# Channel Estimation for UAV Communication Systems Using Deep Neural Networks
## 2 System Model and Problem Formulation
### Problem Formulation  
GBRBM is known as Markov random fields (MRFs) [17] and is an undirected probabilistic graphical model. The input layer represents the observed data consisting of nine nodes, each of which represents a specific kind of input data, N = (N1 \(-\) N9), where N1 represents the latitude, N2 represents the longitude, N3 is the drone ground elevation, N4 and N5 represent the cell latitude and longitude, respectively, n6 represents the cell elevation, N7 is the cell building, N8 is the antenna mast height, and N9 represents the drone altitude. Figure 2 illustrates the diagram enhanced GBRBM with DNN. This architecture is designed to operate systematically on the principle of the proposed algorithm. First, data are collected during the real-time data collecting stage. Next, the raw data will be sent to the pre-training stage. Then, the data are exploited for feature extraction using multi-block enhanced GBRBM; next, we send the pre-trained data to the fine-grained tuning. Finally, the output data of the extracted features are classified using the softmax regression module to obtain the output. The energy function of GBRBM is defined as:  
\[E(v,h|\theta)=\sum_{i=1}^{n_{v}}\frac{(v_{i}-b_{i})^{2}}{2\sigma^{2}}-\sum_{i= 1}^{n_{v}}\sum_{j=1}^{n_{h}}W_{ij}h_{j}\frac{v_{i}}{\sigma_{i}}-\sum_{j=1}^{n_ {h}}c_{j}h_{j} \tag{1}\]  
where \(b_{i}\) represents the bias of the visible layer, \(c_{j}\) represents the bias of the hidden layer, and \(w_{ij}\) represents the weight that connects the visible layer to the hidden layer. Furthermore, \(\sigma_{i}\) accounts for the standard deviation of the visible units. Each block has nine real input data, as mentioned above, while \(n>1\) for hidden layer unit \(h_{j}\). Based on the property of MRFs and the energy function, the joint probability distribution is defined as:  
\[P(v,h)=\frac{1}{Z}e^{-E(v,h)}, \tag{2}\]  
where Z is the partitioning function, as given below  
\[Z=\sum_{v}\sum_{v}-E(v,h), \tag{3}\]By using the joint probability function, the marginal distribution of v can be defined as:  
\[P(v)=\frac{1}{Z}e^{-E(v,h)}, \tag{4}\]  
The hidden and visible layers are both conditionally independent. The condition probability of \(v\) and \(h\) are defined as follows:  
\[P(v_{i}=v|h)=N\left(v|b_{i}+\sum_{j}h_{j}W_{ij}\sigma_{i}^{2}\right), \tag{5}\]  
\[P(h_{j}=1|v)=\text{sigmoid}\left(c_{j}+\sum_{i}W_{ij}\frac{v_{i}}{\sigma_{i}^ {2}}\right), \tag{6}\]  
where \(N\) represents the Gaussian probability density function and denotes the variance with mean. The stochastic maximization of likelihood is used to train GDBM. The likelihood is estimated by marginalizing the hidden neurons. The partial derivative of the maximization log-likelihood function is given by:  
\[\frac{\partial L}{\partial\theta}a\left\langle\frac{\partial\left(-E\left(v^{ (t)},h|\theta\right)\right)}{\partial\theta}\right\rangle_{d}-\left\langle \frac{\partial\left(-E\left(v,h|\theta\right)\right)}{\partial\theta}\right\rangle _{m} \tag{7}\]  
where \(\left\langle.\right\rangle_{d}\) and \(\left\langle.\right\rangle_{m}\) account for the expectation of model distribution \(P(v,h|\theta)\) and data distribution \(P(h|\{v^{(t)}\},\theta)\), respectively. Here, \(\theta\) is the parameter of GBRBM because gradient calculation needs a high computational cost. Reference [18] used a contrastive-divergence (CD) learning that proved to be an efficient approximator for the log-likelihood gradient for GBRBM. The CD learning is recalled as an alternative to calculating the second term of the log-likelihood gradient by iteration a few samples from the data by using Gibbs sampling. As a result, GBRBM parameters are derived as follows:  
Figure 2: Framework architecture of the proposed enhanced GBRBM-based DNN.