# A Joint Denoising Learning Model for Weight Update Space-Time Diversity Method +
## 4 A Joint Denoising Learning Model for Weight Update STD Method
### The STD Model Based on Weight Update Strategy  
The use of CDMA for the transmit signal effectively simplifies the time-domain equalization operation in the STD structure. Still, MAI cannot be avoided by the system, and this is because there is a certain correlation between the signals in each path, while the received signals are mixed in the time- and frequency-domains, resulting in a severe deterioration of the SNR, which causes MAI. The conventional STD model only targets the primary path signal of the desired user, while other path signals are suppressed as interference and cannot effectively suppress MAI. For this purpose, we propose an improved STD model in the second stage, matching a weight matrix for each array element at the receiver end. The model enables the main path signal to be enhanced and effectively reduces the influence of other multipath signals, thus reducing the correlation between signals and suppressing MAI in the system. The received signals of each path are then combined by the maximum SNR principle, which significantly improves the BER of the system. Figure 13 shows the structural composition of the STD model based on the weight update strategy. In the above structure, how to obtain the weight matrix accurately is the key to the algorithm. The weight matrix is used as a constraint that maximizes the signal's weight direction. At the same time, the signals from other multipath directions are suppressed as much as possible. Considering the path \(p\) that first arrives at the receiver unit as the desired signal, assume the existence of a weight update matrix:  
\[\mathbf{w}=\big{[}\mathbf{w}_{p},\mathbf{w}_{2},\ldots,\mathbf{w}_{l}\big{]}^{T}, \tag{17}\]  
Then, the DOA of the desired direction is obtained by the array processing method, denoted as \(\theta_{p}\). To ensure that the desired signal from direction \(\theta_{p}\) is received correctly and to maximize the suppression of the other \(l\)-1 branches' interference, the constraint of \(\mathbf{w}_{p}\) can be written as Equation (18). \[\mathbf{z}=\left\{ \begin{array}{l}\mathbf{w}_{p}\mathbf{A}_{l}(\theta_{p})=1 \\ \mathbf{w}_{i}\mathbf{A}_{l}(\theta_{l})=0;i=1,\ldots,L-1\end{array} \right., \tag{18}\]  
\(L\) indicates the number of beamformers. By weighting and summing the outputs of each array element, the array beam is directed in one direction to achieve directional reception of the signal, thereby enhancing the signal and suppressing interference. The output power \(P_{\text{out}}\) of the received signal is denoted as:  
\[P_{\text{out}}=\mathbf{E}\Big{\{}|\mathbf{z}(t)|^{2}\Big{\}}=\mathbf{E}\Big{\{} \Big{[}\mathbf{w}^{H}\mathbf{y}(t)\Big{]}\Big{[}\mathbf{y}^{H}(t)\mathbf{w} \Big{]}\Big{\}}=\mathbf{w}^{H}\mathbf{R}_{yy}\mathbf{w}, \tag{19}\]  
In the above Equation (19), \(\mathbf{R}_{yy}\) is the covariance matrix of the received signal. In the algorithm of this paper, the linearly constrained minimum variance (LCMV) beamformer is used. The optimal weights are based on the beamformer output power being the minimum. It is also guaranteed that the output of the desired signal is 1 in Equation (18). The weight vector of each receiving array element can be found by Equation (20)  
\[\mathbf{w}_{p}=\arg\min_{\mathbf{w}}P_{\text{out}}\ s.t.\mathbf{w}_{p}^{H} \mathbf{A}\big{(}\theta_{p}\big{)}=1, \tag{20}\]  
According to Equation (20), the output signal of each receiving array element, i.e., the output \(\mathbf{y}_{dv}(k)\) of each diversity receive branch, is  
\[\mathbf{y}_{dv}(k)=\mathbf{w}_{p}^{H}\mathbf{y}(k)=\mathbf{w}_{p}^{H}\left( \sum_{l=1}^{L}\delta_{l}\mathbf{A}_{l}\mathbf{s}_{l}(k)\right)+\mathbf{w}_{p}^ {H}\left(\sum_{l=1}^{L}\mathbf{A}_{l}\mathbf{s}(k)+\mathbf{n}(k)\right), \tag{21}\]  
\(\mathbf{y}(k)\) is a discretized representation of \(\mathbf{y}(t)\), where the continuous-time variables \(t=kT_{b}\), \(p=1,2,\ldots,L\). STD combining follows the space-domain receiving and time-domain weighting processing. It combines the received signals of each diversity branch according to the corresponding strategy.