# SNR-Based Receiver-Type Decision Using Deep Learning for Multiple-Input Multiple-Output Detection
## 3 Proposed Deep Learning-Based MIMO Receiver
### Deep Learning-Based Operation  
We aim to decide the receiver type for the given MIMO channel \(\mathbf{H}\), but it is challenging to analyze \(f_{R}\) in (14) mathematically. Therefore, we utilize deep learning to represent the relationship between \(\mathbf{H}\) and \(f_{R}\). To increase learning accuracy, it is possible to use a feature vector \(\mathbf{p}\) that includes not only MIMO channel coefficients but also various properties of the MIMO channel matrix \(\mathbf{H}\). In other words, our goal is to model the relationship between the feature vector \(\mathbf{p}\) and \(f_{R}(\mathbf{p})\) using DNN. We consider a DNN structure consisting of \(L\) hidden layers, and each layer has \(d_{1},d_{2},\cdots,d_{L}\) nodes, as described in Figure 1. The output of \(l\)th layer, \(l=1,2,\cdots,L+1\) is given by  
\[\mathbf{r}_{l}=\phi_{l}(\mathbf{W}_{l-1}\mathbf{r}_{l-1}+\mathbf{b}_{l-1}), \tag{16}\]  
where \(\phi_{l}(\cdot)\), \(\mathbf{W}_{l}\in\mathbb{R}^{d_{l}\times d_{l+1}}\), \(\mathbf{b}_{l}\in\mathbb{R}^{d_{l+1}}\) are the activation function, weight matrix, and bias vector of \(l\)th layer, respectively. Without a loss of generality, \(\mathbf{r}_{0}\) and \(\mathbf{r}_{L+1}\) indicate the input and output vector of the DNN, respectively, and the number of nodes of each vector is \(d_{0}\) and \(d_{L+1}\), respectively. As the activation functions, we adopt the rectified linear unit (ReLU) for the input layer and hidden layers. Namely, the \(i\)th element of the output vector of \(l^{\prime}\)th layer, \(\{\mathbf{r}_{l^{\prime}}\}_{i}\), \(l^{\prime}=1,2,\cdots,L\), \(i=1,2,\cdots,d_{l^{\prime}}\), is given by  
\[\{\mathbf{r}_{l^{\prime}}\}_{i}=\max\{0,\{\mathbf{W}_{l^{\prime}-1}\mathbf{r} _{l^{\prime}-1}+\mathbf{b}_{l^{\prime}-1}\}_{i}\}. \tag{17}\]  
For the activation function of the output layer, we use the softmax function since the DNN is designed for the classification problem for the receiver-type decision. The \(i\)th element of the output vector, \(\{\mathbf{r}_{L+1}\}_{i}\), \(i=1,2,\cdots,d_{L+1}\), is expressed as  
\[\{\mathbf{r}_{L+1}\}_{i}=\dfrac{\exp(\{\mathbf{W}_{L}\mathbf{r}_{L}+\mathbf{b }_{L}\}_{j})}{\sum_{j=1}^{d_{L+1}}\exp(\{\mathbf{W}_{L}\mathbf{r}_{L}+\mathbf{ b}_{L}\}_{j})}. \tag{18}\]  
Note that \(d_{L+1}=2\) since the number of possible \(f_{R}\) is 2. For the data samples to train the DNN, consider the input and output vector by \(\mathbf{p}_{\text{sample}}=\mathbf{r}_{0}\in\mathbb{R}^{d_{0}}\) and \(\mathbf{q}_{\text{sample}}\in\mathbb{R}^{d_{L+1}}\) for the given MIMO channel \(\mathbf{H}\). As mentioned above, \(\mathbf{p}_{\text{sample}}\) should include properties of \(\mathbf{H}\) as features that help improve learning accuracy. Based on the average SNR of the demodulated signal in (9), we consider the SNR gain for \(N_{T}\) transmitted signals as a feature. Let us define \(G_{p}\) as the SNR gain for \(p\)th transmitted signal. According to (9), \(G_{p}\) can be expressed as  
\[G_{p}=\frac{1}{\left\{\left(\mathbf{H}^{H}\mathbf{H}\right)^{-1}\right\}_{p,p}}. \tag{19}\]  
Then, we construct the input feature vector \(\mathbf{p}_{\text{sample}}\) as follows:  
\[\mathbf{p}_{\text{sample}}=\Big{[}\text{Re}\Big{\{}\text{vec}\left(\mathbf{H }\right)^{T}\Big{\}}\quad\text{Im}\Big{\{}\text{vec}\left(\mathbf{H}\right)^{ T}\Big{\}}\quad G_{1}\quad G_{2}\ \cdots\ G_{N_{T}}\Big{]}^{T}. \tag{20}\]  
Note that we make every element of the feature vector \(\mathbf{p}_{\text{sample}}\) a real number by vectorizing the real and imaginary parts of \(\mathbf{H}\) and stacking them since the weight matrices and bias vectors of the DNN are defined in the real domain. Moreover, we utilize the one-hot-encoding for construct \(\mathbf{q}_{\text{sample}}\), which is determined by \(f_{R}\Big{(}\mathbf{p}_{\text{sample}}\Big{)}\) as follows [26]:  
\[\mathbf{q}_{\text{sample}}= \begin{cases}[1,0];&f_{R}\Big{(}\mathbf{p}_{\text{ sample}}\Big{)}=0,\\ [0,1];&f_{R}\Big{(}\mathbf{p}_{\text{sample}}\Big{)}=1.\end{cases}