# A Joint Denoising Learning Model for Weight Update Space-Time Diversity Method +
## 3 System Model for UWA Communication  
In this section, the transmit-receive model of UWA array communication and the structural characteristics of VURN are described.  
This research focuses on a single-input multiple-output (SIMO) system. A typical SIMO array communication model is shown in Figure 3. When the towing ship is close to the receiving array, the noise is mainly generated by the mechanical vibration of the power source and propeller friction [60]. These noises will be superimposed together with the multipath to be received by the receiving array. The red line in Figure 3 indicates the main path, and the remaining black dashed lines are the multipath signals generated by reflections from the sea surface and the seafloor.  
In the SIMO system, the baseband signal at the transmitter is \(\mathbf{s}(t)\) and the received signal is \(\mathbf{y}(t)\). \(\mathbf{s}(t)\) is reflected to produce \(L\) multipath signals, and the main path is labeled as \(l\). The angle between the principal path and the baseline is called the direction of arrival and is denoted as \(\theta\). \(\mathbf{y}(t)\) can be expressed as:  
\[\mathbf{y}(t)=\left[\begin{array}{c}\mathbf{y}_{1}(t)\\ \mathbf{y}_{2}(t)\\ \vdots\\ \mathbf{y}_{M}(t)\end{array}\right]=\sum_{l=1}^{L}\delta_{l}h_{l}\mathbf{s}(t +\tau_{l})+\mathbf{n}(t), \tag{1}\]  
where \(\tau_{l}\) and \(\delta_{l}\) are the time delay and channel fading of the \(l\)th path signal to the receive array, respectively. \(A_{l}\) is the channel impact response of the \(l\)th path signal arriving at the receive array. The transmit signal uses CDMA, and \(\mathbf{s}(t)\) in Equation (1) can be expressed after spread spectrum processing as:  
\[\mathbf{s}(t)=\sqrt{P}\mathbf{b}(t)\sum_{n=0}^{N_{0}-1}\mathbf{c}(k)\mathbf{p }_{c}[(t-N_{0})T_{c}], \tag{2}\]  
where \(P\) is the transmit power of the signal, \(\mathbf{b}(t)\) is the information bits, \(\mathbf{c}(k)\) is the spreading code sequence, \(T_{c}\) is the spreading code period, \(N_{0}\) is the spreading code length, and \(\mathbf{p}_{c}(t)\) is the pulse waveform of the code slice.  
\(\mathbf{n}(t)\) is the noise, i.e., VURN, which is the focus of this paper. When the tugboat is close to the array, the VURN can radiate through the seawater to the receiving array, causing a severe impact on the communication. The sources of VURN generation are the power system of the tugboat, the propeller blade friction, and hydrodynamic noise. VURN is generally considered as a superposition of linear spectra and continuous spectra [61], and \(\mathbf{n}(t)\) can be expressed as:  
\[\mathbf{n}(t)=\mathbf{L}(t)+\mathbf{C}(t), \tag{3}\]  
where \(\mathbf{L}(t)\) denotes the line spectrum component, \(\mathbf{C}(t)\) denotes the continuous spectrum component, and \(\mathbf{L}(t)\) is composed of multiple sinusoidal functions, which the following equation can obtain:  
\[\mathbf{L}(t)=\sum_{i=1}^{N}A_{i}\sin(2\pi f_{i}t+\varphi_{i}), \tag{4}\]  
Figure 3: A UWA array transmit–receive structure of a SIMO system.  
where \(A_{i},f_{i}\), \(\varphi_{i}\) denote the amplitude, frequency, and initial phase, respectively; \(f_{i}\) satisfies \(f_{i}=n_{b}f_{p}\); \(n_{b}\) is the number of blades of the propeller; \(f_{p}\) is the shaft frequency of the propeller; \(N\) denotes the number of line spectra. Similarly, \(\mathbf{C}(t)\) can be obtained by the following equation:  
\[\mathbf{C}(t)=\sum_{i=1}^{M}\frac{A_{i}}{\sqrt{2\pi}}e^{-\frac{t^{2}}{2\sigma_ {i}}}(t-iT_{b}), \tag{5}\]  
\(M\) denotes the number of continuous spectral duration periods; \(T_{b}\) is the time interval between continuous signals, \(T_{b}=1/f_{i}\); \(\sigma_{i}\) denotes the degree of mixing between continuous signals. For the simulated system, noise \(\mathbf{n}(t)\) and reference noise \(\mathbf{\tilde{n}}(t)\) are the same data. However, the reference noise needs to be collected before the communication for the existing SIMO system. In shorter periods, we consider that the reference noise \(\mathbf{\tilde{n}}(t)\) and \(\mathbf{n}(t)\) are infinitely close to each other.