## 3 Deep-Learning-Based Channel Estimation Methods
### Model-Driven Deep-Learning-Based Channel Estimation Methods
#### 3.4.3 Model-Driven Channel Estimation Combining AMP Algorithm  
The AMP algorithm is a significant iterative algorithm that excels at recovering sparse signals. It can achieve channel estimation in sparse channel environments with low computational complexity [78]. Nevertheless, the AMP algorithm's performance is highly dependent on certain preset parameters, and it presents a challenge to determine the optimal parameters, leading to its limited estimation performance. Therefore, some researchers have combined the AMP algorithm with deep learning. This approach allows the deep learning network to learn and optimize the iterative steps of the AMP algorithm, forming a novel and effective channel estimation method. The learned AMP (LAMP) algorithm is a variant of the AMP algorithm that unfolds and maps the AMP algorithm's iterative steps directly into a DNN, thereby utilizing the DNN's ability to jointly optimize the coefficients of the linear transformations and the parameters of the nonlinear shrinkage function. However, the existing LAMP networks may not achieve the desired performance in beamspace channel estimation. In light of this, [PERSON] et al. [79] investigated an enhanced scheme based on a prior-assisted Gaussian mixture LAMP (GM-LAMP). Specifically, the authors replaced the soft threshold shrinkage function in the original LAMP network with a Gaussian mixture shrinkage function, which is able to reflect more of the beamspace channel's prior information. Compared to the OMP, the AMP, and the LAMP algorithms, the GM-LAMP algorithm has a superior NMSE performance with less pilot overhead. The denoising AMP (DAMP) algorithm is also a variant stemming from the AMP algorithm, replacing the shrinkage function with a denoiser. [PERSON] et al. [80] deeply unfolded the DAMP algorithm by replacing its original denoiser with a DnCNN. They developed the model-driven learned denoising AMP (LDAMP) algorithm for channel estimation with noisy channels in OTFS systems. Then, they predicted the theoretical NMSE performance of the LDAMP algorithm using the state evolution (SE) equation. Owing to the integration of the DAMP algorithm's superior performance and the nonlinear fitting capability of deep learning, the proposed algorithm exhibits effectiveness and high accuracy. In 2022, [PERSON] et al. [81] introduced an AMP-based multi-stage scheme with deep learning for quasi-sparse channel environments. This scheme treats the entire system as an end-to-end DNN, and each iterative process of the sensing matrix, noise introduction, and AMP algorithm is regarded as a DNN layer. Firstly, the AMP sensing matrix is trained to adapt to the quasi-sparse channel and learn the optimal sensing matrix. Secondly, the AMP algorithm's nonlinear shrinkage parameters and linear coefficients are optimized layer-by-layer. Finally, all trainable parameters are jointly optimized, thereby optimizing the entire system. Compared to the AMP, the LAMP, and the LDAMP algorithms, the proposed scheme has a superior NMSE performance while reducing pilot overhead. Table 4 summarizes the reviewed model-driven deep-learning-based channel estimation methods and their performance. The model-driven approach utilizes some prior knowledge of communication systems, which can enhance its interpretability. Based on existing models in the physical layer, it decreases the quantity of parameters to be learned and the sample complexity of the learning model. However, the model-driven approach is less flexible, and the model could be negatively impacted if the prior knowledge is inaccurate in practical scenarios. \begin{table}
\begin{tabular}{p{85.4 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline
**Reference** & **Deep Learning Model** & **Input** & **Output** & **Compared with** & **Performance** & **Complexity** \\ \hline [PERSON] et al.