# Adversarial Sample Generation Method Based on Frequency Domain Transformation and Channel Awareness
## 4 Channel Estimation Method Based on Deep Learning
### The Super-Resolution Denoising Network Model Based on OMP Algorithm
#### 4.2.1 Offline Training
As shown in Figure 6, the output of the entire network can be expressed by the following formula:  
\[y^{(i)}_{SDNet}=f(\theta\,;\,x^{(i)}) \tag{27}\]  
where \(\theta\) represents the weight that plays a key role in determining the performance and performance of the model, and \(f(\cdot)\) represents the forward propagation function of the whole neural network. Training network: we will describe in detail how to use the training set to train the proposed OMP-SDNet framework. First, the weight \(\theta\) of each layer needs to be randomly initialized. The purpose of this step is to ensure that the network has a different initial state at the beginning of training so that it can learn different feature representations. Then it enters the forward propagation stage. During this process, each layer of the neural network processes the input data \(x^{(i)}\) in sequence. Firstly, after three convolution and linear activation operations, the purpose is to accurately extract image blocks from the input low-resolution image and map them to the high-resolution space. Then, these high-resolution image blocks are aggregated and integrated to generate the final high-resolution image. The data then passes successively through the module composed of multiple layers of convolution, batch normalization layers, and rectification linear unit activation functions. Figure 5: Network architecture of the SDNet. Figure 6: Network architecture of the SDNet. This module can effectively extract the features contained in the image, and simultaneously complete the tasks of noise removal and image restoration. The output generated by each layer in the above process will be used as the input for the next layer, thus enabling the information to be smoothly transmitted layer by layer until it reaches the output layer. Ultimately, the output result of the network can be represented by \(f(\theta\,;\,x^{(i)})\). The loss calculation: after each step of forward propagation, we calculate the loss between the predicted output \(f(\theta\,;\,x^{(i)})\) and the actual output \(y^{(i)}\). The goal of training is to minimize losses by adjusting network parameters. In this paper, we use MSE to define the loss function of the network:  
\[MSE=\frac{1}{N}\sum_{i=1}^{N}\left(y_{esti}-y_{real}\right)^{2} \tag{28}\]  
where \(y_{esti}\) represents the predicted value, \(y_{real}\) represents the actual value, and \(N\) represents the sample size. Our training goal is to find a well-trained set to \(\hat{\theta}\) to minimize the loss function, so the optimization problem in this paper can be expressed as:  
\[\min_{\theta}\mathcal{L}_{SDNet}(\theta)=\frac{1}{N}\sum_{i=1}^{N}\left\|f( \theta\,;\,x^{(i)})-y^{(i)}\right\|^{2} \tag{29}\]  
In neural network training, we use backpropagation to calculate the loss gradients and apply the Adam optimization algorithm to update the network parameters according to these gradients. When training, the data are divided into small batches and processed in multiple cycles. Each cycle, the model predicts by forward propagation, calculates MSE losses, backpropagation acquires gradients, and updates parameters to minimize losses. After the training is completed, a model with optimal parameters can be obtained, which can accurately predict new data. The final trained model can be expressed as:  
\[x^{(i)}=F\big{\{}\big{[}\text{Re}\big{(}{S_{i}}^{\prime}\big{)},\text{Im} \big{(}{S_{i}}^{\prime}\big{)}\big{]}\big{\}}\in R^{m_{1}\times m_{2}\times 2} \tag{25}\]0  
where \(x\) represents the input data and \(y_{SDNet}\) represents the output data, which needs to be converted into complex value channel information \(H_{SDNet}\). We can express the estimated channel matrix \(H_{SDNet}\) as:  
\[x^{(i)}=F\big{\{}\big{[}\text{Re}\big{(}{S_{i}}^{\prime}\big{)},\text{Im} \big{(}{S_{i}}^{\prime}\big{)}\big{]}\big{\}}\in R^{m_{1}\times m_{2}\times 2} \tag{25}\]1