# Adaptive Constant False Alarm Detector Based on Composite Fuzzy Fusion Rules
## 4 Adaptive Constant False Alarm Detector Based on Composite Fuzzy Fusion Rules  
A suitable detector can be selected in the adaptive constant false alarm detector according to the type of environmental information. Thus, it can effectively improve the adaptability of the target detector and enhance detection performance. Based on the designed Kaigh-Lachenbruch Quantile constant false alarm rate detector based on composite fuzzy fusion rules, an adaptive constant false alarm detector with composite fuzzy fusion rules (CFVI-CFAR) is further designed. The principle is shown in Figure 1.  
In Figure 1, the variability index (_VI_) and the mean ratio (_MR_) of the leading and the lagging reference windows are calculated, and the CFAR detection method is selected based on environmental clutter type information. The calculation formulas for VI and MR are shown in (6) and (7):  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Fuzzy Fusion Rules** & **Calculation Formula** & **Threshold** \\ \hline “MAX” fusion rules & \(\mu_{\text{FC1}}=\max(\mu_{1}(x),\mu_{2}(x))\) & \(T_{\text{FC1}}=\sqrt{P_{\text{FA}}}\) \\ “MIN” fusion rules & \(\mu_{\text{FC2}}=\min(\mu_{1}(x),\mu_{2}(x))\) & \(T_{\text{FC2}}=1-\sqrt{(1-P_{\text{FA}})}\) \\ “Algebraic product” fusion rules & \(\mu_{\text{FC3}}=\mu_{1}(x)\cdot\mu_{2}(x)\) & \(P_{\text{FA}}=\) \\ “Algebraic sum” fusion rules & \(\mu_{\text{FC4}}=1-(1-\mu_{1}(x))\cdot(1-\mu_{2}(x))\) & \(P_{\text{FA}}=T_{\text{FC4}}\cdot\ln(1-T_{\text{FC4}})\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Fusion rules and judgment thresholds.  
\[VI=1+\frac{\hat{\sigma}^{2}}{\hat{\eta}^{2}}=1+\frac{1}{n-1}\sum\limits_{i=1}^{n} \frac{x_{i}-\tilde{x}^{2}}{\left(\tilde{x}\right)^{2}} \tag{6}\]  
\[MR=\frac{\tilde{x}_{A}}{\tilde{x}_{B}}=\frac{\sum\limits_{i\in A}x_{i}}{\sum \limits_{i\in B}x_{i}} \tag{7}\]  
where \(\hat{\sigma}^{2}\) is estimation of variance, \(\hat{\eta}^{2}\) is the estimation of the mean, \(\tilde{x}\) is the average value of the total reference cell \(N\), \(\tilde{x}_{A}\) and \(\tilde{x}_{B}\) are the sample mean of the leading and lagging windows, respectively.  
The criteria for determining the type of environmental clutter are as follows in (8) and (9):  
\[\left\{\begin{aligned} VI\leq K_{VI}\Rightarrow Homogeneous\\ VI>K_{VI}\Rightarrow Inhomogeneous\end{aligned}\right. \tag{8}\]  
\[\left\{\begin{aligned} K_{MR}^{-1}\leq MR\leq K_{MR}\Rightarrow Same\ means\\ MR<K_{MR}^{-1}orMR>K_{MR}\Rightarrow Different\ means\end{aligned}\right. \tag{9}\]  
where \(K_{VI}\) is the comparison threshold value of statistic _VI_, \(K_{MR}\) is the comparison threshold value of statistic _MR_.  
Equations (8) and (9) can determine whether the leading and lagging windows are homogeneous and whether the mean is the same. When the homogeneous environment is determined, CA-CFAR is selected as the target detector. When the clutter edge environment is determined, GO-CFAR is selected as the target detector. When the multiple-target environment is determined, CFKLQ-CFAR is selected as the target detector. On the one hand, it can improve the probability of target detection in multiple-target environment; on the other hand, it can improve the robustness of target detection when a homogeneous environment or clutter edge environment is misjudged as a multiple-target environment. The specific selection strategy for the CFVI-CFAR algorithm is shown in Table 2.  
Figure 1: The algorithm diagram of CFVI-CFAR.