# Adaptive Constant False Alarm Detector Based on Composite Fuzzy Fusion Rules
## 3 KLQ-CFAR Detector Based on Composite Fuzzy Fusion Rules  
According to Ref. [16], the observed values are mapped between 0 and 1, and the membership function representing the degree of no target and target is obtained as shown in (3):  
\[\mu(x)=\Pr(X>x|H_{0})=1-F_{X}(x) \tag{3}\]  
where \(x\) is the observation, and \(F_{X}(x)\) is the cumulative distribution function of \(X\).  
In this way, the membership function of the KLQ-CFAR detector based on fuzzy fusion rules is shown in (4):  
\[\mu_{\text{KLQ}}(x)=\Pr(X>\frac{D}{\hat{Q}_{\text{KLQ}}(p)}|H_{0})=1-F_{X}(x) \tag{4}\]  
where the numerator D is the value of the unit to be detected, and the denominator is the estimation of the clutter background of Kaigh-Lachenbruch Quantile estimator.  
Because \(\mu(x)\) is a monotonically decreasing function, stronger observations values are assigned smaller membership function values. When \(\mu(x)\) is less than the threshold, it indicates that the target has been detected. According to the physical meaning of (4) [17; 18; 19; 20], the membership function of the KLQ-CFAR detector with fuzzy fusion rules is shown in (5):  
\[\mu_{\text{KLQ}}(x)=M(x) \tag{5}\]  
Two sensors are used to collect environmental information, the membership function values of each KLQ-CFAR are calculated. Four fusion results are obtained by using four fuzzy fusion rules. Currently, there are four commonly used fuzzy fusion rules, and the fuzzy fusion rules and judgment thresholds are provided in Table 1.  
In Table 1, four fuzzy fusion KLQ-CFAR detectors can be implemented according to the above rules. In order to improve the probability of target detection, the four fuzzy fusion KLQ-CFAR detection results are combined. Thus, a KLQ-CFAR detection algorithm based on composite fuzzy fusion rules is proposed. The specific steps are designed as follows:  
Step 1: Use two sensors to collect background information \(\mathbf{X}_{e1}\) and \(\mathbf{X}_{e2}\) of D.  
Step 2: Calculate the values of the membership function \(\mu_{\text{KLQ1}}(x)\) and \(\mu_{\text{KLQ2}}(x)\).  
Step 3: Calculate the results \(\mu_{\text{FC(i)}}\) and thresholds \(T_{\text{FC(i)}}\) of four fuzzy fusion rules, \(i=1,2,3,4\).  
Step 4: Calculate \(y_{i}=\mu_{\text{FC(i)}}-T_{\text{FC(i)}},i=1,2,3,4\).  
Step 5: Calculate \(Z=\sum_{i=1}^{4}w_{i}\cdot y_{i}\).  
Step 6: If \(Z<0\), the target exists. Otherwise, the target does not exist.  
Kaigh-Lachenbruch Quantile constant false alarm rate detector based on composite fuzzy fusion rules employs two independent sensors to collect background information and calculate membership functions \(\mu_{\text{KLQ1}}(x)\) and \(\mu_{\text{KLQ2}}(x)\) to obtain four types of fuzzy fusion results. These results are compared with their respective thresholds. In addition, a composite detection result is obtained by using a weighted voting method to determine the presence or absence of the target. The weighted voting method assigns weights to the detection results of four fuzzy rules and sums them up to obtain a comprehensive detection result. It can reduce the limitations of a single method by combining multiple detection results, thus improving the overall accuracy of detection. When certain detection methods fail or are incompletely detected, other methods can still provide support and improve robustness. The algorithm comprehensively considers four different fusion results and significantly improves the probability of target detection by integrating multiple aspects of information.