## 4 Design and Complexity of Proposed DL Architectures
### DL Architecture  
Figure 4 shows the proposed DL architecture. First, the FNN is utilized so that any nonlinear function can be learned. RNN is also included because FNN was incapable of identifying sequential information in the input data. In contrast, CNN is investigated because it reveals the spatial features of the input data more effectively. All proposed DL networks have \(L\) layers: an input layer, \(L\)-2 hidden layers, and an output layer. Full-connected (FC) layers are chosen as the hidden layers for FNN. Let \(\mathcal{H}\) be the output vector at the last layer. The number of neurons utilized at every layer is denoted by \(N_{t}\), \(l\in 0\cdots L-1\). A normalized feature vector \(\mathbf{v}^{\text{q}}\) of size \(N_{f}\) is sent from the input layer to the first hidden layer for further processing, which is processed as [26]. \[\mathbf{r}_{1}^{q}=max(\mathbf{W}_{1}\mathbf{v}^{\text{q}}+\mathbf{b}_{1},0) \tag{9}\]  
where the output vector of the first hidden layer is represented by \(\mathbf{r}_{1}^{q}\). Both the weight matrix and the bias vector associated with the first hidden layer are denoted by \(\mathbf{W}_{1}\) and \(\mathbf{b}_{1}\) \begin{table}
\begin{tabular}{c c} \hline \hline
**Labels** & **Antenna Subsets** \\ \hline a & \(\beta_{1}=[1,2,3]\) \\ b & \(\beta_{2}=[1,2,4]\) \\ c & \(\beta_{3}=[1,3,4]\) \\ d & \(\beta_{4}=[2,3,4]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Antenna subset-class label mapping for \(N_{r}=4\), \(N_{t}=3\), and \(N_{T}=4\). respectively. The rectified linear unit (ReLU) activation function is utilized at the hidden layers. The output of the ReLU activation function is given by:  
\[f_{ReLU}(s)=max(0,s) \tag{10}\]  
The output vector of the \(l\)th hidden layer is given as [26]  
\[\vec{r}_{l}^{q}=max(\mathbf{W}_{l}\vec{r}_{l-1}^{q}+\mathbf{b}_{l},0) \tag{11}\]  
where the weight matrix and the bias vector associated with the \(l\)th hidden layer are denoted by \(\mathbf{W}_{l}\) and \(\mathbf{b}_{l}\). The FC layer is used as an output layer for all proposed networks. The number of neurons utilized at the output layer corresponds to the number of class labels (\(N_{p}\)). The Softmax activation function (\(\delta\)) is employed at the output layer to perform the multiclass classification task. The output vector of the \(L-1\) layer is given by [26]  
\[\vec{r}_{L-1}^{q}=\delta(\mathbf{W}_{L-1}\vec{r}_{L-2}^{q}+\mathbf{b}_{L-1},0) \tag{12}\]  
where the weight matrix and the bias vector associated with the output layer are denoted by \(\mathbf{W}_{L-1}\) and \(\mathbf{b}_{L-1}\). On the basis of the procedure described above, the entire training mapping rule is given as [26]  
\[\vec{r}^{q}=\delta(\mathbf{W}_{L-1}(max(\mathbf{W}_{L-2}\cdots(\mathbf{W}_{1}\vec{v}_{1}^{ q}+\mathbf{b}_{1},0)\cdots+\mathbf{b}_{L-2},0))+\mathbf{b}_{L-1}) \tag{13}\]  
We use (8) to acquire the feature vector \(\mathbf{f}_{N}\) for a new observation \(\mathbf{H}_{N}\) and the label is retrieved from (13). According to Figure 5a, the classification layer receives its inputs from the output layer via the Softmax layer. Long short-term memory (LSTM) is employed as the hidden layer in the RNN architecture. There are three multiplicative units (input gate, forget gate, and output gate) that make up the memory module of a RNN using the LSTM architecture. For any LSTM unit, given its input and output values (\(\mathbf{x}_{l-1}\) and \(\mathbf{x}_{l}\), respectively), the relationship between them at time step \(t\) is stated as [27].