## 3 Proposed Algorithm
### Algorithm Theory  
The RELAX algorithm estimates the target signal parameters through a nonlinear least square process, affording robustness. Assuming that the received signal has zero mean Gaussian white noise, the underwater sound source sends \(K\) signals of different frequencies. After phase extraction, the phases obtained by demodulation include \(K\) signals of different frequencies and random noise. \[y(n)=\sum_{k=1}^{K}a_{k}\text{e}^{\left|2\pi f_{k}n+e(n)\right|},n=1,\cdots,N \tag{7}\]  
where \(N\) is the sampling point, \(a_{k}\) is the \(k^{\text{th}}\) complex amplitude, and \(f_{k}\) is the \(k^{\text{th}}\) frequency. The noise \(e(n)\) is complex Gaussian noise with zero mean and variance of \(\sigma^{2}\). From the nonlinear least squares criterion:  
\[\{f_{k},a_{k}\}_{k=1}^{K}=\arg\min_{\{f_{k},a_{k}\}_{k=1}^{K}=\left\|\mathbf{y}- \sum_{k=1}^{K}a_{k}\mathbf{\omega}(f_{k})\right\|^{2}} \tag{8}\]  
where \(\mathbf{y}=[y(1),y(2),\cdots,y(N-1)]^{\text{T}}\) and \(\mathbf{\omega}=\left[1,\text{e}^{\left(2\pi f_{k}\right)},\cdots,\text{e}^{\left( 2\pi f_{k}(N-1)\right)}\right]^{\text{T}}\). We estimate the amplitude and frequency of the \(\left(p-1\right)^{\text{th}}\) WSAW, and the remaining signals, including the \(p^{\text{th}}\) WSAW signal, can be expressed as:  
\[\mathbf{y}_{p}=\mathbf{y}-\sum_{k=1}^{p-1}a_{k}\mathbf{\omega}(f_{k}) \tag{9}\]  
By minimizing Equations (8) and (9) simultaneously, we obtain the estimated parameter values of the \(p^{\text{th}}\) WSAW signal:  
\[a_{k}=\left.\frac{\mathbf{\omega}^{H}(f_{k})y_{k}}{N}\right|_{f_{k}=f_{k}} \tag{10}\]  
\[f_{k}=\underset{f_{k}}{\text{argmin}}\bigg{|}\left[I-\frac{\mathbf{\omega}(f_{k}) \mathbf{\omega}^{H}(f_{k})}{N}\right]\mathbf{y}_{k}\|^{2}=\underset{f_{k}}{\text{argmin} }\Big{|}\mathbf{\omega}^{H}(f_{k})\mathbf{y}_{k}\Big{|}^{2} \tag{11}\]  
The convergence condition of Equation (11) is that the difference between the \(\left(p-1\right)^{\text{th}}\) value and the \(p^{\text{th}}\) value of Equation (9) is less than a given threshold \(\varepsilon\). When the side lobes of the strong signals are higher than the main lobes of the weak signals, the RELAX algorithm results will be wrong, because the result of RELAX subtracts the estimated maximum value of each cycle without suppressing the side lobe signal. For example, the underwater sound source transmits four different frequencies, i.e., 100 Hz, 130 Hz, 180 Hz, and 300 Hz signals, and the RELAX algorithm fails to detect the 300 Hz signals in Figure 2. Since the WSAW is a non-stationary signal, the periodogram and the RELAX algorithm are unaware of the time information, and therefore the underwater sound source information cannot be effectively parsed. To solve these problems, we propose the following two improvements. First improvement: when the measured signal contains some significant signals, the strong signal side lobe overwhelms the weak main lobe signal, affecting the estimation accuracy of the WSAW characteristic parameters. Windowing is an effective method to suppress side lobes, which, however, brings two problems simultaneously: main lobe broadening and peak drop. The former affects the estimation accuracy, and the latter affects the gain, leading to a significant error during the next iteration. Adding a window can significantly suppress the side lobes and increase the weak signal parameter estimation accuracy. Let \(w(n)\) be the coefficient of the corresponding window function. Then the normalized amplitude gain is defined as:  
\[G=\frac{1}{N}\sum_{n=0}^{N-1}w(n) \tag{12}\]  
where \(G\) is the signal compensation gain, \(N\) is the number of sampling points, and \(n\) is the sampling moment. Due to the peak drop problem, the amplitude signal estimation is inaccurate when calculating the error function in Equation (9), significantly affecting the subsequent parameter estimation.