# Adversarial Sample Generation Method Based on Frequency Domain Transformation and Channel Awareness
## 4 Channel Estimation Method Based on Deep Learning
### Residual Channel Estimation Framework for Super-Resolution Denoising Based on Deep Learning
#### 4.1.1 Date Preparation  
In the channel estimation task, in order to simulate the diversified channel environment in a real communication scenario and improve the generalization of the model, we generated signals with different signal-to-noise ratios (SNRs). First, an original signal is generated at the transmitter, which can be represented as a complex sequence:  
\[x(t)=x_{real}(t)+j\,x_{imag}(t) \tag{13}\]  
where \(x_{real}(t)\) and \(x_{imag}(t)\) represent the real and imaginary components of the signal. Subsequently, we transmit this signal through the channel and receive a signal with added noise at the receiving end. To simulate various SNRs, Gaussian white noise of varying intensity is added to the received signal for each ratio. Specifically, for each SNR, we add noise and generate the corresponding received signal:  
\[y_{i}(t)=x(t)+z_{i}(t) \tag{14}\]  
where \(z_{i}\) represents Gaussian white noise. \(y_{i}\) represents the signal received by the receiver. Next, the generated signals are longitudinally concatenated according to different signal-to-noise ratios to construct a comprehensive training dataset containing multiple channel environments. Suppose \(N\) signal segments with different signal-to-noise ratios are generated. The operation of vertical splicing can be expressed as:  
\[Y=\left[ \begin{array}{c}y_{1}(t)\\ y_{2}(t)\\ \vdots\\ y_{N}(t)\end{array} \right] \tag{15}\]  
In this way, a matrix containing signal segments with different signal-to-noise ratios was generated, which can simulate the diverse channel conditions in the actual communication environment more comprehensively. In channel estimation, the orthogonal matching pursuit (OMP) algorithm is widely used as an effective sparse signal recovery technique. Therefore, the OMP estimator will be used next to estimate the generated and processed signals. It is applicable to the estimation problem of sparse signals, with low computational complexity and good noise robustness. This algorithm adopts the idea of the greedy algorithm. Essentially, it selects the atoms with the greatest correlation to the signal from a group of atoms called the atomic set. Next, it uses the known atoms to construct estimation coefficients, gradually reduces the residuals, and obtains a sparse signal representation. This algorithm is easy to understand and can accurately restore high-dimensional sparse information. First, the residual \(r\) is initialized, and the atom with the highest correlation to the residual is selected from the set of atoms:  
\[\lambda_{k}=\underset{i=1,\ldots,N}{\arg\max}|\langle r_{k},A(i)\rangle|,k=0,1,\ldots,K \tag{16}\]where \(\lambda_{k}\) is the \(k\)th column index determined by the maximum absolute value of the correlation; \(r_{0}=b\) represents the residual of the first iteration; \(A(i)\) is the \(i\)th term of \(A\), representing the estimated value of the channel matrix. Next, the column index with the highest correlation is added to the index set \(\Lambda^{k}\), and the data corresponding to these indicators in the observation matrix is updated to the reconstructed set of atoms \(A_{\Lambda^{k}}\):  
\[\Lambda^{k}=\Lambda^{k-1}\cup\{\lambda_{k}\} \tag{17}\]  
\[A_{\Lambda^{k}}=A_{\Lambda^{k-1}}\cup A(\Lambda^{k}) \tag{18}\]  
Using the LS method, we obtain an approximate solution:  
\[x_{k}=\operatorname*{arg\,min}_{x}\left\|A_{\Lambda^{k}}x-b\right\|_{2}=A_{ \Lambda^{k}}^{-1}b \tag{19}\]  
Calculate the new approximation of the data and the new residual:  
\[r_{k}=b-b_{k} \tag{20}\]  
\[b_{k}=Ax_{k} \tag{21}\]  
This algorithm selects the vector that best matches the current residual through repeated iterations, and performs orthokerization processing on the selected vectors at each step to gradually construct an approximate solution. When the residuals are small enough or reach the preset sparsity, the iteration terminates. The final solution is a linear combination of all the previously selected vectors.