## 3 Method
### CA-CFAR Detector  
The two-parameter CFAR target detection algorithm is a well-established and widely used method in SAR image target detection. It is based on Gaussian background model for clutter, uses a sliding window to obtain mean and standard deviation for calculating the \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Numerical Value** \\ \hline Center frequency & 17\(\sim\)17.5 GHz \\ Signal bandwidth & 500 MHz \\ Radar transmit signal & FMCW \\ Synthetic aperture & 360 \({}^{\circ}\)C \\ Radius of arm & 1.2 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of CS-GBSAR system. Figure 1: Schematic diagram of CS-GBSAR. threshold [21; 22; 23]. Under the assumption that the clutter background follows a Gaussian distribution, the probability density function can be represented as:  
\[f(x)=\frac{1}{\sqrt{2\pi}\sigma}exp\left[-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right] \tag{1}\]  
The \(\sigma\) is the standard deviation, \(\mu\) is the mean of the distribution, and the distribution function is:  
\[F(x)=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}\sigma}exp\left[-\frac{(t-\mu)^{2} }{2\sigma^{2}}\right]\mathrm{d}t \tag{2}\]  
The \(Z=\frac{t-\mu}{\sigma}\), then:  
\[F(x)=\int_{-\infty}^{\frac{x-\mu}{\sigma}}\frac{1}{\sqrt{2\pi}\sigma}exp\left[ -\frac{Z^{2}}{2}\right]\mathrm{d}t \tag{3}\]  
The standard probability density function of the normal distribution is:  
\[f_{0}(x)=\frac{1}{\sqrt{2\pi}}exp\left[-\frac{x^{2}}{2}\right] \tag{4}\]  
Then, the false alarm probability \(P_{fa}\) can be derived [24]:  
\[P_{fa}=\int_{l_{c}}^{\infty}f_{0}(x)\mathrm{d}x=1-F_{0}(l_{c}) \tag{5}\]  
Then, the threshold is given as [25]:  
\[l_{c}=\Phi^{-1}(1-P_{fa}) \tag{6}\]  
where:  
\(\sigma\): is the standard deviation;  
\(\mu\): is the mean of the distribution;  
F(\(x\)): is the distribution function;  
\(P_{fa}\): is the false alarm probability;  
\(l_{c}\): is the threshold. The algorithm involves comparing each pixel in the image with a detection threshold \(l_{c}\). Pixels with values \(X_{i}\geqslant l_{c}\) are identified as target points, while those with \(X_{i}<l_{c}\) are identified as clutter points. The probability distributions of target and background in SAR images can be observed from Figure 2, where the statistical distribution of target amplitude x is represented by p(x), and the noise and background amplitude distribution is represented by q(x). It is necessary to determine the optimal value of \(l_{c}\). False alarm probability corresponds to the probability of erroneously judging that an event has occurred due to interference caused by noise during target detection [26]. Radar signal detection often employs the Neyman-Pearson (NP) criterion, where the sufficient statistics are compared with adaptive thresholds through statistical hypothesis testing [27]. The aim of the NP criterion is to ensure optimal detection performance while maintaining a certain false alarm probability. In theory, the division of false alarms needs to integrate the distribution of background and target. In practice, the number of target units is usually less than that of background units, so the detection can adopt a suboptimal scheme, to determine whether the signal to be detected is the background. The general process of CA-CFAR detection is as follows. It first utilizes the size of the detection window based on the size of the target to be detected. The classic hollow sliding window is shown in Figure 3, comprises the target area, protected area, and background area. The blue region represents the range of values, which is used to calculate the threshold \(I_{c}\). The white area represents the protected area, and the data in this area do not do any calculation. Then, compute the mean and variance based on a suitable number of pixel values around the target area selected by the sliding window. Then, using the given false alarm probability and the computed mean and variance, the threshold \(I_{c}\) is determined.