## 3 Problem Formulation and the Proposed Path Loss Model  
In urban environments, the direct connection line between the Tx and Rx is very likely to be blocked by buildings. Figure 2 shows the conceptual overview of the Tx-Rx pair and the path profile along the direct connection line, where \(h_{Tx}\) is the Tx height and \(D\) is the horizontal distance. The obstruction of buildings results in excessive path loss, in addition to free space propagation. Therefore, this paper incorporates the path profile in path loss modeling. The proposed model combines LOS and NLOS propagation scenarios. For LOS cases, where the radiation from the Tx is not blocked by buildings, the path loss is determined using the log-distance path loss model (2). For NLOS cases, this paper assumed that the excessive path loss in addition to the free space propagation is caused by the obstruction of buildings. Figure 3 shows the proposed deep-learning-based path loss model. The input data include the Tx height, \(h_{Tx}\), the distance between the Tx-Rx pair, \(D\), and the path profile. Assuming the cell radius is \(Q\), the path profile is represented as a vector of length \(Q\), with a sample distance of 1 m. The values in the profile vector stand for the building height along the path. As a result, the length of the input layer for the neural network is \(Q+2\). Note that \(D\leq Q\), given that the Rx is within the coverage of the Tx. In the output layer, the proposed model requires only one neuron, which is used to predict the path loss. In between the input and output layers, three hidden layers, namely, L1, L2, and L3, are used to extract essential features that cause path loss. Given the \(k\)th Tx-Rx pair, the input to the proposed deep neural network is represented as:  
\[\mathbf{x}_{k}=\left[ \begin{array}{cccccc}h_{Tx,k}&D_{k}&q_{1,k}&q_{2,k}& \cdots&q_{Qk}\end{array} \right]^{T} \tag{4}\]  
Figure 3: The network architecture of the proposed deep-learning-based path loss modeling. Figure 2: Illustration of the Tx–Rx pair and its path profile. where \(q_{i,k}\), for \(1\leq i\leq Q\), is the building height of the \(k\)th path profile. The input should be normalized before being fed into the network because normalization has been proven to be able to accelerate the training process and improve the performance of neural networks [19]. The normalized inputs are denoted as \(\widetilde{\mathbf{x}}_{k}\) and \((\theta_{1},\mathbf{b}_{1})\), \((\theta_{2},\mathbf{b}_{2})\), and \((\theta_{3},\mathbf{b}_{3})\) as the weights to be optimized at the hidden layers L\({}_{1}\), L\({}_{2}\) and L\({}_{3}\), respectively. The dimensions of \(\theta_{i}\) and \(\mathbf{b}_{i}\), for \(i=1,\ 2,\ 3\), are the hyper-parameters and can be denoted as \(N_{i}\times N_{i-1}\) and \(N_{i}\times 1\), respectively. Note that \(N_{0}\) is the length of the input vector, i.e., \(Q+2\), and \(N_{1}\), \(N_{2}\) and \(N_{3}\) are typically tuned manually by trial and error. The features extracted by the three hidden layers can be expressed in a vector form as:  
\[\mathbf{F}_{k}(\mathbf{\Theta},\mathbf{B})=\varphi_{3}\Big{(}\theta_{3} \varphi_{2}\Big{(}\theta_{2}\varphi_{1}\Big{(}\theta_{1}\widetilde{\mathbf{x }}_{k}^{*}+\mathbf{b}_{1}\Big{)}+\mathbf{b}_{2}\Big{)}+\mathbf{b}_{3}\Big{)} \tag{5}\]  
where \(\mathbf{\Theta}=(\theta_{1},\theta_{2},\theta_{3})\), \(\mathbf{B}=(\mathbf{b}_{1},\mathbf{b}_{2},\mathbf{b}_{3})\), and \(\varphi_{1}\), \(\varphi_{2}\) and \(\varphi_{3}\) are the activation functions for the hidden layers. In this study, the rectified linear Unit (ReLU) activation function was adopted. Subsequently, the extracted features were used to predict the path loss based on non-linear regression. The resulting path loss, \(\hat{y}_{k}\), is given by:  
\[\hat{y}_{k}(\mathbf{\Theta},\mathbf{B},\mathbf{\vartheta},b)=\varphi(\vartheta \mathbf{F}_{k}(\mathbf{\Theta},\mathbf{B})+b) \tag{6}\]  
where \(\vartheta\) is a row vector, \(b\) is a scaler, and \(\phi\) is the sigmoid activation function.