## 3 A Variational Modal Decomposition Target Signal Processing and Recognition Algorithm
If this relationship surpasses the threshold, wavelet threshold denoising is applied; otherwise, the component is deemed noise and subsequently eliminated, followed by signal reconstruction. According to the above algorithm, the algorithm's essence lies in decomposing the target acoustic signal into a finite number of modal components. By utilizing the alternating method of the penalty operator, the optimal solution for Formula (3) can be achieved. Assuming the VMD mode number is \(k\), a finite number of modal components decompose the target acoustic signal, as illustrated in Figure 2.  
The parameter limit value of the second step in Figure 2 can be obtained by Formula (9). \[\left\{ \begin{array}{l}{{x_{k}}^{n+1}(s)=f(s)-\sum\limits_{i\
eq k}{{x_{i}}(s) +\frac{\lambda(s)}{{2+4\sigma\left({s-{s_{k}}}\right)^{2}}}}}\\ {{s_{k}}^{n+1}=\frac{{\int_{0}^{\infty}{s|{x_{k}}(s)|}2 ds}}{{\int_{0}^{\infty} {|{x_{k}}(s)|}2 ds}}}\end{array} \right. \tag{9}\]  
It is not difficult to see from Figure 2 that the selection of decomposition parameters for VMD is the core of the whole processing algorithm. If the selection of parameters is not appropriate, it will cause the target acoustic signal to be mixed in the noise signal and diagnosis of effective target acoustic information will not be possible [28]. According to the above algorithm flow, we obtained the initial signal of any acoustic sensor in the five-element acoustic array target positioning system and subjected it to the processing algorithm that was designed, using VMD to decompose the original signal containing noise, and the time domain waveforms corresponding to the five modal components are obtained, as shown in Figure 3.  
Figure 2: The process of target acoustic signal decomposition into a finite number of modal components. Figure 3 displays the various modal components, namely IMF1-IMF5. The outcomes indicate that the variational modal decomposition signal processing algorithm results in a smoother presentation of target information in the output signal. Target Information Time Extraction Method Based on Cross-Correlation Function in Basic Five-Element Acoustic Array  
Upon acquiring the reconstructed signal, the target signal's distinctive points are recognized based on the peak point. The low-frequency component is eliminated, and after assessing the threshold of the low-frequency part, the remaining components are reconstructed. The peak value of the reconstructed signal is determined, and the peak mutation point is recorded as the target's time point. As illustrated in Figure 1, the target signals output by any two acoustic sensors exhibit correlation for the same target. To find the five time parameters of the acoustic position model in basic arrays A, B, and C, we use the correlation of two acoustic sensor signals to construct a pairwise cross-correlation function in each basic array. Here, we use basic array A as an example to illustrate the time extraction algorithm. Assuming that \(x_{0}(t)\sim x_{4}(t)\) are the output of five acoustic sensors as \(S_{0}\sim S_{4}\) and \(S_{0}\) is the center, looking for the cross-correlation between \(S_{0}\) and the other four acoustic sensors, the cross correlation function of \(x_{0}(t)\) and \(x_{1}(t)\sim x_{4}(t)\) is:  
\[\left\{ \begin{array}{l}R_{x_{0}x_{1}}(\tau)=\int_{0}^{\Delta T_{01}}x_{0}(t )x_{1}(t+\tau_{01})dt\\ R_{x_{0}x_{2}}(\tau)=\int_{0}^{\Delta T_{02}}x_{0}(t)x_{2}(t+\tau_{02})dt\\ R_{x_{0}x_{3}}(\tau)=\int_{0}^{\Delta T_{03}}x_{0}(t)x_{3}(t+\tau_{03})dt\\ R_{x_{0}x_{4}}(\tau)=\int_{0}^{\Delta T_{04}}x_{0}(t)x_{4}(t+\tau_{04})dt\\ \end{array} \right. \tag{10}\]  
where \(\Delta T_{01}\sim\Delta T_{04}\) indicate the pulse width of the signal from \(S_{0}\sim S_{4}\). The target signal is collected in a certain sampling period using the acquisition card in practice. Figure 3: Original signal and IMF component by VMD. This necessitates the discretization of Formula (10), resulting in the transformation of Formula (10) into Formula (11). \[\left\{