## 3 Proposed Algorithm
### Implementation Steps
#### 3.2.2 Detection of the Number of WSAW  
Before executing the algorithm, it is essential to estimate the WSAW number \(K\). Typical approaches include experience value, FFT, and entropy estimation. The first method heavily relies on subjective human experience and is suitable for scenarios where the number of signals is determined. For example, the focus is only on breathing and heartbeat signals for vital signals. The latter two schemes are suitable for most situationsbut are inefficient for low SNR signals, especially when the WSAW amplitude detected by radar is in the order of microns and thus can easily be overwhelmed by other interference signals. An example of this strategy was developed by [PERSON] [31], who proposed an autocorrelation matrix eigendecomposition method based on chi-square statistics that accurately calculated the number of sine waves under low SNR. The signals \(\mathbf{y}=[y_{1},y_{2},\cdots,y_{N-1}]\) of the wave height change are demodulated from the phase of the beat signals. These signals are calculated to obtain the autocorrelation matrix \(\hat{\mathbf{R}}_{y}\), by performing eigendecomposition on the matrix to obtain the corresponding eigenvalues and eigenvectors and arrange them in descending order. We use the eigenvalues of the ranking number and the covariance matrix \(\hat{\mathbf{\sigma}}_{\varepsilon}^{2}(k)\) of the estimated noise to obtain the covariance matrix \(\hat{\mathbf{Q}}_{k}\) of the signal:  
\[\mu_{k}=\left(\frac{T-N_{e}+1}{\hat{\mathbf{\sigma}}_{\varepsilon}^{2}(k)}\right) \hat{\mathbf{V}}_{k}^{T}\hat{\mathbf{Q}}_{k}^{-1}\hat{\mathbf{V}}_{k} \tag{19}\]  
where \(k\) represents the minimum number of eigenvalues, \(T\) is the maximum order of perturbation analysis, \(N_{e}\) is the data length, and \(V\) is the corresponding eigenvector, which is compared with the chi-square statistical result of the WSAW number. In Figure 3, the upper part illustrates the estimation result of the autocorrelation matrix decomposition method using chi-square statistics, and the lower part is the periodogram estimation result. Figure 3a indicates that under high SNR, the estimation results of both methods are five, which is consistent with the simulation results. Figure 3b reveals that under low SNR, the first method's estimation result is consistent with the simulation, and the periodogram-based result is eight due to because of the noise. The proposed algorithm first requires a Hilbert transform to convert the real signal to a complex. Then we calculate the compensation gain and the number of WSAW. After that, we use both critical improvements presented in Section 3.1 to calculate the non-stationary signal interval points. Finally, we employ the output of the improvement modules to estimate the WSAW frequency of each interval point. Algorithm 1 introduces the improved algorithm for estimating the frequency of WSAW. Figure 3: Estimation of the WSAW number, where the simulated underwater sound source excites five signals of different frequencies. (**a**) It shows the estimation results of the above algorithm and periodogram under high SNR and (**b**) low SNR conditions. ```
INPUT: the demodulated WSAW signals;
1. Hilbert transform;
2. Adding window;
3. Computing the gain from Equation (11);
4. Calculating the number of WSAW by Equation (19);
5. Decomposing non-stationary signals into segmented stationary signals
6. Calculating \(x\) by the difference between the adjacent points from the windowed signal;
7. Calculating \(R\) from Equation (16) and computer \(GIP(x_{i})\) from Equation (17);
8. Calculating \(\alpha_{k}\) and \(f_{k}\) from Equation (18);
9. Calculating the average value and standard deviation of \(GIP\) to get the interval point;
10. Estimating the parameters of each stationary signal
11. Initializing the parameter \(\alpha_{k}\) and \(f_{k}\);
12. Estimating the component to be estimated of \(\hat{y}_{k,W}\) from Equation (13);
13. Calculating \(\hat{\alpha}_{k}\) and \(\hat{f}_{k}\) from Equations (14) and (15);
14. OUTPUT: interval point; \(\hat{f}_{k}\).