## 3 A Variational Modal Decomposition Target Signal Processing and Recognition Algorithm  
The characteristic waveform of explosive sound signals typically includes rapidly rising peaks and relatively short attenuation. In contrast to stationary signals, explosive sound signals show sudden and intense amplitude jumps, with high-intensity signals across a broad frequency range in the spectrum. Within the unit acoustic array, there is a degree of similarity among the explosive sound signals collected by each acoustic sensor. More precisely, the five acoustic sensors are relatively close, resulting in the information obtained in the unit acoustic array being largely consistent. However, due to the varying distances from each acoustic array to the sound source, directional attenuation of explosive information occurs during the transmission process. Consequently, each acoustic sensor receives different levels of acoustic energy. This disparity is primarily attributed to the gradual attenuation of projectile explosive information along the transmission path. Considering the presence of various interference noises in the test environment, such as wind sounds, speech, and other non-projectile-related information, these noises also impact the explosive sound signal. Nevertheless, the acoustic signals collected during each explosive event exhibit a certain correlation within each acoustic array. This indicates that despite the influence of noise, the explosive signal can still maintain a certain level of identifiability and similarity within the acoustic array. As the target traverses the test area of the five-element acoustic positioning system, the acoustic sensor captures information at the moment the target enters the test area, and the amplification and processing circuits then generate a sudden change signal. This signal not only carries information about the target but also may be affected by background noise, vibration signals, and other interference signals resulting from environmental changes. To address the impact of existing interference signals on extracting genuine target information, we employ a target signal processing algorithm based on variational modal decomposition (VMD) [24,25]. VMD decomposes the output signal from the infrared sky screen into a series of modal components with distinct center frequencies. Treating the target signal as a mutation signal, it reconstructs a signal enriched with additional frequency and amplitude components. By iteratively searching for the optimal solution of the variational model, the center frequency and bandwidth of each intrinsic mode function are alternately updated to achieve adaptive signal separation in the frequency domain. This iterative process results in \(n\) modal components, forming the basis for a more refined decomposition of the signal. The following outlines the specific steps of the algorithm:  
Step 1: it is presumed that the signal undergoes decomposition, denoted as \(f(t)\), which consists of \(n\) modal components \(x_{n}\) with different center frequencies, and each modal component converges around its respective center frequency, denoted as \(s_{n}\). In order to estimate the bandwidth of the frequency-shift signal, we use Gaussian smoothness to evaluate the frequency-shift signal to ensure that the estimated bandwidth of each subsignal is the smallest. Formula (3) is used to construct a variational problem with constraints. \[\left\{ \begin{array}{l}\min\biggl{\{}\sum\limits_{n=1}^{n}\left\|\partial \right[\left(\delta(t)+\frac{\mathrm{i}}{\pi t}\right)*x_{n}(t)\right]\exp(- \mathrm{j}s_{n}t)\left\|_{2}^{2}\right\}\\ \text{s.t.}\sum\limits_{n=1}^{n}x_{n}(t)=f(t)\end{array} \right. \tag{3}\]  
where \(x_{n}(t)\) is the output acoustic signal of the target, \(f(t)\) is the discrete subsignal with distinct frequencies obtained through VMD decomposition, and \(\delta(t)\) is the unit impulsefunction. In order to solve the optimal solution of the variational problem, we introduce Formula (4). \[