# An Improved Jacobi-Based Detector for Massive MIMO Systems
## 3 Proposed Algorithm for MIMO Detection
### Convergence Proof  
Taking the iterative process into account, the overall convergence performance is split into two parts for theoretical analysis. For the first part, we research the convergence performance of the combined SD and Jacobi iterative process and compare it with the conventional Jacobi algorithm. **Theorem 1**. : _A necessary and sufficient condition for the iterative algorithm \(\mathbf{x}^{(k)}=\mathbf{B}\mathbf{x}^{(k-1)}+\mathbf{c}\) to be convergent for all initial vectors \(\mathbf{x}^{(0)}\) is that \(\rho\left(\mathbf{B}\right)<1\)[25]._  
As the theorem shows, the convergence velocity of the iterative algorithm is closely related to the spectral radius of the iteration matrix. Therefore, we derive the iteration matrix of the combined algorithm firstly. The iterative form of iterative process of the combined algorithm is concluded as below:  
\[ \begin{split}\bar{\mathbf{x}}_{2}^{(k)}&=\bar{ \mathbf{x}}^{(k-1)}+\alpha_{k-1}\mathbf{r}^{(k-1)}+\mathbf{D}^{-1}\left( \mathbf{r}^{(k-1)}-\alpha_{k-1}\mathbf{p}^{(k-1)}\right)\\ &=\left(\mathbf{I}-\alpha_{k-1}\mathbf{A}-\mathbf{D}^{-1}\mathbf{ A}+\alpha_{k-1}\mathbf{D}^{-1}\mathbf{A}^{2}\right)\bar{\mathbf{x}}^{(k-1)}+ \alpha_{k-1}\mathbf{b}+\mathbf{D}^{-1}\mathbf{b}-\alpha_{k-1}\mathbf{D}^{-1} \mathbf{A}\mathbf{b},\end{split}  \tag{17}\]  
where \(\mathbf{B}=\mathbf{I}-\alpha_{k-1}\mathbf{A}-\mathbf{D}^{-1}\mathbf{A}+\alpha _{k-1}\mathbf{D}^{-1}\mathbf{A}^{2}\) is considered as the iteration matrix. Afterward, analyze the spectral radius of \(\mathbf{B}\), hereinafter \(\rho\left(\mathbf{B}\right)\):  
\[ \begin{split}\rho\left(\mathbf{B}\right)&=\rho \left(\mathbf{I}-\alpha_{k-1}\mathbf{A}-\mathbf{D}^{-1}\mathbf{A}+\alpha_{k- 1}\mathbf{D}^{-1}\mathbf{A}^{2}\right)\\ &=\rho\left(\left(\mathbf{I}-\alpha_{k-1}\mathbf{A}\right)\left( \mathbf{I}-\mathbf{D}^{-1}\mathbf{A}\right)\right).\end{split}  \tag{18}\] \begin{table}
\begin{tabular}{c c c} \hline \hline \(m\) **Value** & **Signal-to-Noise Ratio (Target \(10^{-5}\))** & **Computational Complexity of Moore–Penrose Inverse** \\ \hline \(m=2\) & 11.56 decibel (dB) & \(O\left(N_{t}\right)\) \\ \(m=3\) & 11.44 dB & \(O\left(3N_{t}^{2}\right)\) \\ \(m=4\) & 11.55 dB & \(O\left(4N_{t}^{2}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance comparison for different \(m\). Figure 2: Bit error rate (BER) curves of the proposed signal detection algorithm with different value \(m\), where \(K=3\). Since the MMSE filtering matrix \(\mathbf{A}\) is a positive definite matrix, the diagonal matrix \(\mathbf{D}\) of \(\mathbf{A}\) can be considered as the positive definite. According to that, we can infer that the norm of \(\rho\left(\mathbf{I}-\mathbf{D}^{-1}\mathbf{A}\right)\) and \(\rho\left(\mathbf{I}-\alpha_{k-1}\mathbf{A}\right)\), \(k=1:K\) are both less than one [26]. Furthermore, we can get \(\rho\left(\mathbf{I}-\mathbf{D}^{-1}\mathbf{A}\right)<1\) and \(\rho\left(\mathbf{I}-\alpha_{k-1}\mathbf{A}\right)<1\), respectively. Thus, it is easy to infer \(\rho\left(\mathbf{B}\right)<1\). According to Theorem 1, we can assert that the iteration of the combined algorithm is convergent.