# Fast Converging Gauss-Seidel Iterative Algorithm for Massive MIMO Systems
## 4 Simulation Results and Analysis
### Complexity Analysis  
In massive MIMO communications, computational cost is a critical metric for signal-detection algorithms [30]. The sophistication of the CJGS iterative scheme consists of three parts, initialization, conjugate gradient and Jacobi co-processing, and GS iteration estimation, and only the number of true multiplicative operations of the method is considered in the analysis. The computation of \(A\) and \(y_{MF}\) is only once required, which is the same as the  
Figure 6: BER performance for fully correlated channel.  
Figure 7: BER performance comparison of the CJGS iterative algorithm with different initial values.  
MMSE detection algorithm [31]. Therefore, this part of the calculation is ignored and only the complexity of the subsequent steps is analyzed. One real multiplication is noted as one real multiplication time, the multiplication of a complex number with a constant is noted as two real multiplication times, and the multiplication of two complex numbers is noted as four real multiplication times.  
The complexity of \(s^{(0)}\): this involves multiplying a constant \(\frac{1}{B+U}\) and a vector \(y_{MF}\) of \(B\times 1\) with a complexity of \(2U\).  
The complexity of conjugate gradient and Jacobi co-processing:  
The complexity of \(r^{(0)}\): this involves multiplying a matrix \(A\) of \(U\times U\) and a vector \(y_{MF}\) of \(U\times 1\), with a complexity of \(4U^{2}\).  
The complexity of \(u^{(0)}\): this concerns multiplying a vector \(\left(r^{(0)}\right)^{H}\) of \(1\times U\) and a vector \(r^{(0)}\) of \(U\times 1\), with a complexity of \(2U\). And it involves multiplying a matrix \(A\) of \(U\times U\) and a vector \(p^{(0)}\) of \(U\times 1\), with a complexity of \(4U^{2}\). It concerns multiplying a vector \(\left(r^{(0)}\right)^{H}\) of \(1\times U\) and a vector \(p^{(0)}\) of \(U\times 1\), with a complexity of \(2U\).  
The complexity of \(s^{(1)}\): this involves multiplying a constant \(u^{(0)}\) and a vector \(p^{(0)}\) of \(U\times 1\), with a complexity of \(2U\). The complexity of \(D^{-1}(y_{MF}+(E+F)(s^{(0)}+u^{(0)}p^{(0)}))\) is \(8U^{2}\).  
The complexity of the GS iteration: the calculation of \(s^{(i+1)}\) involves multiplying twice a matrix of \(\mathsf{U}\times\mathsf{U}\) and a vector of \(U\times 1\) with a complexity of \(8U^{2}\).  
Hence, the total complexity of the CJGS algorithm is \(16U^{2}+8U+8 iU^{2}\). Taken together, the computational complexity of the various algorithms is shown in Table 1.  
In Figure 8, the complexity of the different algorithms for MIMO systems are compared. The base station antenna count is 128 and the user-side antenna count ranges from 0 to 30 with three iterations. The suggested algorithm is much less complex than the MMSE scheme, whose complexity is \(O(U^{2})\).  
\begin{table}
\begin{tabular}{c c} \hline
**Algorithm** & **Complexity** \\ \hline MMSE [6] & \(8U^{3}+8 BU^{2}+8 BU+2U\) \\ \hline GS [13] & \(4U^{2}+8 iU^{2}\) \\ \hline CPGS [19] & \(20U^{2}+2U+8 iU^{2}\) \\ \hline JAGS [18] & \(12U^{2}+8 iU^{2}\) \\ \hline PGS [20] & \(20U^{2}+2U+8 iU^{2}\) \\ \hline CJGS & \(16U^{2}+8U+8 iU^{2}\) \\ \hline \end{tabular}
\end{table}
Table 1: Complex comparison of different algorithms.  
Figure 8: Comparison of algorithmic complexity.