# Optimized Accelerated Over-Relaxation Method for Robust Signal Detection: A Metaheuristic Approach
## 3 Proposed Method for Signal Detection
### Convergence Analysis of OAOR Method  
In order to thoroughly examine the convergence characteristics of the proposed method, we start with an in-depth convergence proof, which is then formalized in the form of a theorem. Proof of OAOR Convergence  
To analyze the convergence of the proposed method, we define the error at iteration \(k\) from Equation (6) of the defined system model as  
\[\mathbf{e}^{(k)}=\mathbf{\hat{x}}-\mathbf{x}^{(k)} \tag{29}\]  
where \(\mathbf{\hat{x}}\) is the true MMSE solution. We can express the error propagation as  
\[\mathbf{e}^{(k+1)} =\mathbf{\hat{x}}-\mathbf{x}^{(k+1)} \tag{30}\] \[=\mathbf{\hat{x}}-[\mathbf{x}^{(k)}+\omega^{*}(\mathbf{D}^{-1} \mathbf{G}^{H}\mathbf{r}-\mathbf{D}^{-1}\mathbf{G}^{H}\mathbf{G}\mathbf{x}^{(k )}+\gamma^{*}(\mathbf{D}-\mathbf{L}-\mathbf{U})\mathbf{x}^{(k)})] \tag{31}\]  
Substituting \(\mathbf{G}^{H}\mathbf{r}=(\mathbf{G}^{H}\mathbf{G}+\beta\mathbf{I})\mathbf{ \hat{x}}\) from the MMSE equation,  
\[\mathbf{e}^{(k+1)} =\mathbf{e}^{(k)}-\omega^{*}[\mathbf{D}^{-1}(\mathbf{G}^{H} \mathbf{G}+\beta\mathbf{I})\mathbf{\hat{x}}-\mathbf{D}^{-1}\mathbf{G}^{H} \mathbf{G}\mathbf{x}^{(k)}+\gamma^{*}(\mathbf{D}-\mathbf{L}-\mathbf{U}) \mathbf{x}^{(k)}] \tag{32}\] \[=\mathbf{e}^{(k)}-\omega^{*}[\mathbf{D}^{-1}\mathbf{G}^{H} \mathbf{G}(\mathbf{\hat{x}}-\mathbf{x}^{(k)})+\beta\mathbf{D}^{-1}\mathbf{ \hat{x}}+\gamma^{*}(\mathbf{D}-\mathbf{L}-\mathbf{U})\mathbf{x}^{(k)}]\] (33) \[=\mathbf{e}^{(k)}-\omega^{*}[\mathbf{D}^{-1}\mathbf{G}^{H} \mathbf{G}\mathbf{e}^{(k)}+\beta\mathbf{D}^{-1}\mathbf{\hat{x}}+\gamma^{*}( \mathbf{D}-\mathbf{L}-\mathbf{U})(\mathbf{\hat{x}}-\mathbf{e}^{(k)})]\] (34) \[=[\mathbf{I}-\omega^{*}\mathbf{D}^{-1}\mathbf{G}^{H}\mathbf{G}+ \omega^{*}\gamma^{*}(\mathbf{D}-\mathbf{L}-\mathbf{U})]\mathbf{e}^{(k)}- \omega^{*}[\beta\mathbf{D}^{-1}+\gamma^{*}(\mathbf{D}-\mathbf{L}-\mathbf{U})] \mathbf{\hat{x}} \tag{35}\]  
Let \(\mathbf{M}=\mathbf{I}-\omega^{*}\mathbf{D}^{-1}\mathbf{G}^{H}\mathbf{G}+ \omega^{*}\gamma^{*}(\mathbf{D}-\mathbf{L}-\mathbf{U})\) and \(\mathbf{c}=-\omega^{*}[\beta\mathbf{D}^{-1}+\gamma^{*}(\mathbf{D}-\mathbf{L }-\mathbf{U})]\mathbf{\hat{x}}\). Then:  
\[\mathbf{e}^{(k+1)}=\mathbf{M}\mathbf{e}^{(k)}+\mathbf{c} \tag{36}\]  
This is a linear recurrence relation. Its solution is given by  
\[\mathbf{e}^{(k)}=\mathbf{M}^{k}\mathbf{e}^{(0)}+(\mathbf{I}-\mathbf{M})^{-1}( \mathbf{I}-\mathbf{M}^{k})\mathbf{c} \tag{37}\]  
For convergence, we require \(\lim_{k\to\infty}\mathbf{e}^{(k)}=\mathbf{0}\). This occurs if and only if the following hold:1. \(\lim_{k\to\infty}\mathbf{M}^{k}=\mathbf{0}\), which is equivalent to \(\rho(\mathbf{M})<1\). 2. \((\mathbf{I}-\mathbf{M})^{-1}\) exists, which is guaranteed if \(\rho(\mathbf{M})<1\). Therefore, the necessary and sufficient conditions for convergence are  
\[\rho(\mathbf{M})=\rho(\mathbf{I}-\omega^{*}\mathbf{D}^{-1}\mathbf{G}^{H} \mathbf{G}+\omega^{*}\gamma^{*}(\mathbf{D}-\mathbf{L}-\mathbf{U}))<1 \tag{38}\]  
Now, we can state the convergence theorem for the OAOR method. **Theorem 1** (Convergence of OAOR). : _The OAOR method converges to the MMSE solution if and only if_  
\[\mathbf{e}^{(k+1)} =\mathbf{\hat{x}}-\mathbf{x}^{(k+1)} \tag{30}\]0  
_where \(\rho(\cdot)\) denotes the spectral radius of the matrix._  
Proof. : From the error propagation equation derived earlier,  
\[\mathbf{e}^{(k+1)} =\mathbf{\hat{x}}-\mathbf{x}^{(k+1)} \tag{30}\]1  
For convergence, we need \(\lim_{k\to\infty}\mathbf{e}^{(k)}=\mathbf{0}\). This requires the following:  
1. \(\lim_{k\to\infty}\mathbf{M}^{k}=\mathbf{0}\)
2. \((\mathbf{I}-\mathbf{M})^{-1}\) exists  
Condition 1 is satisfied if and only if \(\rho(\mathbf{M})<1\), where \(\rho(\mathbf{M})\) is the spectral radius of \(\mathbf{M}\). Condition 2 is also satisfied when \(\rho(\mathbf{M})<1\), as this ensures that \(\mathbf{I}-\mathbf{M}\) is invertible. Therefore, the necessary and sufficient conditions for convergence is  
\[\mathbf{e}^{(k+1)} =\mathbf{\hat{x}}-\mathbf{x}^{(k+1)} \tag{30}\]2  
This completes the proof. **Lemma 1** (Convergence Rate).