## 3 A Variational Modal Decomposition Target Signal Processing and Recognition Algorithm
\begin{array}{l}L(\{x_{n}\},\{s_{n}\},\lambda)=\sigma\sum\limits_{n=1}^{n}\parallel \partial_{t}\bigg{[}\bigg{(}\delta(t)+\frac{\mathrm{j}}{\pi t}\bigg{)}*x_{n}(t )\bigg{]}\exp(-\mathrm{j}s_{n}t)\parallel\limits_{2}^{2}\\ +\parallel f(t)-\sum\limits_{n=1}^{n}x_{n}(t)\parallel\limits_{2}^{2}+\bigg{<} \lambda(t),f(t)-\sum\limits_{n=1}^{n}x_{n}(t)\bigg{>}\end{array}  \tag{4}\]  
where \(\sigma\) is the penalty operator and \(\lambda(t)\) is the Lagrange multiplier. Employing the penalty operator alternating direction method, we determine the optimal solution for Formula (2). Taking into account the convergence conditions and applying the optimal solution, we iteratively update each intrinsic mode function \(x_{n}(t)\), center frequency \(s_{n}\) and \(\lambda(t)\) using Formulas (5)-(7). Finally, the target output signal to be decomposed is successfully decomposed into a finite number of modal components. \[x_{n}^{k+1}(\omega)=\frac{f(\omega)-\sum\limits_{i=1,i\
eq n}^{n}x_{i}(\omega )+\frac{\lambda(\omega)}{2}}{1+2\pi(\omega-\omega_{k})^{2}} \tag{5}\]  
\[s_{n}^{k+1}=\frac{\int_{0}^{\infty}\omega\abs{x_{n}(\omega)}^{2}d\omega}{\int _{0}^{\infty}\abs{x_{n}(\omega)}^{2}d\omega} \tag{6}\]  
\[\lambda^{n+1}(\omega)=\lambda^{n}(\omega)+\tau\bigg{(}f(\omega)-\sum\limits_{ n=1}^{n}x_{n}^{k+1}(\omega)\bigg{)} \tag{7}\]  
where \(\tau\) is the noise tolerance parameter, k is the number of iterations, \(f(\omega)\), \(x_{i}(\omega)\), \(\lambda(\omega)\), and \(x_{n}^{k+1}(\omega)\) represent the Fourier transform of \(f(t)\), \(x_{i}(t)\), \(\lambda(t)\), and \(x_{n}^{k+1}(t)\), respectively. Step 2: Wavelet threshold processing is applied to the acoustic sensor's output signal. The fundamental concept involves filtering the signal. This method transforms the signal into different spatial scales through wavelet transform, filters out noise through threshold processing, and ultimately reconstructs the signal. In the process of filtering and denoising the target signal, the threshold processing in the wavelet threshold method is crucial, encompassing the selection of the threshold and the choice of the threshold function. Threshold quantization rules generally fall into two categories: hard threshold and soft threshold. The hard threshold function may induce local mutations at the threshold \(\lambda\) after signal processing. Although the signal processed by the soft threshold function may lose a small portion of the high-frequency signal, it is smoother and eliminates the local mutation caused by the hard threshold function. By utilizing the weighted average method of hard threshold and soft threshold functions, a new threshold function can be derived. The specific formula is shown in (8). \[s_{j,k}=\left\{ \begin{array}{l}\bigg{(}\text{sgn}\Big{(}s_{j,k}\Big{)} \bigg{)}\bigg{(}\Big{(}A\cdot\abs{s_{j,k}}\Big{)}^{1/B}-(A\lambda)^{1/B}\bigg{)} ^{B},\Big{|}s_{j,k}\Big{|}\geqslant\lambda\\ 0,\abs{s_{j,k}}<\lambda\end{array} \right. \tag{8}\]  
where \(A\) and \(B\) are regulatory factors, \(A=[0,1]\), \(B>0\). This threshold function incorporates an adjustment coefficient \(A\) to amalgamate the benefits of both hard and soft thresholds. Additionally, an adjustment factor \(B\) is introduced to augment the shrinkage of wavelet coefficients [26; 27]. Step 3: The time information is obtained when the target passes through the test area instantaneously. Utilizing a five-element acoustic array, the target positioning system captures target signals via an acquisition card. During target recognition, variations in the target's position within the test area and discrepancies in the frequency of the mutatedsignal are observed. This divergence may arise due to the incomplete decomposition of the target signal in a specific component. Consequently, it is essential to evaluate the outcomes in each component, establishing the relationship between the threshold in the initial modal component and the peak point in the remaining components.