# Low-Complexity GSM Detection Based on Maximum Ratio Combining
## 5 Simulation Results and Computational Complexity Analysis  
In this section, the BER performance and the computational complexity of the proposed algorithms for GSM systems are simulated under the assumption of perfect channel state information. In all simulations, the SNR represents the ratio of the symbol power to the noise power. In this paper, the computational complexity is defined as the total number of the real-valued multiplications and divisions involved in an algorithm.  
Simulation Result and Computational Complexity Analysis for M-QAM Constellation 5.1.1. Simulation Result  
Figures 2 and 3 show the BER performance comparison of the proposed algorithms and ML algorithm for a GSM system in two scenarios with (1) \(N_{t}=32\), \(N_{r}=16\), \(M=8\), \(N_{p}=2\) and (2) \(N_{t}=16\), \(N_{r}=8\), \(M=16\), \(N_{p}=3\). It can be seen from Figures 2 and 3 that the BER performance of the above-mentioned algorithms is almost the same in the whole SNR range.  
It needs to be verified whether the proposed two algorithms can greatly reduce the number of TACs detected by the receiver. Figures 4 and 5 depict the average number of TACs detected by the MRC-RS and MRC-MP algorithms at the receiver in two scenarios: (1) \(N_{t}=32\), \(N_{r}=16\), \(M=8\), \(N_{p}=2\) and (2) \(N_{t}=16\), \(N_{r}=8\), \(M=16\), \(N_{p}=3\).  
When \(N_{t}=32\) and \(N_{p}=2\), the number of TACs for a conventional GSM system is 256. It can be seen from Figure 4 that the MRC-RS algorithm finally detects about 30 TACs under different SNRs, which is about 12% of the total TACs. The detected TACs by MRC-MP algorithm decrease gradually with the increase of SNR. About 190 TACs are detected when the SNR is 20 dB, which is about 74% of the total TACs.  
\begin{table}
\begin{tabular}{l l} \hline \hline
1: & Input : \(\mathbf{y},\mathbf{H},N_{r}\), \(N_{p}\), \(N_{t}\). \\
2: & Initialize \(\beta=1\), \(\mathbf{r}_{\beta}=\mathbf{y}\), \(\psi\) is an empty set. \\
3: & While \(\beta\leq N_{p}\) do \\
4: & for \(l=1\) to \(N_{t}\) \\
5: & \(k_{\beta}^{l}=\frac{\left|\mathbf{h}_{i}\mathbf{r}_{\beta}\right|^{2}}{\left|\mathbf{h}_{i} \right|^{2}}\) \\
6: & end for \\
7: & \(\frac{\sum_{i=1}^{N_{t}}k_{\beta}^{l}}{N_{t}}\) \\
8: & for \(l=1\) to \(N_{t}\) \\
9: & if \(k_{\beta}^{l}>\eta_{aav}\) then \\
10: & \(\psi_{\beta}=\{l\}\) \\
11: & end if \\
12: & end for \\
13: & Estimate the signal \(\widetilde{x}_{\beta}=\left(\mu_{1},\cdots,\mu_{\left|\psi_{\beta}\right|} \right)^{T}\), where \(\mu_{i}\)= \(\mathbf{h}_{\psi_{\beta}}^{l}\mathbf{y}=\left(\mathbf{h}_{\psi_{\beta}}^{H}\mathbf{h}_{\psi_{ \beta}}\right)^{-1}\mathbf{h}_{\psi_{\beta}}^{H}\mathbf{y}\). \\
14: & Update residual \(\mathbf{r}_{\beta}=\mathbf{y}-\mathbf{H}_{\psi_{\beta}}\widetilde{x}_{\beta}\). \\
15: & end while \\
16: & The \(N_{p}\) antenna index sets \(\left\{\psi_{\beta}\right\}_{\beta=1}^{N_{p}}\) are obtained through \(N_{p}\) iterations, and delete the duplicate indices. \\
17: & The qualified TACs is obtained according to the \(\left\{\psi_{\beta}\right\}_{\beta=1}^{N_{p}}\) antenna index set. \\
18: & Quantitated modulation symbol. \\
19: & Output the transmit antenna combination and modulation symbol \((\hat{l},\hat{s})\). \\ \hline \hline \end{tabular}
\end{table}
Table 2: Proposed MRC-MP detector.  
Figure 3: BER performance comparison of the proposed detectors and ML detector for GSM system in the presence of Rayleigh fading channel and perfect channel state information at the receiver with \(N_{t}=16\), \(N_{r}=8\), \(M=16\) and \(N_{p}=3\).  
Figure 2: BER performance comparison of the proposed detectors and ML detector for GSM system in the presence of Rayleigh fading channel and perfect channel state information at the receiver with \(N_{t}=32\), \(N_{r}=16\), \(M=8\) and \(N_{p}=2\).  
When \(N_{t}=16\) and \(N_{p}=3\), the number of TACs for the conventional GSM system is 512. We can see from Figure 5 that the number of TACs detected by MRC-RS and MRS-MP algorithms is 158 and 467, which is about 30% and 90% of the total TACs, respectively.