# Research on Laser Dual-Mode Fusion Detection Method of Ship Wake Bubbles
## 3 Simulation of Ship's Wake Bubble Laser Dual-Mode Detection Based on Monte Carlo
### Ship's Wake Laser Dual-Mode Fusion Detection Signal Processing Method
#### 4.2.1 Multi-Timescale Fusion Hierarchical Signal Processing Method
The amplitude SBR of the two signals in Figure 12 is as follows:  
\[SBR_{b}=\frac{V_{s}}{V_{b}} \tag{4}\]  
Calculating the ratio of the pulse widths of the two signals in the same way, the SBR of the pulse widths is as follows:  
\[SBR_{p}=\frac{P_{s}}{P_{b}} \tag{5}\]  
Approximating the two signals as sinusoids, the tail flow signal expression is as follows:  
\[y_{s}(t)=V_{s}\sin(\frac{\pi t}{P_{s}}) \tag{6}\]  
The water background signal expression is as follows:  
\[y_{b}(t)=V_{b}\sin(\frac{\pi t}{P_{b}}) \tag{7}\]  
Integrate the two signals separately in time provides the following:  
\[\int_{0}^{t}y_{s}(t)dt=\int_{0}^{P_{s}}V_{s}\sin(\frac{\pi t}{P_{s}})dt=\frac {2V_{s}P_{s}}{\pi} \tag{8}\]  
\[\int_{0}^{t}y_{b}(t)dt=\int_{0}^{P_{b}}V_{b}\sin(\frac{\pi t}{P_{b}})dt=\frac {2V_{b}P_{b}}{\pi} \tag{9}\]  
The SBR of the integral values of the two signals is as follows:  
\[SBR_{I}=\frac{\int_{0}^{t}y_{s}(t)dt}{\int_{0}^{t}y_{b}(t)dt}=\frac{\frac{2V_ {s}P_{b}}{\pi}}{\frac{2V_{b}P_{b}}{\pi}}=\frac{V_{s}}{V_{b}}\cdot\frac{P_{s}} {P_{b}}=SBR_{b}\cdot SBR_{p} \tag{10}\]  
\[\therefore\cdot SBR_{v}>1,\ SBR_{p}>1,\ \therefore\Firstly, the signal is processed based on the single-cycle integral circuit, and the processed signal is cached. When it is accumulated to a certain period, the envelope detection and mean filtering signal processing are performed on the multi-cycle echo signal. Combined with the envelope detection and mean filtering signal processing method, according to the set window size, the multi-cycle cumulative signal is processed to obtain the continuous change in the signal and determine whether the specified target is detected.  
In this paper, a ship's wake detection device is constructed using a high-frequency laser. When the motion speed of the detection device matches the pulse period of the laser, the signals of consecutive \(M\) cycles can be approximated as the echoes of the same wake bubble. Therefore, the method first splits the signal into \(n\) intervals of length \(M\): \([(x_{0},\ x_{1}),(x_{1},\ x_{2}),\cdots,(x_{n-1},\ x_{n})]\). The signal amplitude corresponding to the endpoints of each interval is \(y_{i},\ i=0,1,2,\cdots,n\). The maximum value in each interval is \((a_{j},\ m_{j})\), \(j=1,2,\cdots,n\). The two endpoints and the maximum point within each interval are used to fit the envelope of the signal by employing cubic spline interpolation, which maximizes the retention of the echo signal characteristics of the wake. Cubic spline interpolation is a numerical analysis method that fits discrete points into smooth curves by means of cubic polynomials. Let the cubic spline function in each interval be as follows:  
\[S_{j}(x)=a_{j}x^{3}+b_{j}x^{2}+c_{j}x+d_{j} \tag{12}\]  
Then, the cubic spline function satisfies the following three conditions:  
1. Interpolation condition, i.e., \(S_{j}(x_{i})=y_{i},\ S_{j}(a_{j})=m_{j}\);
2. Equal values of the spline function at the common endpoints of adjacent intervals, i.e., \(S_{j}(x_{i})=S_{j+1}(x_{i})\);
3. The curve is smooth, i.e., \(S_{j}(x)^{\prime},\ S_{j}(x)^{\prime\prime},\ S_{j}(x)^{\prime\prime}\) is continuous. From these three conditions, the value of the cubic spline function coefficients \(a_{i},\ b_{i},\ c_{i},\ d_{i}\) can be derived, which in turn fits the envelope of the original echo signal. The amplitude distribution of the fitted envelope curve still has a large dispersion affected by the original signal amplitude jitter. In order to reduce the standard deviation of the signal envelope amplitude, a sliding mean filtering algorithm is used to further smooth it. It is a filtering algorithm that takes the average of \(N\) consecutive points instead of the \(\frac{N+1}{2}\)rd point value [14]. \(N\) is the size of the sliding window and must be an odd number.