# Efficient Deep Learning-Based Detection Scheme for MIMO Communication Systems
## 4 Results and Discussion
#### Complexity Analysis  
The ML detector defined by Equation (4) computes the Euclidean distance between the received vector \(\mathbf{r}\) and the products of the CSI matrix \(\mathbf{H}\) with all the combinations possible \(M^{N_{t}}\) received of the modulated symbols \(\mathbf{s}\). This procedure involves \(M^{N_{t}}\) products, \(N_{t}\cdot N_{r}\) subtractions, and norm evaluation that results in a complexity of \(\mathcal{O}(N_{t}\cdot N_{r}\cdot M^{N_{t}})\). In the case of DL detectors, we add the multiplications produced at each layer to obtain their complexity. The operations produced by activation functions can be neglected, as the calculations of the outputs are not affected by the input size. Table 8 lists the resulting complexities for all DL-based detection schemes. For this evaluation, we consider only the forward propagation since the backward propagation has a similar complexity. Therefore, the total complexity for each system is approximately twice the values listed in Table 8 if the training phase is considered. Figure 5 shows the resulting complexity in flops for the signal-labeling and detection strategies designed in this paper.  
The results show that complexity is directly affected by the number of hidden units in each layer \(d_{\ell+1}\). This value is multiplied by twice the number of Rx antennas \(N_{r}\) due to the input size of \(\mathbf{x}_{0}\), and it will not affect the final result. The number of Tx antennas \(N_{t}\) and the modulation order \(M\) have different variations in complexity regarding the selected labeling strategy.  
The OH labeling scheme provides the highest complexity and the best BER performance, F1-score, precision, and recall metrics. On the other hand, the lowest complexity is achieved when using a DSE strategy to label signals, reducing the last factor to \(\log_{2}(M)\) and obtaining the lowest BER performance but not having the worst classification score. The OHA provides an alternative to reduce the complexity from exponential \(M^{N_{t}}\) to a product \(M\cdot N_{t}\) with promising results in BER performance.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**MIMO Configuration** & **Labeling Approach** & **Precision** & **Recall** & **F1-Score** \\ \hline \(N_{t}=2\) & OH & 0.97 & 0.98 & 0.97 \\ \(N_{r}=2\) & OHA & 0.93 & 0.93 & 0.93 \\  & DSE & 0.96 & 0.98 & 0.96 \\ \hline \(N_{t}=4\) & OH & 0.96 & 0.94 & 0.95 \\ \(N_{r}=4\) & OHA & 0.89 & 0.89 & 0.91 \\  & DSE & 0.91 & 0.92 & 0.93 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Classification performance for the analyzed MIMO detection schemes.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**MIMO Configuration** & **Labeling Approach** & **Complexity** \\ \hline \(N_{t}=2\) & OH & \(\mathcal{O}((2(N_{r}+d_{\ell+1})+M^{N_{t}}))\) \\ \(N_{r}=2\) & OHA & \(\mathcal{O}((2(N_{r}+d_{\ell+1})+M\cdot N_{t}))\) \\  & DSE & \(\mathcal{O}((2(N_{r}+d_{\ell+1})+\log_{2}(M)\cdot N_{t}))\) \\ \hline \(N_{t}=4\) & OH & \(\mathcal{O}(d_{\ell+1}(2N_{r}+M^{N_{t}}))\) \\ \(N_{r}=4\) & OHA & \(\mathcal{O}(d_{\ell+1}(2N_{r}+M\cdot N_{t}))\) \\  & DSE & \(\mathcal{O}(d_{\ell+1}(2N_{r}+\log_{2}(M)\cdot N_{t}))\) \\ \hline \hline \end{tabular}
\end{table}
Table 8: Complexity of the analyzed DL-based MIMO signal detection schemes.