# 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  
The architecture of the SDNet framework consists of the following parts. Training data generation: prepare a large amount of labeled training data represented as:  
\[(x,y)=(x^{(1)},y^{(1)}),\cdots,(x^{(N)},y^{(N)}) \tag{24}\]where \(N\) represents the number of training samples, each \((x^{(i)},y^{(i)})\) represents the \(i\)th \((i\in{1,2,\cdots,N})\) training samples in the dataset, where \(x^{(i)}\) represents the input data (features) and \(y^{(i)}\) represents the output data used to train the neural network. According to Equation (3), for the \(i\)th user, the input data should be the sent signal \(S_{i}\) and set the corresponding output as the channel impulse response \(H_{i}^{emp}\). Since the inputs and outputs of the neural network must be real numbers, we need to convert the complex-valued matrix into a real-valued matrix. For the input \(S_{i}\in C^{m\times 1}\), we reshape it into a 2D matrix \({S_{i}}^{\prime}\in C^{m_{1}\times m_{2}}\), where \(m=m_{1}\times m_{2}\), and both \(m_{1}\) and \(m_{2}\) are integers. Then, by superimposing the real part and the imaginary part into two channels, the input of OMP-SDNet can be represented as:  
\[x^{(i)}=F\big{\{}\big{[}\text{Re}\big{(}{S_{i}}^{\prime}\big{)},\text{Im} \big{(}{S_{i}}^{\prime}\big{)}\big{]}\big{\}}\in R^{m_{1}\times m_{2}\times 2} \tag{25}\]  
where \(F(\cdot\;,\;\cdot)\) represents conversion to a real function. For the label \(X_{reali}\), the real part and the imaginary part are superimposed in turn and reconstructed into a vector with a real value of \(2m\), which can be expressed as:  
\[y^{(i)}=\Big{[}\text{Re}\Big{\{}H_{i}^{emp}\Big{\}},\text{Im}\Big{\{}H_{i}^{ omp}\Big{\}}\Big{]}\in R^{2m\times 1} \tag{26}\]  
Network architecture SRCNN\(\rightarrow\)DnCNN\(\rightarrow\)FC: the proposed network is shown in Figure 5. Compared with the classical convolutional layer, the network has better performance in extracting channel features and de-noising. The network architecture adopts a three-level series structure of SRCNN\(\rightarrow\)DnCNN\(\rightarrow\)FC. First, through the three-layer convolution of SRCNN (Conv1: \(3\times 3\times 64\), ReLU; Conv2: \(3\times 3\times 128\), ReLU; Conv3: \(3\times 3\times 256\), ReLU) to achieve super-resolution reconstruction of channel features and extract high-frequency details; subsequently, the five-layer residual module of DnCNN (each module contains \(3\times 3\) convolution, batch normalization and ReLU activation) is connected to sup  
Figure 4: Deep neural network-based channel estimation framework. press the noise layer by layer and retain the effective signal. Finally, the features are mapped to the channel estimates through the fully connected layer (FC), and the dual-channel results of the real part and the imaginary part are output. First comes the input layer \(x^{(i)}\), which is responsible for receiving and carrying the input data. The second, third, and fourth layers immediately following are convolutional layers, each equipped with a different number of filters with varying sizes of convolution kernels. After each convolutional layer, a rectification linear unit activation function is connected to perform initial feature extraction and transformation of the data. At this point, the obtained intermediate output is saved. Subsequently, the data flows through multiple convolutional layers of size (\(3\times 3\)) with the same fill settings. Each Conv layer here is successively connected to a batch normalization layer and a rectification linear unit activation function. Through this series of processing, the noisy image is obtained. Ultimately, subtract the resulting noisy image from the previously saved output, and the difference obtained is the final output result of the entire network. It is worth emphasizing that each connection in the above architecture corresponds to a specific weight.