# SNR-Based Receiver-Type Decision Using Deep Learning for Multiple-Input Multiple-Output Detection
## 3 Proposed Deep Learning-Based MIMO Receiver
### Threshold for Receiver-Type Decision  
The ZF receiver is one of the most widely investigated linear receivers in which the multi-stream interference can be eliminated. However, as shown in (2), the post-processing signal suffers from noise enhancement depending on the MIMO channel matrix \(\mathbf{H}\). In (2), \(\left\{\mathbf{z}^{ZF}\right\}_{p}=z_{p}^{ZF}\), which corresponds to the transmitted signal of the \(p\)th transmit antenna, is given by  
\[z_{p}^{ZF}=x_{p}+\sqrt{\frac{N_{T}}{E_{s}}}\bigg{\{}\left(\mathbf{H}^{H} \mathbf{H}\right)^{-1}\mathbf{H}^{H}\mathbf{n}\bigg{\}}_{p}. \tag{7}\]  
Denoting \(\mathbf{G}=\left[\mathbf{g}_{1}\cdots\mathbf{g}_{p}\cdots\mathbf{g}_{N_{T}} \right]=\mathbf{H}\big{(}\mathbf{H}^{H}\mathbf{H}\big{)}^{-1}\), (7) can be rewritten by  
\[z_{p}^{ZF}=x_{p}+\sqrt{\frac{N_{T}}{E_{s}}}\mathbf{g}_{p}^{H}\mathbf{n}. \tag{8}\]Then, the average SNR for the \(p\)th demodulated signal can be calculated as follows [25]:  
\[ \begin{split}\rho_{p}&=\frac{\mathbb{E}\Big{[}\abs{x_{ p}}^{2}\Big{]}}{\mathbb{E}\Big{[}\frac{N_{T}}{E_{s}}\Big{[}\mathbf{\upbeta}_{p}^{H}\mathbf{ \upbeta}_{p}^{H}\Big{]}\Big{]}}\\ &=\frac{E_{s}}{N_{T}N_{0}}\frac{1}{\big{\|}\mathbf{\upbeta}_{p}\big{\|} _{F}^{2}}\\ &=\frac{\rho}{N_{T}}\frac{1}{\big{\{}\mathbf{G}^{H}\mathbf{G} \big{\}}_{p,p}}\\ &=\frac{\rho}{N_{T}}\frac{1}{\big{\{}\big{(}\mathbf{H}^{H}\mathbf{ H}\big{)}^{-1}\big{\}}_{p,p}}.\end{split}  \tag{9}\]  
Applying the singular value decomposition (SVD) of the MIMO channel \(\mathbf{H}=\mathbf{U}\Sigma\mathbf{V}^{H}\), \(\rho_{p}\) in (9) can be rewritten as  
\[ \begin{split}\rho_{p}&=\frac{\rho}{N_{T}}\frac{1}{ \Big{\{}\sum_{i=1}^{N_{T}}\sigma_{i}^{-2}\mathbf{v}_{i}\mathbf{v}_{i}^{H} \Big{\}}_{p,p}}\\ &=\frac{\rho}{N_{T}}\frac{1}{\sum_{i=1}^{N_{T}}\sigma_{i}^{-2} \Big{|}\{\mathbf{v}_{i}\}_{p}\Big{|}^{2}},\end{split}  \tag{10}\]  
where \(\mathbf{U}=\big{[}\mathbf{u}_{1}\cdots\mathbf{u}_{q}\cdots\mathbf{u}_{N_{R}} \big{]}\) and \(\mathbf{V}=\big{[}\mathbf{v}_{1}\cdots\mathbf{v}_{p}\cdots\mathbf{v}_{N_{T}} \big{]}\) are matrices including left singular vectors and right singular vectors, respectively, and \(\sigma_{p}=\{\mathbf{\Sigma}\}_{p,p}\) denotes the \(p\)th largest singular value of \(\mathbf{H}\). As shown in (10), the average SNR for each demodulated signal is determined by the distribution of singular values and right singular vectors. For ease of analysis, we assume that \(\abs{\{\mathbf{v}_{1}\}_{p}}^{2}=\cdots=\abs{\{\mathbf{v}_{N_{T}}\}_{p}}^{2}=v^ {2}\). Then, the derived result in (10) is a harmonic mean of \(\frac{\rho}{N_{T}^{2}\sigma_{1}^{2}}\cdots\frac{\rho}{N_{T}^{2}\sigma}\sigma_{ N_{T}}^{2}\). Applying the inequality between the arithmetic mean and harmonic mean, we can obtain the upper-bound of (10) as  
\[\rho_{p}\leq\frac{\rho}{N_{T}^{3}v^{2}}\sum_{i=1}^{N_{T}}\sigma_{i}^{2}, \tag{11}\]  
where the equality holds when \(\sigma_{1}=\cdots=\sigma_{N_{T}}\). As mentioned at the beginning of this section, it implies that there are poorly conditioned MIMO channels, which indicates that \(\sigma_{N_{T}}\ll\sigma_{1}\) experience SNR loss due to critical noise enhancement. In (10), it can be stated that the error performance of the ZF receiver depends on the MIMO channel \(\mathbf{H}\). Therefore, we propose a receiver that decides the receiver type based on whether the ZF receiver can guarantee a certain level of symbol error rate (SER). To this end, we define a criterion for the receiver-type decision that compares the average SER of the ZF receiver with a threshold \(\eta\). Consider a function \(f_{R}\in\{0,1\}\) for the receiver type, where \(f_{R}=1\) and \(f_{R}=0\) indicate the use of the ZF receiver and ML receiver, respectively. Then, \(f_{R}\) can be expressed as  
\[f_{R}= \begin{cases}0;&\quad\bar{P}_{S}^{ZF}(\rho,\mathbf{H})>\eta,\\ 1;&\quad\bar{P}_{S}^{ZF}(\rho,\mathbf{H})\leq\eta.\end{cases}