# Underwater-Acoustic-OFDM Channel Estimation Based on Deep Learning and Data Augmentation
## 3 The UWA-OFDM-Channel-Estimation Framework Based on Deep Learning
### GAN-Based UWA-OFDM Channel Sample Enhancement  
The GAN structure is shown in Figure 3, including the generator and the discriminator. In the generator, to learn the distribution \(p_{g}\) on the UWA channel data \(h\), we randomly define a prior noise variable \(z\) as input that satisfies a normal distribution or other arbitrary distribution. The generator network \(G(z;\theta_{g})\) maps the latent distribution \(p_{z}\) of the latent variables \(z\) into the data distribution \(p_{g}\), thus obtaining a simulated sample that fits the distribution \(p_{g}\), which is a differentiable function characterized by the multilayer perceptron \(G\). In the discriminator, we need to define a multilayer perceptron \(D(z;\theta_{d})\). \(D(z;\theta_{d})\) denotes the probability of converting \(h\) into a sample of actual data that fits the distribution \(p_{data}\) while different from \(p_{g}\). The role of the generator is to generate false data similar to the real data, while the discriminator is responsible for classifying the real data and the false data. We train \(D\) to maximize the probability of correctly assigning true samples and generating samples, so we can train \(G\) simultaneously by minimizing the \(\log(1-D(G(z)))\). That is to say, the discriminator \(D\) and the generator \(G\) play a minimax game on the value function \(V(G,D)\):  
\[ \begin{split}\min_{\theta_{g}}\max_{\theta_{d}}V(D,G)=& E_{h\sim p_{data}(h)}[\log D(h;\theta_{d})]+\\ & E_{z\sim p_{z}(z)}[\log(1-D(G(z;\theta_{g});\theta_{d}))],\end{split}  \tag{5}\]  
where \(\theta_{g}\) and \(\theta_{d}\) are the training parameters of the network. Furthermore, the generative and discriminant networks alternatively optimize the parameters \(\theta_{g}\) and \(\theta_{d}\), based on their respective loss functions. That is, the parameters of one network (i.e., \(G\) or \(D\)) are calculated by using the error-Back-Propagation-(BP) algorithm and the optimization method (e.g., the gradient-descent method), while the parameters of the other network are fixed. By iteratively optimizing the parameters of the GAN network, the generative and discriminant networks learn reasonable mapping functions. This means that the distribution \(p_{g}\) of the simulated samples generated by the generator network converges to the actual data distribution \(p_{data}\), and the discriminator network cannot distinguish the simulated samples that match the distribution \(p_{g}\) from the actual sample (matching the distribution \(p_{data}\)). Figure 2: The UWA-OFDM-channel-estimation framework based on deep learning. GANs are very clever in theory. We should first train the discriminator as well as possible during the training process. But, in practice, the better the discriminator is trained, the more difficult is the generator to optimize, which eventually leads to a GAN training crash. Therefore, we adopted the Wasserstein-GAN-with-Gradient-Penalty-(WGAN-GP) architecture, to alleviate the instability during GAN training [24], as shown in Figure 4. By introducing the Wasserstein distance, which has smoothing properties superior to the Jensen-Shannon (JS) divergence, the gradient-vanishing problem can be theoretically solved. Then, we used a discriminator neural network with a limited numerical range to maximize the mathematical transformation of the Wasserstein distance in a solvable form, and the Wasserstein distance could be approximated. In the regime of this approximate optimal discriminator, optimizing the generator serves to decrease the Wasserstein distance, effectively bringing the generated distribution closer to the real distribution. The WGAN not only addresses the issue of unstable training but also provides a reliable metric for tracking training progress, with this metric being highly correlated to the quality of the generated samples. The WGAN-GP incorporates a gradient penalty [25], replacing the weight-clipping method used in the WGAN. The gradients of the discriminator are restricted by directly imposing an additional penalty term.