# Computationally Efficient Channel Estimation in 5G Massive Multiple-Input Multiple-output Systems
## 5 Simulation Results  
We deployed Matlab (R2017a, Mathworks, Natick, MA, USA) for performing analysis and experimentation. In order to verify the soft and hard detection performance of the SDGS algorithm \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Real Multiplications** & **Performance** \\ \hline NS [11] & \(\left(\mathbf{8K^{3}}-\mathbf{8K^{2}}+\mathbf{2K}\right)\left(i+2\right)+\left(\mathbf{4K^{2} }-\mathbf{4K}\right)\)\(\left(i>\mathbf{1}\right)\) & General \\ CG [15] & \(\mathbf{4K^{2}}(i+\mathbf{1})+\mathbf{10 Ki}\) & Better \\ GS [17] & \(\mathbf{4K^{2}i}+\mathbf{2K}\) & Better \\ SDGS & \(\mathbf{2K^{2}}(i+\mathbf{2})+\mathbf{12 Ki}\) & Optimal \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Complexity comparison of four kinds of detection algorithms for hard decision calculation. proposed in this paper, this section presents Monte Carlo simulation results based on Matlab. The main simulation parameters configured are in Table 2.  
Figure 2 compares the bit error rate (BER) based on Neumann series (NS) expansion, the conjugate gradient (CG) detection algorithm, the Gauss-Seidel iterative detection algorithm, the MMSE exact inversion detection algorithm, and the proposed SDGS joint algorithm under different antenna configurations. The decision mode is a hard decision; that is, estimated signal vector \(\hat{s}\) is directly judged. The simulation results showed that the detection performance of the various algorithm increased with the number of iterations or the number of items expanded by the Neumann series. For example, when the number of iterations \(i=2\), the BER performance of the SDGS algorithm was much better than the BER when the number of items expanded by the Neumann series was 2. By comparing the performance in Figure 2a,b, it can be seen that with the increase in the ratio of the number of base station antennas to the number of users \((N/K)\), the BER performance of the various algorithms was greatly improved. For example, if the BER was to reach \(10^{-3}\), the MMSE algorithm and the proposed algorithm require an SNR of about 13 dB when the antenna configuration is \(64~{}\times~{}16\), and only 8 dB when the configuration is \(128~{}\times~{}16\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Number of transmitter antennas \((N)\) & 64–128 \\ Number of receiver antennas \((K)\) & 16 \\ Baseband modulation mode & 16 QAM \\ Signal-to-noise ratio (SNR) & 10–12 dB \\ Code rate & 1/2 \\ Channel characteristics & Uncorrelated Rayleigh fading \\ Number of iterations \((i)\) & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. Figure 2: Hard decision bit error rate (BER) performance comparison. (**a**) Analysis at \(64~{}\times~{}16\) antenna configuration; (**b**) Analysis at \(128~{}\times~{}16\) antenna configuration. Figure 3 shows the soft decision simulation results for the two antenna configurations. With BER based on the NS, CG, and GS iterative detection algorithms, the MMSE exact inversion detection algorithm and the SDGS joint algorithm were compared. We set the system's convolutional code rate to 1/2, and the LLR calculation used the approximate calculation method described in this paper. Simulation results showed that no matter what kind of MMSE receiver was used, the soft decision was checked. The measured performance was much better than the hard decision. For example, when the BER reached \(10^{-4}\), when the antenna was configured, the MMSE algorithm and the proposed algorithm required an SNR of 10 dB for hard decisions and only 5 dB for soft decisions. In addition, for the same number of iterations, the BER performance of the SDGS algorithm proposed in this paper was better than the other three simplified algorithms, and after a few iterations, the detection performance could quickly approach the detection performance of the ideal MMSE filter matrix inversion.