## 2 Automatic Tracking Method of Weak Acoustic Targets within Jamming Environment
### Template Generation  
This part illustrates the matching template generation using the Gabor filter. The Fourier transform is the most widely used transform in signal processing; however, it lacks local time and position information. [PERSON] introduced the renowned \"window\" Fourier transform, particularly short-time Fourier transform (STFT) [22], in the article \"theory of communication\" in 1946 in order to examine the frequency characteristics of the signal in the local range. The Gabor filter [23] is a short-time Fourier transform with a Gaussian window function that is extensively employed as a linear filter for edge enhancement in image processing. The two-dimensional form of the Gabor function is  
\[ \begin{split} Gabor_{2D}(x,y,x_{0},y_{0},\sigma_{x},\sigma_{y},u_{x}, u_{y})\\ &=\rho e^{i\phi}s(x,y)l(x,y,\sigma_{x},\sigma_{y})\end{split}  \tag{26}\]with  
\[l(x,y,\sigma_{x},\sigma_{y})=e^{-\pi((x-x_{0})^{2}/\sigma_{x}^{2}+(y-y_{0})^{2}/ \sigma_{y}^{2})} \tag{27}\]  
\[s(x,y)=e^{i(2\pi(u_{x}x+u_{y}y))} \tag{28}\]  
where \((x_{0},y_{0})\) represents the center point of the Gaussian function, \(\phi\) represents the initial phase, \((\sigma_{x},\sigma_{y})\) is the scale in two directions, \(u_{x}\) and \(u_{y}\) are frequency domain coordinates, and \(\rho\) is the amplitude ratio of the Gaussian function. In a comprehensive way, the Gaussian function also has rotation (clockwise):  
\[(x-x_{0})_{r}=(x-x_{0})\cos\varphi+(y-y_{0})\sin\varphi \tag{29}\]  
\[(y-y_{0})_{r}=-(x-x_{0})\sin\varphi+(y-y_{0})\cos\varphi \tag{30}\]  
where \(\varphi\) is the rotation direction of the Gaussian function. The final Gabor function therefore can be rewritten as  
\[ \begin{gathered} gabor_{2D}(x,y,x_{0},y_{0},\varphi,\sigma_{x}, \sigma_{y},u_{x},u_{y})\\ =\rho e^{-\pi((x-x_{0})^{2}/\sigma_{x}^{2}+(y-y_{0})^{2}/\sigma_ {y}^{2})}e^{i(2\pi(u_{x}x+u_{y}y)+\delta)}\end{gathered}  \tag{31}\]  
The real and imaginary components of a 2D Gabor filter are shown in Figure 3a,b, with the real component utilized to simulate the desired template. Furthermore, in this study, it is assumed that the template has the same scale \(\sigma_{x}\) (\(\sigma=\sigma_{x}=\sigma_{y}\)) and frequency coefficient \(u\) (\(u=u_{0}=v_{0}\)) in both dimensions. The real component of the Gabor function is therefore expressed as  
\[ \begin{gathered} gabor_{real}(x,y,\lambda,\delta,\phi,\varphi, \gamma)\\ =exp(-\frac{x_{r}^{2}+\gamma^{2}y_{r}^{2}}{2\sigma^{2}})\times \cos(\frac{2\pi x_{r}}{\lambda}+\phi)\end{gathered}  \tag{32}\]  
with  
\[x_{r}=x\cos\varphi+y\sin\varphi \tag{33}\]  
\[y_{r}=-x\sin\varphi+y\cos\varphi \tag{34}\]  
where \(\gamma\) is the spatial aspect ratio of stripes, \(\delta\) is the standard deviation of the Gaussian factor, \(\lambda\) is the wavelength used to adjust the width of stripes. The transformations between parameters are such as \(\frac{1}{2\rho^{2}}=\frac{1}{\sigma_{x}^{2}},\frac{\gamma^{2}}{2\delta^{2}}= \frac{1}{\sigma_{y}^{2}}\). A 2D Gabor filter is the spatial domain product of a sinusoidal plane wave with a Gaussian kernel function. It has the ability to achieve excellent localization in both the spatial and frequency domains, which is similar to human biological visual features, allowing it to adequately characterize the local structural information of spatial frequency (scale),  
Figure 3: An example of Gabor filter. (**a**) Real component in the spatial domain. (**b**) Imaginary component in the spatial domain. spatial position, and direction selectivity. Assigning the time and frequency parameters \(<\theta_{r},t_{r}>\) to \(<x_{r},y_{r}>\), the final template generating model is described as  
\[ \begin{split} G(\theta,t,\lambda,\delta,\phi,\varphi,\gamma)=gabor _{real}(\theta,t,\lambda,\delta,\phi,\varphi,\gamma)\\ =exp(-\frac{\theta_{r}^{2}+\gamma^{2}t_{r}^{2}}{2\delta^{2}}) \times\cos(\frac{2\pi\theta_{r}}{\lambda}+\phi)\end{split}