## 2 Methodology
### Improved VI-CFAR
#### 2.2.1 Fosta-Cfar
When \(K_{MR}^{-1}\leq MR\leq K_{MR}\) and \(VI\leq K_{VI}\), the current environment is judged as homogeneous environment, and CA-CFAR is used to detect targets. When \(K_{MR}^{-1}\leq MR\leq K_{MR}\) and \(VI>K_{VI}\), the current environment is judged as a multiple-target environment, and KLQ-CFAR is used to detect targets. When \(MR>K_{MR}\) or \(MR<K_{MR}^{-1}\), the current environment is judged as clutter edge environment. In order to improve the robustness of detection, a new detection method based on fuzzy inference is proposed. In the fuzzy inference machine of FOSTA-CFAR, the difference \(e\) between \(MR\) and \(K_{MR}\) (\(K_{MR}^{-1}\)) is used as the input, and the fuzzy inference output is \(m\), which is used to determine the background clutter estimation \(X_{k+mN}\) of FOSTA-CFAR and the threshold coefficient \(T_{F}\). Then, the product of \(X_{k+mN}\) and \(T_{F}\) is compared with the CUT \(X_{0}\). If \(X_{0}\)\(>\)\(T_{F}X_{k+mN}\), a target is considered to be present in the current environment. Otherwise, it is regarded as nonexistent. The value of \(e\) is defined as:  
\[e\ =\ \left\{ \begin{array}{ll}MR-K_{MR},\ MR>K_{MR}\\ K_{MR}^{-1}-MR,\ MR<K_{MR}^{-1}\end{array} \right. \tag{5}\]  
From (5), the larger the \(e\), the greater the difference in the clutter power between the leading window and the lagging window. Therefore, when \(e\) is larger, a larger \(m\) should be selected to maintain the false alarm control capability. Instead, the smaller the \(e\), the smaller the difference in the clutter power between the leading window and the lagging window. Therefore, when \(e\) is smaller, the probability of misjudgment is higher, and a smaller \(m\) should be selected to maintain its detection performance. According to Ref. [24], the optimum background clutter estimation of OSTA-CFAR should be determined near the 3/4 quartile point. However, if the range of the neighborhood centered on the 3/4 quartile point is too large, the detection performance of the detection algorithm will decrease. Instead, if the range is too small, the improvement in detection performance will be inconspicuous. In practice, based on experimental tests, the domain of \(e\) and \(m\) are determined as [0, 24] and [\(-\)3, 3], respectively. Figure 2: Calculation block diagram of FOSTA-CFAR. Five fuzzy subsets on the domain are defined as: {NB (Negative Big), NS (Negative Small), ZO (Zero), PS (Positive Small), PB (Positive Big)}. Fuzzy subsets, NS, ZO, and PS, are defined by a triangular membership function as:  
\[trim(x)=\left\{ \begin{array}{l}0,\ x\leq a\\ \frac{x-a}{b-a},\ a\leq x\leq b\\ \frac{c-x}{c-b},\ b\leq x\leq c\\ 0,\ x\geq c\end{array} \right. \tag{6}\]  
where, \(x\in[a,c]\), and \(b=(a+c)/2\). And fuzzy subsets, NB and PB, are defined by a Gaussian membership function as:  
\[gaussmf(x)=e^{-\frac{(x-d)^{2}}{2\sigma^{2}}} \tag{7}\]  
where, \(d\) and \(\sigma\) are the position parameter and the shape parameter, respectively. The fuzzy rules between the input \(e\) and the output \(m\) are shown in Table 2:  
The \(k_{F}\)-th reference cell value is determined as the background clutter estimation of FOSTA-CFAR, and \(k_{F}\) is calculated as:  
\[k_{F}=k+m \tag{8}\]  
where, \(k=\frac{3}{4}N\). Thus, the false alarm probability of FOSTA-CFAR is determined as:  
\[P_{FA}=k_{F}\left( \begin{array}{c}N\\ k_{F}\end{array} \right)\frac{\Gamma(T_{F}+N-k_{F}+1)\Gamma(k_{F})}{\Gamma(k_{F} +N+1)} \tag{9}\]  
The threshold coefficient \(T_{F}\) of FOSTA-CFAR can be solved by (9).