
For the convergence of this method, the error at iteration \(k\) can be defined as \(\mathbf{e}^{(k)}=\mathbf{\hat{x}}-\mathbf{x}^{(k)}\). Thus, the error propagation expression can be written as  
\[\mathbf{G}^{H}\mathbf{G}=\mathbf{D}-\mathbf{L}-\mathbf{U} \tag{5}\]1  
Let \(\mathbf{M}=\mathbf{I}-\omega^{*}\mathbf{D}^{-1}\mathbf{G}^{H}\mathbf{G}+ \omega^{*}\gamma^{*}\mathbf{D}^{-1}(\mathbf{D}-\mathbf{L}-\mathbf{U})\). Then, we can write:  
\[\mathbf{G}^{H}\mathbf{G}=\mathbf{D}-\mathbf{L}-\mathbf{U} \tag{5}\]2  
For the method to converge, we require that the spectral radius of \(\mathbf{M}\) be less than 1:  
\[\mathbf{G}^{H}\mathbf{G}=\mathbf{D}-\mathbf{L}-\mathbf{U} \tag{5}\]3  
where \(\rho(\mathbf{M})\) denotes the spectral radius of \(\mathbf{M}\). The choice of optimal relaxation parameters \(\omega^{*}\) and \(\gamma^{*}\) is crucial for the performance of the OAOR method. To determine these parameters, we formulate an optimization problem:  
\[ \begin{array}{ll}\text{minimize}&f(\omega^{*},\gamma^{*})=\mathbb{E}[\| \mathbf{x}-\mathbf{\hat{x}}(\omega^{*},\gamma^{*})\|^{2}]\\ \text{subject to}&0<\omega^{*}<2,\quad 0<\gamma^{*}<1\\ &\rho(\mathbf{I}-\omega^{*}\mathbf{D}^{-1}\mathbf{G}^{H}\mathbf{G}+\omega^{*} \gamma^{*}\mathbf{D}^{-1}(\mathbf{D}-\mathbf{L}-\mathbf{U}))<1\end{array}  \tag{18}\]  
This optimization problem aims to minimize the mean square error while ensuring the convergence of the method. To solve this optimization problem and find the optimal \(\omega^{*}\) and \(\gamma^{*}\), we employ a novel PSO algorithm which will be discussed in detail in the subsequent section.