# Adversarial Sample Generation Method Based on Frequency Domain Transformation and Channel Awareness
## 3 Related Work  
### Mathematical Model of OFDM Channel Estimation System  
The channel estimation system model consists of two parts: the transmitting end and the receiving end. At the transmitting end, the input bit stream is modulated and mapped onto the OFDM subcarriers, and pilot symbols are inserted at specific subcarrier positions for subsequent channel estimation. Then, the frequency-domain signal is converted into a time-domain signal through the inverse fast Fourier transform (IFFT), and a cyclic prefix is inserted. CP is used to resist Inter-symbol interference (ISI), and finally the signal is transmitted through the additive white Gaussian noise (AWGN) channel. At the receiving end, the signal successively removes the cyclic prefix, undergoes fast Fourier transform (FFT) to convert back to the frequency domain, extracts the pilot symbol, and uses the pilot signal for channel estimation, thereby obtaining the channel state information (CSI). Subsequently, the channel influence is eliminated through frequency-domain equalization, the amplitude and phase of the signal are restored, and then the original bit stream is restored through subcarrier demapping and demodulation. The specific process is shown in Figure 1, which shows the training data sources of SDRNet (such as OFDM pilot signals) and the processing flow. According to the above system model block diagram, we can further establish the above signal processing flow as a mathematical model as follows. Suppose the total number of subcarriers in an OFDM system is \(N\), and a subframe contains \(I\) OFDM symbols in total. For the \(i\)th OFDM symbol, suppose the \(n\)th subcarrier symbol on the \(i\)th symbol is \(s_{i}(n)\), and there is \(s_{i}=[s_{i}(0),\ldots,s_{i}(n),\ldots,s_{i}(N-1)]^{T}\). After OFDM modulation of the frequency-domain symbol through discrete Fourier transform (IFFT):  
\[S_{i}=F^{H}s_{i} \tag{1}\]  
where \(S_{i}\in\mathbb{C}^{N\times 1}\) represents the transmitted time-domain sequence and \(F^{H}\) is the Fourier transform matrix, specifically expressed as:  
\[[F]_{n,k}=\frac{1}{\sqrt{N}}\exp(-j\frac{2\pi}{N}kn) \tag{2}\]  
The OFDM transmission model can be further constructed as follows:  
\[y_{i}=H_{i}s_{i}+z_{i} \tag{3}\]  
where the frequency-domain symbol vector received on the \(i\)th OFDM symbol block is \(y_{i}=[y_{i}(0),\ldots,y_{i}(n),\ldots,y_{i}(N-1)]^{T}\), \(z_{i}\) is the additive white Gaussian noise of the channel, the covariance matrix is \(Q_{z}=\sigma^{2}I_{N}\), \(\sigma^{2}\) is the noise variance, \(I_{N}\) represents the \(N\times N\) dimensional identity matrix, and \(H_{i}\in\mathbb{C}^{N\times\text{N}}\) represents the frequency-domain response matrix of the channel on:  
\[H_{i}=FG_{i}F^{H} \tag{4}\]  
where \(G_{i}\in\mathbb{C}^{N\times\text{N}}\) represents the impulse response matrix of the \(i\)th symbolic time channel, expressed as:  
\[G_{i}= \begin{bmatrix}\text{h}_{i}(0,0)&0&\ldots&\text{h}_{i}(0,\text{L}-1)& \ldots&\text{h}_{i}(0,1)\\ \text{h}_{i}(1,1)\,\text{h}_{i}(1,0)&0&\ldots&\ldots&\text{h}_{i}(1,2)\\ \vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\ 0&\ldots&0&\text{h}_{i}(\text{N}-1,\text{L}-1)\ldots\text{h}_{i}(\text{N}-1,0 )\end{bmatrix}  \tag{5}\]  
where \(h_{i}(k,l)\) represents the \(k\)th sampling point of the channel impulse response at the \(l\)th tap at the \(i\)th symbol time. Equation (3) can be considered as the mathematical model of baseband signal transmission in the OFDM system, and \(H_{i}\) is the parameter to be estimated in the channel estimation link. For the channel estimation method of OFDM systems, the first step is to estimate the channel response at the pilot symbol position, and then further obtain the channel response on the complete resource block through time-domain interpolation and frequency-domain interpolation.