# 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
However, the time-series characteristics between signals are destroyed after using the noise-reduction learning process. To reduce the loss of temporal characteristics and make the estimation of time delay in the STD operation more accurate, the TDNN is used after the CGAN noise-reduction process. The TDNN is a deep learning network dedicated to time-series correlation processing, which can effectively restore the temporal characteristics of the data [70]. The TDNN stores the above temporal information through a time delay neuron (TDN). The network structure of the TDNN is shown in Figure 12.  
The TDNN is a DL model with a simple structural form; the proposed model is structured as a three-layer network with input, hidden, and output layers. The input is the data after RCL processing, i.e., \(\mathbf{TR}_{g}\), and the output is the time-series-adjusted \(\mathbf{TR}_{g}\). \(\mathbf{TR}_{g}\) and \(\mathbf{TR}_{tg}\) are \(4\times 128\) matrices. The input layer can be thought of as a time delay block for adjusting the time delay. The hidden layer consists of 64 TDNs; each TDN contains two convolutional layers and an activation function, and the convolutional layer size is \(3\times 3\). The TDNN has 128 convolutional kernels for convolutional operations in the hidden layer. The output layer is used to implement a nonlinear regression with time delay. Take \(\mathbf{TR}_{g}\) as an example. When it is input into the TDNN structure, its current moment and the previous \(N\) moments are stored in the TDNN. The output of each TDN can be represented as Equation (12). Figure 8.0  
\(O_{TDN}(t)\) is the output; \(M\) denotes the number of received signals; \(N\) represents the Nth moment. \(\mathbf{TR}_{gi}\) indicates the \(i\)th input; _weight_ is the TDN weight, where \(\mathbf{TR}_{gi}(t-d)\)  
Figure 8.1 Structural composition of the RCL module. Figure 8.2 Network structure composition of TDNN. \(d=1,\ldots,N\) is the \(i\)th input layer node and \(b_{i}\) is the bias of the \(i\)th input. \(fun\) is the excitation function. From Equation (12), it can be seen that the neuron output is determined by the current moment of each input and its data from the last \(N\) moments together. Therefore, the TDNN can effectively handle nonlinear dynamic timing problems. After the time-series adjustment, the output of each neuron in the output layer of the TDNN is shown in Equation (13):  
Figure 8.3  
where \(r\) is the output layer node number and \(R\) is the number of output layers. \(v_{jd}^{r}\) is the TDNN weight of the connection between the \(r\)th output layer node and the implied layer node. \(v_{j}^{r}\) is the bias of the \(r\)th output layer node. \(\mathbf{H}^{i}\) indicates the input of the \(i\)th hidden layer. \(O^{\prime}(t)\) is the output, which is the result of each node being restored by the time-series, and this result is \(\mathbf{TR}_{ij}\). Similarly, when the input is \(\mathbf{TI}_{g}\), the output is \(\mathbf{TI}_{ig}\). Because compression is performed in the pre-processing stage, decompression is required during signal reduction, which the following equation can achieve:  
Figure 8.4 Figure 8.5  
\(\mathbf{\hat{R}}(t,f)\) and \(\mathbf{\hat{I}}(t,f)\) represent the real and imaginary parts after time-series restoration and decompression with the noise-reduction effect. \(\mathbf{\hat{R}}(t,f)\) and \(\mathbf{\hat{I}}(t,f)\) are restored to the time-domain by the inverse Fourier transform and combined to obtain Equation (16), where \(j\) denotes the imaginary part. Figure 8.6  
\(\mathbf{\hat{y}}(t)\) is the result of the first stage of noise-reduction about \(\mathbf{y}(t)\).