# Adversarial Sample Generation Method Based on Frequency Domain Transformation and Channel Awareness
## 2 Background Introduction  
Channel estimation is essential for ensuring communication quality, with traditional OFDM-based methods exhibiting distinct advantages and limitations. The least squares (LSs) method [5, 6], widely used for its computational simplicity, suffers from severe performance degradation in low-SNR scenarios due to its disregard for noise. Discrete Fourier transform (DFT)-based techniques [7] effectively suppress noise through frequency-domain filtering while maintaining low complexity; however, their performance rapidly deteriorates when frequency offsets exceed 5% of the subcarrier spacing. The linear minimum mean square error (LMMSE) method [8, 9], though capable of achieving near-optimal estimation under moderate-to-high SNRs, depends on prior knowledge of channel covariance and exhibits cubic computational complexity, limiting its scalability in large-scale MIMO systems. Recent developments in deep learning have introduced data-driven alternatives. Deep neural networks (DNNs) [10, 11, 12] demonstrate strong performance across varying pilot lengths through end-to-end learning, while convolutional neural networks (CNNs) [13] utilize pilot position information to enhance estimation accuracy. Advanced frameworks such as CsiNet and CsiNet-LSTM [14] improve the robustness of CSI feedback, and tensor-train DNNs (TT-DNNs) [15] reduce parameter dimensionality for high-dimensional CSI, albeit with slower convergence. Hybrid approaches that integrate traditional techniques with DNNs [13, 16] seek to balance complexity and adaptability, for instance, by leveraging spectral-time averaging to track channel variations. For sparse channel estimation, traditional algorithms such as orthogonal matching pursuit (OMP) [17] remain prevalent, though their effectiveness heavily relies on accurate sparsity level estimation. Enhanced OMP variants [14, 15, 18, 19] exploit angular sparsity or introduce adaptive mechanisms to improve robustness, often at the cost of increased complexity. In contrast, deep learning-based methods, including CNN-based MIMO-OFDM estimators [20] and DNN-enhanced OTFS systems [21, 22] have shown superior performance over OMP, particularly in delay-Doppler domains. Channel estimation guarantees communication quality and is related to the accuracy and stability of signal transmission. Modulation classification, as the basis of communication systems, is a key prerequisite for subsequent signal processing and information interpretation. The two are closely related and jointly promote the development of communication technology. Under this broad background, the security and stability of communication systems have become key research directions, and adversarial attacks and channel estimation are precisely the core issues among them. In the field of adversarial attacks, many research achievements have been remarkable. In the early days, after [PERSON] et al. [23] discovered that deep neural networks were vulnerable to slight adversarial interference, [PERSON] et al. [24, 25] proposed the adversarial transformation-enhanced transfer Attack (ATTA), which constructs an adversarial transformation network through adversarial learning to generate adversarial noise and thereby resist the distortion problem caused by the network. The high-frequency component semantic similarity attack proposed by [PERSON] et al. [26] focuses on the high-frequency noise of the image. [PERSON] et al. 's [27] adversarial attack method based on adversarial generative networks (GANs), adGAN, reduces the performance of intelligent systems by leveraging adversarial generative networks. [PERSON] et al. [28] studied the perturbation pattern analysis of radio frequency signals. [PERSON] et al. [29] utilized Hamiltonian Monte Carlo to generate a series of adversarial samples. With the in-depth research on the security of communication systems, the correlation between modulation classification and adversarial attacks has gradually emerged. In the context of modulation classification, [PERSON] et al.