# A Joint Denoising Learning Model for Weight Update Space-Time Diversity Method +
## 4 A Joint Denoising Learning Model for Weight Update STD Method
### Noise-Reduction Learning Models Based on Complex-CGAN
After the above feedback learning, we can assume that the generative ability of the GM and the discriminative ability of the DM will be improved. The purpose of this stage is to estimate a noise component \(\mathbf{TR}_{e}\) by learning the mapping so that it infinitely approximates \(\mathbf{TR}_{hi}\). Figure 10 shows the detailed structural setup of the DM in the complex-CGAN. The input \(\mathbf{TR}_{y}\) of the GM is collapsed into \(\big{\{}\mathbf{TR}_{y1},Id_{y1}\big{\}},\ldots,\big{\{}\mathbf{TR}_{yi},Id_{yi} \big{\}}\), where \(Id\) is to label the input features, and the input feature size is \(4\times 32\). Similarly, the set of features of DM is obtained as \(\big{\{}\mathbf{TR}_{hi},Id_{hi}\big{\}},\ldots,\big{\{}\mathbf{TR}_{hi},Id_{hi} \big{\}}\). In the GM, the feature set is adjusted to a \(4\times 32\) vector of data features after reshaping. Then, after the pooling layer, it will be adjusted to a \(4\times 4\) downsampling matrix. In the DM, although there are two pooling layers, they will still be adjusted to a \(4\times 4\) feature matrix. We used categorical conditional batch normalization (CatBN) as a criterion in batch normalization, which can further reduce orthogonality loss. The activation function CReLU is a complex-type ReLU that satisfies the Cauchy-Riemann equation [67]. In this paper, a loss function with the \(L1\) loss term \(\zeta_{L1}(G)\) is used to optimize the network [68] with the following objective function:  
\[G^{*}=\arg\min\max V_{CGAN}(G,D)+\lambda\zeta_{L1}(G), \tag{9}\]  
\[\zeta_{L1}(G)=E_{n,y}\Big{[}\big{\|}G\big{(}\mathbf{TR}_{yi},\mathbf{TR}_{hi} \big{)}\big{\|}_{1}\Big{]}, \tag{10}\]  
where \(\lambda\) is the weight of the \(L1\) loss term and \(G^{*}\) denotes the optimized GM. \(E[\cdot]\) denotes the estimated value after each round of calculation, and for \(\mathbf{TR}_{y}\), the estimated value is \(\mathbf{TR}_{e}\). \(G[\cdot]\) and \(D[\cdot]\) denote the outputs of the GM and DM [69], respectively, which are both upsampled to a matrix of size \(4\times 128\). The GM has \(3\) convolutional layers of \(1\times 1\), \(3\) convolutional layers of \(3\times 3\), and \(1\) convolutional layer of \(4\times 4\). The DM has \(6\) convolutional layers of \(1\times 1\) and \(3\) convolutional layers of \(3\times 3\). \(V_{CGAN}(G,D)\) denotes the standard CGAN loss function:  
\[\min_{G}\max_{D}V_{CGAN}(D,G)=E_{n,y}\big{[}\log D\big{(}\mathbf{TR}_{yi}, \mathbf{TR}_{hi}\big{)}\big{]}+E_{n,y}\big{[}\log\big{(}1-D\big{(}\mathbf{TR} _{yi},\mathbf{TR}_{hi}\big{)}\big{)}\big{]}, \tag{11}\]  
When Equation (9) reaches the Nash equilibrium, the GM and DM stop their operations, and the estimated value of the last round is obtained as \(\mathbf{TR}_{e}\). Similarly, for the imaginary part input feature \(\mathbf{TI}_{y}\), \(\mathbf{TI}_{e}\) is also obtained after the training of this model. The RCL module references the traditional ANC noise cancellation technique for residual learning, and it is worth noting that RCL is a dynamic process. When Equation (9) reaches the Nash equilibrium, \(\mathbf{TR}_{y}\) and \(\mathbf{TR}_{e}\) are processed by RCL, and the output obtains \(\mathbf{TR}_{y}\) with a noise-reduction effect. The proposed model uses ResNet as the residual network and designs it as a combined inception and residual structure. The residual learning  
Figure 10: The detailed structural setup of the DM in the complex-CGAN. module is also equipped with the same arithmetic function for complex-type data. The RCL structure used is defined in Figure 11. In the RCL module, for the real part, the inputs are \(\mathbf{TR}_{y}\) and \(\mathbf{TR}_{e}\), both of which are \(4\times 128\) matrices. After RCL processing, \(\mathbf{TR}_{g}\) is obtained. Similarly, for inputs \(\mathbf{TI}_{y}\) and \(\mathbf{TI}_{e}\), \(\mathbf{TI}_{g}\) is obtained. Since the input to the system is a UWA signal with time-series characteristics, the time delay between the received signals needs to be obtained when STD processing is performed on the noise-reduction results.