Improved Adaptive Constant False Alarm Rate Detector Based on Fuzzy Theory for Multiple-Target Scenario

Abstract

An improved adaptive constant false alarm rate (CFAR) detector based on fuzzy theory is proposed to address the issue of poor detection performance and robustness of the variability index (VI) class CFAR detectors due to the misjudgment of the background environment and other reasons. The integration of the order statistic threshold adjustable detection algorithm (OSTA) into the adaptive CFAR detector has the potential to address the aforementioned issue. Therefore, in a clutter edge environment, the ratio of the means of the leading and lagging windows is calculated separately, and the differences between these mean ratios and predefined thresholds are used as inputs to the fuzzy inference machine, and the background clutter estimation of the OSTA is determined based on the fuzzy output, which can extend the range of values of the background clutter estimation, and improve the detection performance of the OSTA in this environment. The Kaigh–Lachenbruch quantile detection algorithm (KLQ) exhibits robust detection performance in multiple-target environments. Therefore, KLQ is used to detect targets in this environment, further improving the detection performance of the detector. The experimental results show that in multiple-target environments with an average misjudgment rate of 27.48%, the proposed detector increases the detection probability by 15.58% compared to the recently proposed variability index heterogeneous clutter estimate modified ordered statistics CFAR detector (VIHCEMOS-CFAR), and in a clutter edge environment, the false alarm rate of the proposed detector was reduced by approximately 89.64% compared to VIHCEMOS-CFAR. Additionally, the effectiveness of the proposed detector is also validated by real clutter data. It can be seen that the proposed adaptive CFAR detector has better robustness to the misjudgment of the background environment and better overall detection performance regardless of the environment.

1. Introduction

The constant false alarm rate (CFAR) detector determines the existence of a target by comparing the sample value of the cell under test (CUT) with the threshold. As an important component of radar, it is applied widely in advanced assisted driving, ship navigation, and electronic reconnaissance. A CFAR-enhanced ship detection method based on YOLOv5 is proposed in Ref. [1], which improves the detection performance of ship targets in SAR images by incorporating traditional features, especially enhancing robustness in complex sea conditions. Ref. [2] combines CFAR with dual-polarization data to propose a novel radar target detection framework, demonstrating excellent performance in large-scale SAR ship detection, particularly in the detection of small and near-shore vessels. Ref. [3] introduces an imaging technique based on Multi-Input Multi-Output Synthetic Aperture Radar (MIMO-SAR), which enhances target detection capabilities for radar in intelligent driving systems by improving side-view resolution. Ref. [4] presents a deep learning-based radar object detection network (RODNet) for millimeter-wave radar, achieving efficient real-time target detection by integrating camera and radar data, addressing the challenge of target recognition in complex environments. The performance of the CFAR detector will be directly impacted by the background environment in the reference window and the type of detector. It usually contains three different background environments: multiple target, clutter edge, and homogeneous. CFAR detectors can be categorized into mean-based, order-statistic-based, and adaptive types based on how background clutter is estimated. The mean-based category includes algorithms such as cell-averaging CFAR (CA-CFAR), smallest-of CFAR (SO-CFAR), and greatest-of CFAR (GO-CFAR). In CA-CFAR, the clutter level is estimated by taking the average of reference cells from both the leading and lagging windows [5]. CA-CFAR performs well in homogeneous environments, but its detection capability decreases in scenarios involving multiple targets or clutter edges [6]. Furthermore, the SO-CFAR and GO-CFAR detectors were proposed in Ref. [7]. In SO-CFAR, the background clutter is estimated by selecting the smaller total from the reference cells in the leading and lagging windows. In contrast, GO-CFAR estimates the background clutter using the larger total from these two windows. In both detectors, the thresholds are obtained by multiplying their background clutter estimations and the threshold coefficients. SO-CFAR demonstrates improved detection performance over CA-CFAR in multiple-target environments but performs poorly in homogeneous and clutter edge scenarios [8]. Likewise, GO-CFAR outperforms CA-CFAR in clutter edge environments, while its effectiveness diminishes in homogeneous and multiple-target environments [9]. To enhance detection capability under multiple-target conditions, an order statistic CFAR detector (OS-CFAR) was introduced in Ref. [10]. In OS-CFAR, the sample quartile (SQ) estimator is used to select the k-th reference cell in ascending order for background clutter estimation. OS-CFAR exhibits better detection performance than CA-CFAR in multiple-target environments; however, its performance degrades in homogeneous and clutter edge environments. Furthermore, Ref. [11] proposed an order statistic class CFAR detector based on the Kaigh-Lachenbruch quantile (KLQ) estimator, known as KLQ-CFAR. Compared to OS-CFAR, KLQ-CFAR offers better detection performance in both homogeneous and multiple-target environments by substituting the KLQ estimator for the SQ estimator. Overall, when the prior parameters in the quantile estimator are properly selected, order statistic class CFAR detectors can generally achieve favorable detection performance in multiple-target environments [12]. However, the environment changes in real-world applications, and the detectors mentioned above usually only adapt to one environment’s detection requirements. To meet the detection requirements under various environmental conditions, an adaptive variability index CFAR detector (VI-CFAR) was proposed in Ref. [13], which integrates the advantages of CA-CFAR, GO-CFAR, and SO-CFAR. It selects the appropriate detection algorithm by computing the variability index (VI) and mean ratio (MR) from reference cells in both the leading and lagging windows. However, due to the limited robustness of CA-CFAR, GO-CFAR, and SO-CFAR, the detection performance of VI-CFAR significantly degrades when the environmental type is misjudged, when a multiple-target environment is mistaken for clutter edge environments [14]. To enhance robustness, several improved strategies based on VI-CFAR have been developed [15,16,17,18,19,20]. Among them, the VIHCEMOS-CFAR detector was introduced in Ref. [15], in which the GO algorithm of VI-CFAR is replaced with the heterogeneous clutter estimate (HCE) algorithm, and in multiple-target environments, the CA detection algorithm is substituted by the Modified Ordered Statistics (MOS) detection algorithm. Although VIHCEMOS-CFAR performs better than VI-CFAR in multiple-target environments, it is less effective at controlling false alarms in environments with clutter edges. The NVI-CFAR was designed by Ref. [16]. In NVI-CFAR, the SO detection algorithm in VI-CFAR is replaced by the OSSO detection algorithm, and in multiple-target environments, the CA detection algorithm is replaced by the OS detection algorithm. Compared to VI-CFAR, the performance of NVI-CFAR is improved in multiple-target environments, as the OS detection algorithm performs better than the CA detection algorithm in this context. However, its robustness remains poor when the environmental type is misjudged, as the threshold selection method in the OSSO detection algorithm is fixed and cannot adapt to the detection requirements in various environments. Additionally, an adaptive CFAR detector with fuzzy rules is introduced in Ref. [17], which is designed to enhance detection in multiple-target environments by utilizing the KLQ-CFAR and the CFVI-CFAR. Although it is found to be effective in complex environments, its performance is degraded in uniform backgrounds. Ref. [18] combines VI-CFAR with CNN for high-frequency surface wave radar, which improves detection in complex environments through dual-detection map fusion. However, misclassification may occur under low Signal-to-Noise Ratio (SNR) conditions. Ref. [19] presents RWVI-CFAR, a robust VI-CFAR for Weibull clutter, which incorporates AOCML-CFAR to improve detection in multi-target and clutter-edge scenarios. Its performance is, however, lower in uniform clutter environments. Ref. [20] proposes the CM-CM CFAR algorithm, which combines CMLD-CFAR and CM-CFAR to efficiently handle clutter environments with low resource consumption on an FPGA. Nevertheless, challenges such as target masking in complex multiple-target environments or clutter edge environments remain.

In recent years, fuzzy theory has demonstrated powerful modeling capabilities in handling uncertainty, vagueness, and incomplete information. These methodologies have been widely applied in fields such as data mining, control systems, and intelligent decision-making. For example, Ref. [21] proposed a method to incorporate fuzziness into the rough approximation framework, providing a theoretical foundation for bridging continuous and discrete information. In addition, Ref. [22] explored the algebraic structures derived from rough approximations in incomplete information systems, offering a new perspective for information fusion using fuzzy systems in complex environments. Furthermore, Ref. [23] demonstrated that Type-2 fuzzy set theory possesses good capabilities in representing uncertainty boundaries and managing the complexity of fuzzy inference. These studies indicate that fuzzy theory is particularly well-suited for addressing problems characterized by vague boundaries and high uncertainty. In the context of CFAR target detection, performance degradation caused by abrupt changes and non-uniformity in the clutter background represents a typical manifestation of such fuzziness.

Therefore, an improved CFAR detector is proposed, which improves the OSTA detection method by integrating fuzzy theory and designing an adaptive FOSTA detection algorithm. In the FOSTA algorithm, the mean ratio of the leading and lagging windows is calculated, and the difference between this ratio and the preset threshold is then used as input to the fuzzy inference machine. This allows for the dynamic adjustment of the background clutter estimation range, effectively enhancing the adaptability and robustness of the OSTA algorithm under varying environmental conditions. Meanwhile, KLQ is adopted to improve the detection performance in a multiple-target environment. Thus, the overall detection performance of the system can be enhanced effectively. Compared with the traditional VI-class CFAR detectors, the proposed CFAR detector adopts an improved detection algorithm in a clutter edge environment. This algorithm can adaptively adjust its own threshold to adapt to a variety of environmental conditions and improve the detection performance of the detector in the case of environmental misjudgment, thereby enhancing the robustness of the VI class CFAR detectors.

2. Methodology

2.1. VI-CFAR

VI-CFAR is an adaptive CFAR detector, and its calculation block diagram can be shown in Figure 1.

In Figure 1, the homogeneous signal I and the orthogonal signal Q are processed by the envelope detector and stored in the samples X_i, i = 1, 2, …, N, in the leading window A and the lagging window B, respectively, sorted in ascending order. Based on the sample values in the leading window A and the lagging window B, the variability index (VI) and the mean ratio (MR) are calculated. By comparing VI and MR with threshold values, the current environmental type can be determined. Based on this determination, VI-CFAR selects different detection algorithms for target detection. The selected detection algorithm calculates the background clutter estimate $T·Z_{VI}$, and the threshold coefficient T is determined based on the false alarm probability (P_FA). If X₀ is greater than or equal to $T·Z_{VI}$, it indicates the presence of a target. Conversely, if X₀ is less than $T·Z_{VI}$, it indicates no target.

The VI is a second-order statistic and its value is jointly determined by the overall mean and the overall variance. For a reference window, VI can be calculated as:

$$VI=1+{\displaystyle \frac{{\widehat{\sigma }}^2}{{\widehat{\mu }}^2}}=1+{\displaystyle \frac{1}{n-1}}·\sum {\displaystyle \frac{{\left(X_i-\overline{X}\right)}^2}{{\left(\overline{X}\right)}^2}}$$

(1)

where, n = N/2, and $\overline{X}$ is the arithmetic mean of n cell samples in the reference window.

The variability index VI_A of the leading window and the variability index VI_B of the lagging window are calculated, respectively. The homogeneity of the background environments in the leading window or the lagging window is assessed by the hypothesis testing in (2):

$$\left\{\begin{array}{c}VI\le K_{VI}\Rightarrow Nonvariable\hfill \\ VI>K_{VI}\Rightarrow Variable\hfill \end{array}\right.$$

(2)

where, K_VI is a threshold. If the test result is nonvariable, it indicates that there is no interference target in the reference window. In contrast, if the test result is variable, it indicates the existence of interference targets in the reference window.

The MR is the sample mean ratio between the leading window and the lagging window, and it can be calculated as:

$$MR={\displaystyle \frac{\overline{X_A}}{\overline{X_B}}}={\displaystyle \frac{{\sum }_{i\subseteq A}{X}_i}{{\sum }_{i\subseteq B}{X}_i}}$$

(3)

Whether sample means of reference cells in the leading window and the lagging window are the same, is determined using (4):

$$\left\{\begin{array}{c}{K}_{MR}^{-1}\le MR\le K_{MR}\Rightarrow SameMeans\hfill \\ MR<K_{MR}^{-1}orMR>K_{MR}\Rightarrow DifferentMeans\hfill \end{array}\right.$$

(4)

where, K_VI is a threshold. If the test result is the same, it indicates that the clutter levels in the leading window and the lagging window are the same. In contrast, if the test result is different, it indicates that the clutter levels in the leading window and the lagging window are different.

Algorithm selection rules of VI-CFAR are given in

Table 1. Algorithm selection rules of VI-CFAR.

Leading Window Variable?	Lagging Window Variable?	Different Means?	VI-CFAR Adaptive Threshold	Equivalent CFAR Method
No	No	No	$T_N·\left(\sum X_A+\sum X_B\right)$	CA
No	No	Yes	$T_n·max\left(\sum X_A,\sum X_B\right)$	GO
No	Yes	-	$T_n·\sum X_A$	CA
Yes	No	-	$T_n·\sum X_B$	CA
Yes	Yes	-	$T_n·min\left(\sum X_A,\sum X_B\right)$	SO

.

In Table 1, T_N and T_n, respectively, represent the threshold coefficient of the entire sliding window and the half sliding window.

From Table 1, the adaptability of VI-CFAR is effectively improved by using different detection algorithms in different background environments. However, in VI-CFAR, homogeneous environments and multiple-target environments are frequently misjudged as clutter edge environments, and the detection performance of the GO algorithm is sensitive to the background environment. Therefore, the use of the GO algorithm in a clutter edge environment may potentially reduce the detection performance of VI-CFAR. In addition, the detection performance of CA and SO algorithms in a multiple-target environment is not satisfactory, which will further reduce the detection performance of VI-CFAR.

For this, an improved VI-CFAR is proposed. Combining fuzzy theory, a fuzzy order statistic threshold-adjustable CFAR detection algorithm (FOSTA) is proposed to improve the robustness of the detector. Meanwhile, KLQ is used to enhance the detection performance of the system in multiple-target environments.

2.2. Improved VI-CFAR

Improved VI-CFAR is composed of CA-CFAR, FOSTA-CFAR, and KLQ-CFAR. In FOSTA-CFAR, the difference between the mean ratio of the leading window and the lagging window and its threshold is used as the input of the fuzzy inference machine, and the clutter estimation is determined based on the fuzzy output, which can extend the range of the background clutter estimation of OSTA, and FOSTA-CFAR is used as a detection strategy in clutter edge environment to enhance overall detection performance and robustness. In addition, KLQ-CFAR is used to enhance the detection performance of the detector in multiple-target environments.

2.2.1. FOSTA-CFAR

When the environmental type is misjudged, the detection performance of VI-CFAR significantly decreases due to the poor robustness of the detection algorithm used in a clutter edge environment. Therefore, the detection algorithm with strong robustness should be used in clutter edge environments. OSTA-CFAR is a threshold-adjustable OS-CFAR [1]. In OSTA-CFAR, the homogeneous clutter power level estimation a is calculated by the prior knowledge, the cells in the reference window are sorted in ascending order, and the l-th reference cell X_l:N is determined as the clutter edge environment judgment sample. If X_l:N is greater than a, the environment is judged as the non-clutter edge environment, and the background clutter estimation is determined as the kth reference cell sample value. In contrast, if X_l:N is less than or equal to a, the environment is judged as the clutter edge environment, and the background clutter estimation is determined as the k + 1-th reference cell sample value.

OSTA-CFAR has better false alarm control ability than OS-CFAR. Furthermore, as a member of the order statistic class detection algorithms, OSTA-CFAR has stronger robustness to the environment. However, in the actual detection process, it is very difficult to obtain prior knowledge and calculate the value of a due to the variability of the environment. In addition, even if the prior knowledge can be obtained and the value of a can also be calculated, OSTA-CFAR only replaces the background clutter estimation from the k-th reference cell sample value to the k+1-th reference cell sample value in clutter edge environment, which constraints the performance improvement of this detection algorithm.

In the process of target detection, background clutter often exhibits significant uncertainty and fuzziness. The OSTA detection algorithm, characterized by limited threshold variability, struggles to respond accurately to transitional environmental states, making it susceptible to local disturbances that can degrade detection performance. Fuzzy theory excels at handling such uncertainty and ambiguity. It enables the modeling of input variables as fuzzy sets and allows flexible reasoning through fuzzy rules to produce continuous, adaptive outputs. This capability addresses a deficiency in the OSTA-CFAR detector when operating in complex environments. Compared to tools such as neural networks, fuzzy theory offers a simpler mathematical structure, so it not only resolves the aforementioned issues but also maintains the real-time responsiveness of the detector.

Therefore, to further enhance the adaptability of the OSTA-CFAR to compensate for the performance degradation of VI class CFAR detectors caused by environmental misjudgment, combining fuzzy theory, FOSTA-CFAR is proposed, and its calculation block diagram can be shown in Figure 2.

In Figure 2, the homogeneous signal I and the orthogonal signal Q are processed by the envelope detector and input into FOSTA-CFAR. The samples X_i in the leading window A and the lagging window B are arranged in ascending order to calculate MR. The difference between MR and the threshold is input into the fuzzy inference system to obtain the appropriate m value, and then the clutter estimation suitable for the current environment is obtained for target detection. From (4), it can be seen that the higher the value of MR, the higher the mean ratio of the clutter power of the leading window and the lagging window, that is, the higher the amplitude of the clutter power jump, and the lower the probability of misjudgment in this case. Instead, it can be concluded that the lower the value of MR, the higher the probability of misjudgment in this case. Therefore, the clutter power jump amplitude and misjudgment can be reflected by MR in the current environment. Also, the thresholds K_MR and $K_{MR}^{-1}$ are determined before target detection. When $K_{MR}^{-1}\le MR\le K_{MR}$ and $VI\le K_{VI}$, the current environment is judged as homogeneous environment, and CA-CFAR is used to detect targets. When $K_{MR}^{-1}\le MR\le K_{MR}$ 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+m:N of FOSTA-CFAR and the threshold coefficient T_F. Then, the product of X_k+m:N and T_F is compared with the CUT X₀. If X₀ > T_FX_k+m:N, a target is considered to be present in the current environment. Otherwise, it is regarded as nonexistent.

The value of e is defined as:

$$e=\left\{\begin{array}{c}MR-K_{MR},MR>K_{MR}\hfill \\ K_{MR}^{-1}-MR,MR<K_{MR}^{-1}\hfill \end{array}\right.$$

(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.

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:

$$trimf\left(x\right)=\left\{\begin{array}{c}0,x\le a\hfill \\ {\displaystyle \frac{x-a}{b-a}},a\le x\le b\hfill \\ {\displaystyle \frac{c-x}{c-b}},b\le x\le c\hfill \\ 0,x\ge c\hfill \end{array}\right.$$

(6)

where, x ∈ [a, c], and b = (a + c)/2.

And fuzzy subsets, NB and PB, are defined by a Gaussian membership function as:

$$gaussmf\left(x\right)=e^{-{\displaystyle \frac{{\left(x-d\right)}^2}{2{\sigma }^2}}}$$

(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. Algorithm selection rules of FOSTA-CFAR.

e	$NB$	$NS$	$ZO$	$PS$	$PB$
m	$NB$	$NS$	$ZO$	$PS$	$PB$

:

The k_F-th reference cell value is determined as the background clutter estimation of FOSTA-CFAR, and k_F is calculated as:

$$k_F=k+m$$

(8)

where, $k={\displaystyle \frac{3}{4}}N$.

Thus, the false alarm probability of FOSTA-CFAR is determined as:

$$P_{FA}=k_F\left(\begin{array}{c}N\\ k_F\end{array}\right){\displaystyle \frac{\Gamma \left(T_F+N-k_F+1\right)\Gamma \left(k_F\right)}{\Gamma \left(k_F+N+1\right)}}$$

(9)

The threshold coefficient T_F of FOSTA-CFAR can be solved by (9).

2.2.2. KLQ-CFAR

VI-CFAR performs poorly in multiple-target environments because of its weakly robust detection algorithm. Therefore, the detection algorithm with strong robustness should be used in multiple-target environments. KLQ-CFAR is an order statistic class detector proposed in Ref. [11]. In KLQ-CFAR, the SQ estimator used in OS-CFAR is replaced by the KLQ estimator with higher estimation efficiency. As a result, KLQ-CFAR outperforms OS-CFAR in detection with multiple targets.

According to [11], when r₁ = 6 and r₂ = 3, KLQ-CFAR has the best detection performance in multiple-target environments, where r₁ and r₂ denote the number of cells trimmed from the upper and lower end of the reference window, respectively. Therefore, the KLQ detection algorithm with r₁ = 6 and r₂ = 3 is used in multiple-target environments in the improved VI-CFAR.

2.2.3. Composition of Improved VI-CFAR Detection Algorithm

The proposed improved VI-CFAR which is named as Kaigh–Lachenbruch quantile variability index constant false alarm rate detector (KLQVI-CFAR) is composed of CA-CFAR, FOSTA-CFAR, and KLQ-CFAR. The selection rules for the detection algorithm are shown in

Table 3. The selection rules for the detection algorithm in KLQVI-CFAR.

Leading Window Variable?	Lagging Window Variable?	Different Means?	VI-CFAR Adaptive Threshold	Equivalent CFAR Method
No	No	No	$T_N·\left(\sum X_A+\sum X_B\right)$	CA
No	No	Yes	$T_F·FOSTA\left(k_F\right)$	FOSTA
No	Yes	-	$\tau h_{KLQ}\left(X_e\right)$	KLQ
Yes	No	-	$\tau h_{KLQ}\left(X_e\right)$	KLQ
Yes	Yes	-	$\tau h_{KLQ}\left(X_e\right)$	KLQ

.

In Table 3, T_N is the threshold coefficient of CA-CFAR, which is calculated as in (10):

$$T_N={\left(P_{FA}\right)}^{-{\displaystyle \frac{1}{N}}}-1$$

(10)

FOSTA(k_F) is the background clutter estimation of FOSTA-CFAR.

As can be seen from Table 3, it is anticipated that KLQVI-CFAR will perform better in detection in homogeneous and multiple-target environments than VI-CFAR because more dependable FOSTA and KLQ detection algorithms are used.

3. Simulation Results

The effectiveness of KLQVI-CFAR is demonstrated through experimental validation, where its detection performance is evaluated using a Monte-Carlo experiment. The amplitudes of the background clutter signals I and Q follow a Rayleigh distribution, the power of the signal passing through the square law detector follows an exponential distribution, and the CUT under H₀ has a probability density function:

$$f\left(x|H_0\right)={\displaystyle \frac{1}{\mu }}exp\left(-{\displaystyle \frac{x}{\mu }}\right)$$

(11)

where H₀ indicates X₀ < S, and $\mu$ is the scale parameter.

The CUT and the interferences in the reference cells follow the Swerling II model, and the CUT or interferences under H₁ has a probability density function:

$$f\left(x|H_1\right)={\displaystyle \frac{1}{\mu \left(1+SNR\right)}}exp\left(-{\displaystyle \frac{x}{\mu \left(1+SNR\right)}}\right)$$

(12)

where H₁ indicates X₀ > S.

The scale parameter is set as $\mu$ = 1, the number of reference cells is set as N = 32, and the thresholds are set as K_VI = 4.76 and K_MR = 1.806 according to Ref. [12].

Choose CA-CFAR [24], OS-CFAR [10], KLQ-CFAR [11], VIHCEMOS-CFAR [15], and IBQ-CFAR [25] as comparative methods for performance analysis. Unless stated otherwise, the SNR is varied from 0 dB to 35 dB in steps of 1 dB, for each CFAR detector, set P_FA = ${10}^{-6}$, and 10⁴ Monte-Carlo experiments are performed to evaluate performance.

3.1. Homogeneous Environment

Comparative analysis experiments on detection performance are conducted in a homogeneous environment. The average detection probability of each detector in a homogeneous environment with SNR ranging from 10 dB to 25 dB is shown in

Table 4. Comparison of the average detection probability of each detector in homogeneous environments.

Detector	CA	KLQ(6,3)	OS(28)	IBQ(6,3)	VIHCEMOS	KLQVI(6,3)
Average detection probability	0.6755	0.6613	0.6519	0.6623	0.6705	0.6710

.

In Table 4, KLQVI(6,3) denotes KLQVI-CFAR with r₁ = 6 and r₂ = 3, OS(28) denotes OS-CFAR with the ordinal value k = 28, KLQ(6,3) denotes KLQ-CFAR with r₁ = 6 and r₂ = 3, and IBQ(6,3) denotes IBQ-CFAR with r₁ = 6 and r₂ = 3.

Table 4 shows that KLQVI-CFAR achieves an average detection probability that is 1.47% higher than KLQ-CFAR, 2.93% higher than OS-CFAR, and 1.31% higher than IBQ-CFAR. This indicates that KLQVI-CFAR exhibits superior detection performance in homogeneous environments compared to the other three methods. Moreover, the higher average detection probability of KLQVI-CFAR compared to VIHCEMOS-CFAR suggests that the incorporation of the FOSTA detection algorithm improves the detector’s robustness across diverse background scenarios.

To assess the significance of performance differences between KLQVI-CFAR, KLQ-CFAR, and VIHCEMOS-CFAR, Analysis of Variance (ANOVA) is employed. ANOVA is used to test whether there are significant differences in the means of different groups. F represents the ratio of between-group variance to within-group variance, with a larger F indicating greater between-group differences. P is used to determine whether the differences between groups are statistically significant, and P less than 0.05 indicates that the differences are significant. The ANOVA results for KLQVI-CFAR, KLQ-CFAR, and VIHCEMOS-CFAR are shown in

Table 5. ANOVA in homogeneous environment.

	Sum of Squares	Degree of Freedom	Mean Square	F	P
Between groups	8.1 × ${10}^{-4}$	2	4.1 × ${10}^{-4}$	30.8367	1.0671 × ${10}^{-7}$
Within group	3.5 × ${10}^{-4}$	27	1.3 × ${10}^{-5}$
Total	1.2 × ${10}^{-3}$	29

.

As shown in Table 5, in homogeneous environment, the sum of squares between groups for KLQVI-CFAR, KLQ-CFAR, and VIHCEMOS-CFAR is 8.1 × ${10}^{-4}$, the sum of squares within groups is 3.5 × ${10}^{-4}$, F is 30.8367, and P is 1.0671 × ${10}^{-7}$. Since P is less than 0.05, it indicates that the differences between groups are statistically significant.

The environmental misjudgment probability refers to the probability that the current environment is incorrectly classified as another environment by the detector. When environmental misclassification occurs, detection algorithms that are unsuitable for the current environment may be selected by traditional detectors, leading to a deterioration in detection performance. Therefore, the environmental misjudgment probability is considered a critical metric closely associated with the performance of the detector. This probability is calculated by performing 10⁴ Monte-Carlo experiments at each SNR level and recording the frequency of incorrect environment classifications made by the detector. VIHCEMOS-CFAR and KLQVI-CFAR use the same environmental classification method, so their environmental misjudgment probabilities are the same, as shown in Figure 3.

From Figure 3, the average probability of misjudging a homogeneous environment as a clutter edge environment is as high as 9.89%; as a result, the detection performance of VIHCEMOS-CFAR is worse than that of CA-CFAR in a homogeneous environment. In comparison, the FOSTA detection algorithm is used in KLQVI-CFAR in a clutter edge environment, which has better detection performance than that of the HCE detection algorithm used in VIHCEMOS-CFAR in homogeneous environments. Therefore, KLQVI-CFAR can still maintain a high detection probability when the environmental type is misjudged.

The detection probability of each detector is reflected in its target recognition capability. A higher detection probability is associated with a stronger detection ability, while a lower detection probability is indicative of weaker performance, potentially resulting in the failure to effectively recognize targets. Therefore, in order to more intuitively compare the detection performance of various detectors in a homogeneous environment, the detection probability of each detector is given as shown in Figure 4.

As can be seen in Figure 4, the curves from top to bottom are CA, KLQVI(6,3), VIHCEMOS, IBQ(6,3), KLQ(6,3) and OS(28), respectively. The order of the above curves represents the performance of different detectors from high to low in homogeneous environments. Thus, it can be seen that in homogeneous environments, CA-CFAR has the best detection performance, while OS(28) has the worst detection performance. Experimental results show that the FOSTA algorithm can compensate for the performance degradation of VI-class CFAR detectors in homogeneous environments caused by environmental misjudgment.

3.2. Multiple-Target Environment

For ease of analysis, it is assumed that the power of the CUT is equal to that of the interference target, and there is no interference target in the CUT. Generally, multi-target interference can be divided into two situations: single window (leading window or lagging window) interference and double windows interference.

The average detection probability of each detector in a multiple-target environment is shown in

Table 6. Comparison of average detection probability of each detector in multiple-target environments.

Environment	CA	KLQ(6,3)	OS(28)	IBQ(6,3)	VIHCEMOS	KLQVI(6,3)
Three interferences in the leading window	0.1189	0.3492	0.3128	0.3382	0.3233	0.3497
Four interferences in the lagging window	0.1068	0.5270	0.5003	0.5145	0.4582	0.5296
One interference in each of the leading and lagging windows	0.4029	0.9414	0.9341	0.9395	0.9305	0.9427

.

As presented in Table 6, KLQVI-CFAR achieves an average detection probability that exceeds that of VIHCEMOS-CFAR by 8.17% when three interferences are present in the leading window and SNR ranges from 5 dB to 20 dB. With four interferences in the lagging window and SNR range of 0 dB to 35 dB, the improvement reaches 15.58%. Additionally, when one interference exists in both the leading and lagging windows and SNR is between 20 dB and 35 dB, KLQVI-CFAR outperforms VIHCEMOS-CFAR by 1.31%. These results demonstrate that the KLQ algorithm implemented in KLQVI-CFAR offers superior detection performance in multi-target environments compared to the MOS algorithm used in VIHCEMOS-CFAR. Furthermore, under conditions of environmental misclassification, the FOSTA algorithm in KLQVI-CFAR provides greater robustness than the HCE algorithm in VIHCEMOS-CFAR.

Moreover, when three interferences are located in the leading window, KLQVI-CFAR’s average detection probability is 3.40% and 11.78% higher than that of IBQ-CFAR and OS-CFAR, respectively. In the case of four interferences in the lagging window, the respective improvements are 2.93% and 5.86%. When one interference exists in each of the leading and lagging windows, KLQVI-CFAR consistently outperforms IBQ-CFAR, OS-CFAR, and KLQ-CFAR. Owing to its adaptability, KLQVI-CFAR also achieves significantly better detection performance than CA-CFAR in multi-target scenarios.

Among the three scenarios, the case with one interference in both the leading and lagging windows is selected as a representative to validate the significance of performance differences between KLQVI-CFAR, KLQ-CFAR, and VIHCEMOS-CFAR. The ANOVA results for KLQVI-CFAR, KLQ-CFAR, and VIHCEMOS-CFAR are shown in

Table 7. ANOVA in multiple-target environments when there is one interference in each of the leading windows and lagging windows.

	Sum of Squares	Degree of Freedom	Mean Square	F	P
Between groups	6.9 × ${10}^{-4}$	2	3.5 × ${10}^{-4}$	618.5880	2.812 × ${10}^{-23}$
Within group	1.5 × ${10}^{-5}$	27	5.6 × ${10}^{-7}$
Total	7.1 × ${10}^{-4}$	29

.

As shown in Table 7, in this environment, F for KLQVI-CFAR, KLQ-CFAR, and VIHCEMOS-CFAR is 618.5880, and P is 2.812 × ${10}^{-23}$, with P being less than 0.05, indicating that the superior performance of KLQVI-CFAR is not due to chance.

In order to further illustrate the negative effect of environmental misjudgment in multiple-target environments, environmental misjudgment probabilities are given as shown in Figure 5.

Figure 5a illustrates that when the SNR is between 5 dB and 20 dB, the average rate at which multiple-target environments are misclassified as clutter edge environments reaches 28.87%. Similarly, Figure 5b reveals that this misclassification rate can average 27.48%, peaking at 66% when the SNR is 9 dB. Additionally, Figure 5c shows that when a single interference exists in both the leading and lagging windows, the probability of misclassification is relatively low, when the SNR exceeds 20 dB, the impact of environmental misjudgment becomes minimal, and the performance of KLQVI-CFAR and VIHCEMOS-CFAR is largely dependent on the detection algorithm used for multiple-target scenarios.

In order to more intuitively demonstrate the advantages of KLQVI-CFAR in detection performance in multiple-target environments, the detection probabilities of various detectors are shown in Figure 6.

From Figure 6a–c, KLQVI(6,3) consistently exhibits the best detection performance, indicating that KLQVI-CFAR has stronger detection capability and robustness compared to detectors such as VIHCEMOS-CFAR and KLQ-CFAR.

For different CFAR detectors, the lower SNR when the detector reaches the same detection probability, the better its detection performance. In order to analyze the performance of the detectors in a multiple-target environment, four interferences are added to the reference window, and Pd is set as 0.5. The SNR required for each detector to meet the detection probability is shown in

Table 8. Comparison of signal-to-noise ratios between various detectors.

Pd = 0.5	CA	KLQ(6,3)	OS(28)	IBQ(6,3)	VIHCEMOS	KLQVI(6,3)
SNR/dB	–	15.58	16.84	15.83	17.71	15.49

.

From Table 8, CA detection method cannot meet the detection requirement of Pd = 0.5. Furthermore, when Pd = 0.5 dB, the SNR required for KLQVI (6,3) is 2.22 dB lower than that required for VIHCEMOS, and 0.09 dB lower than that required for KLQ (6,3). Therefore, KLQVI(6,3) has better detection performance than VIHCEMOS and KLQ(6,3), and the designed KLQVI-CFAR is effective and reasonable.

In order to illustrate the superiority of the anti-interference ability of KLQVI-CFAR in multiple-target environments, the number of interferences is increased from 0 to 5, and the detection probability of various CFAR detectors at SNR = 20 dB are shown in

Table 9. Detection probabilities of various detectors with different numbers of interferences.

Number of Interferences	0	1	2	3	4	5
OS(28)	0.8276	0.8135	0.7886	0.7534	0.7052	0.3203
KLQ(6,3)	0.8345	0.8237	0.8122	0.7997	0.7475	0.6443
CA	0.8475	0.5441	0.3503	0.2306	0.1611	0.1016
IBQ(6,3)	0.8369	0.8206	0.8055	0.7833	0.7387	0.5901
VIHCEMOS	0.8377	0.7836	0.7819	0.7748	0.7661	0.6869
KLQVI(6,3)	0.8414	0.8186	0.8141	0.8021	0.7684	0.6958

.

As shown in Table 9, when the number of interferences is less than or equal to 3, the detection probability of KLQVI-CFAR is similar to that of KLQ-CFAR, IBQ-CFAR, and VIHCEMOS-CFAR. However, when the number of interferences exceeds 3, the detection probability of KLQVI-CFAR outperforms all other compared detectors. Thus, it can be seen that KLQVI-CFAR has better anti-interference ability than VIHCEMOS-CFAR in multiple-target environments.

The Receiver Operating Characteristic (ROC) curve is commonly used to evaluate the performance of CFAR detectors, illustrating the relationship between false alarm probability and detection probability for each detector under a fixed SNR. A higher detection probability at the same false alarm probability is associated with better detector performance. Four interferences are added to the reference window, SNR is set as 15 dB, and the ROC curves of various detectors in a multiple-target environment are shown in Figure 7.

As shown in Figure 7, the ROC curves from top to bottom are KLQVI(6,3), KLQ(6,3), IBQ(6,3), OS(28), VIHCEMOS and CA, respectively. Therefore, it can be seen that KLQVI-CFAR has stronger robustness to the detection results and shows better detection performance.

When there are interferences in the reference window, the detection threshold of the detector will unexpectedly increase, which will result in a lower observed false alarm probability P_fa than the pre-set false alarm probability P_FA. At this time, the detection performance of the detector will be affected. Therefore, the false alarm control capability in a multiple-target environment is also an important evaluation indicator for measuring the performance of detectors. The value $log\left(P_{fa}/P_{FA}\right)$ can be used to evaluate the false alarm control ability of each detector in the multiple-target environment. The closer the value $log\left(P_{fa}/P_{FA}\right)$ is to 0, the stronger the false alarm control ability. The false alarm control capability of each detector in a multiple-target environment is shown in Figure 8.

As shown in Figure 8, because the detection algorithm used in KLQVI-CFAR is more robust than all other comparative methods, it is able to control false alarms better when there are interferences in the reference window.

3.3. Clutter Edge Environment

In a clutter edge environment, the clutter power level in the reference window will suddenly change, which will lead to a sharp increase in the false alarm probability and the formation of the false alarm peak. Generally, the lower the false alarm peak, the better the false alarm control ability of the detector. When the background power of the weak clutter region is 1 dB, and the background power of the strong clutter region is 10 dB, set P_FA = ${10}^{-4}$, and each detector undergoes ${10}^6$ Monte-Carlo experiments, respectively. The P_fa of VIHCEMOS-CFAR and KLQVI-CFAR in a clutter edge environment is shown in Figure 9.

In Figure 9, NC indicates the location of the clutter edge. When the clutter edge appears at the 18th reference cell, the false alarm rate of the proposed detector is reduced by approximately 89.64% compared to VIHCEMOS-CFAR. Thus it can be seen that the false alarm control capability of the FOSTA detection algorithm used in KLQVI-CFAR is significantly stronger than that of the HCE detection algorithm used in VIHCEMOS-CFAR. From the experimental results, it can be seen that KLQVI-CFAR has better false alarm control ability than VIHCEMOS-CFAR.

3.4. Performance of KLQVI-CFAR Based on Real Clutter Data

McMaster IPIX radar clutter data (http://soma.ece.mcmaster.ca/ipix/dartmouth/, accessed on 20 April 2025) is used to test the performance of KLQVI-CFAR. Set P_FA = ${10}^{-3}$, target and interferences follow the Swerling II model, and the parameters of model are the same as (12).

The detection probabilities of various detectors in a homogeneous environment are shown in Figure 10.

From Figure 10, the comparison results of the detection performance based on real data are similar to those based on simulation data mentioned above. Therefore, the FOSTA algorithm used in KLQVI-CFAR can improve the detection performance.

When 6 interferences are added to the reference window, the detection probabilities of various detectors in a multiple-target environment are shown in Figure 11.

Similarly, from Figure 11, in multiple-target environments, the comparison results of the detection performance based on real data are also similar to those based on simulation data mentioned above. Thus, it can be seen that KLQVI-CFAR has good potential application capabilities.

4. Discussion

To address the performance degradation and lack of robustness in VI class CFAR detectors caused by background misjudgment in complex environments, KLQVI-CFAR is proposed. This detector integrates fuzzy theory and the OSTA detection method, and a designed adaptive FOSTA detection algorithm. In FOSTA, the mean ratio of the leading and lagging windows is calculated separately, and their deviation from a preset threshold is used as input to the fuzzy inference machine. This allows for dynamic adjustment of the background clutter estimation range, effectively enhancing the adaptability and robustness of the OSTA algorithm under various environmental conditions. Moreover, to address the issue of missed detections in multiple-target environments common to traditional VI class CFAR detectors, KLQVI-CFAR introduces the KLQ detection algorithm, which improves detection accuracy in high-density multiple-target environments. By integrating the FOSTA and KLQ algorithms, KLQVI-CFAR demonstrates excellent target detection capability and false alarm control across different complex environments. Simulation results show that, under various typical non-homogeneous environments, KLQVI-CFAR outperforms detection algorithms such as VIHCEMOS-CFAR and KLQ-CFAR in both detection probability and false alarm probability, effectively reducing the impact of background misjudgment on detection performance. Furthermore, its effectiveness and robustness in practical applications are validated through experiments on real clutter data. Although the KLQVI-CFAR detector performs excellently in various complex environments, its performance in extremely low SNR conditions still needs to be further improved. Additionally, its robustness may be affected in extreme clutter environments. This phenomenon is consistent with the observations of CFAR performance degradation in complex scenarios in recent literature. Future work will therefore concentrate on enhancing the algorithm’s ability to adapt to harsh environments. This will involve enhancing computational efficiency, streamlining the algorithm’s structure, and investigating the integration of lightweight deep learning mechanisms into the detection process to raise its level of intelligence and real-time.