Multi-Target CFAR Detection Method for HF Over-The-Horizon Radar Based on Target Sparse Constraint in Weibull Clutter Background

Abstract

High frequency radar has a wide monitoring range and low range resolution. It may contain multiple targets or outlier interference phenomena in different clutter regions of the range–Doppler (RD) spectrum in detected background. The key to the performance of target detection in multi target backgrounds is the ability to determine the attributes of targets or outliers. Our previous research shows that the number of targets belongs to an absolute minority compared to the number of background units. In this paper, we propose a new method for multi-target detection building on the ordered statistics constant false alarm detector (OS-CFAR). The new method fully utilizes the sparse characteristics of the target and uses the idea of introducing regularization processing to eliminate interfering targets, and obtain an estimate of shape parameters for target detection. To further improve the performance of the algorithm, a correction method is proposed for the inaccurate selection of the k value. Upon estimating the distribution parameters, the detection threshold is calculated, and the target’s constant false alarm detection is completed. Simulation and measured data show that our algorithm can effectively counter the interference of multiple targets and maintain a constant false alarm characteristic under different conditions, providing a reliable target detection method.

1. Introduction

High-frequency over-the-horizon radar (OTHR) uses the diffraction or reflection of high-frequency electromagnetic waves to detect over-the-horizon long-range targets and provide wide-range sea area monitoring. OTHR works in a complex electromagnetic environment that introduces noise into the processed azimuth–range–Doppler (ARD) spectra from various sources, including but not limited to target noises, atmospheric noises, ionospheric, and sea clutter noises. These signals are statistically non-uniform and widely distributed (Figure 1), representing a significant challenge in effectively detecting the target of interest in a complex detection background.

The target detection processing of high-frequency radar is usually carried out in the RD spectrum of the echo. In the area of atmospheric noise and different uniform clutter in the actual RD spectrum of the HF over-the-horizon radar, the target detection and processing methods will face the impact of multiple targets and similar targets. In the traditional radar detector, the constant false alarm rate (CFAR) method mainly uses the statistical characteristics of the reference unit amplitude to calculate the threshold of the cell under test (CUT) and predict the probability of target signal detection and false alarm detection. Therefore, CFAR is applied to detect targets in noise/clutter environments by calculating the adaptive detection threshold of its detection unit based on the neighboring reference units. Under the condition of statistical independence and consistency, the more reference units there are, the higher the accuracy of the estimated detection threshold will be. However, the more reference units the CFAR uses, the more targets/ jamming targets the environment may contain, which reduce the estimation accuracy. Detection performance in multi-target backgrounds can be improved by distinguishing targets from interference targets in the background adaptively, allowing for more units to be selected without impacting the CUT detection threshold estimation. However, due to the lack of prior information on the number of targets/jamming targets and the background distribution parameters, it is difficult to adaptively determine these strong signals in the multi-target detection background. Many scholars at home and abroad have carried out relevant research on high-frequency radar detection methods for multi-target background in recent years [1,2,3,4,5,6,7]. The differences in statistical characteristics between jamming targets in multi-target backgrounds and in the detection background requires improvements in the robustness of the algorithm to estimate the average power level of the background clutter. Alternatively, the jamming target in the reference unit should be eliminated first to create a processed reference unit that can be used to estimate the detection threshold.

Multiple targets are also a cause of interference for the cell-averaging (CA) method, that Barboy proposed to address with a censored cell-averaging (CCA)—CFAR detector using adaptive elimination algorithms, such as VTM-CFAR and VI-CFAR [8,9,10]. Aiming to automatically remove jamming targets in the sliding window reference unit, Himonas introduced a series of CFAR detection methods, such as ACMLD-CFAR [11]. Farrouki took a different approach based on the characteristics of the ordered data variability (ODV) to develop an ACCA-ODV-CFAR detection method [12]. Further, Zaimbashi proposed an ADCCA-CFAR detection method based on the fuzzy membership function that does not need any prior information on background detection in an exponential background leading to a better detection performance than ACCA-ODV-CFAR in a multi-target environment [13]. However, all of the aforementioned detection methods are based on the assumption of a Gaussian/Rayleigh clutter model background. The clutter distribution in the actual environment may obey the non-Gaussian/non-Rayleigh distribution, and the distribution parameters are typically unknown prior to detection. These differences create difficulties in effectively detecting and removing strong targets/jamming targets, directly affecting the threshold estimation of the detector, and greatly reducing target detection performance [3,14,15,16,17,18,19]. Zhou Weizhen proposed a weighted amplitude iteration (WAI)-CFAR method for gamma (GM) distribution, which improves the detection performance under non-uniform clutter by weighting [20]. Santamaria used the subspace relationship of signals and adopted the generalized likelihood ratio (GLR) to detect the statistical model [21]. Hua Xiaoqiang proposed four linear discriminant analysis (LDA)-based matrix information geometry (MIG) detectors, which used the principle of popular projection (manifold projection) to improve the robustness of the detector to anomalies [22].

In continuation of previous research, our team and many other researchers found that the actual RD spectrum detection unit of the high frequency OTHR had a good fitting effect on the Weibull distribution [23,24,25]. In this paper, we highlight how the Weibull distribution can be used as the statistical model of the RD spectrum detection unit amplitude. In the early stages of our research, we found a significantly lower number of detection units occupied by the target compared to the background units. We refer to this feature of the target as “sparsity”. Considering that the OS-CFAR detection method has good performance for target detection in a multi-target environment, we propose a new method of high-frequency radar multi-target detection based on target sparse constraint in a Weibull clutter background [4,26,27]. This method adaptively identifies the targets/jamming targets in the detection background by sparsely limiting the target in the parameter estimation process, and then estimates the distribution parameters using all uniform background elements to improve the accuracy of parameter estimation. Our proposed detection method does not need prior information on the number of targets and distribution parameters, and can identify multi-interference targets adaptively, effectively achieving CFAR detection of targets in a multi-target environment. However, the algorithm has an issue with the accurate selection of the k value that we have addressed with a correction method. Simulation and measured data verification show that the proposed method using target sparsity constant false alarm detection has better detection performance and robustness compared to traditional detection methods. At the same time, it also verifies the positive role of sparse characteristics in target recognition, which can provide support for target/jamming target recognition in a complex background.

Here, we introduce the OS-CFAR detection method based on background distribution parameter estimation and the basic theory of target/jamming target identification using sparse characteristics. The Weibull statistical model applicable to the actual RD spectrum of OTHR is selected based on existing research results and analysis of actual data. A distributed parameter estimation model is constructed, and the model is optimized to estimate the shape parameters of the background. To further improve the performance of the algorithm, a correction method is proposed to address the inaccurate selection of the k value. Upon completing the distribution parameter estimation, the detection threshold is calculated, and the target detection is carried out. Finally, the feasibility and performance of the proposed method are verified and analyzed by simulation and experiment, which provides support for the identification of interference targets in the subsequent research.

2. Basic Theory

2.1. RD Spectrum Statistical Characteristics of High-Frequency Over-The-Horizon Radar

In the actual RD spectrum of high-frequency OTHR target detection will face the problem of multiple targets. The key to accurately identify the target in the background is to select the appropriate threshold, which, in turn, depends on the statistical model that the amplitude of the background unit obeys. In the non-Rayleigh envelope clutter, the Weibull distribution is the statistical model with the closest fit to the measured data, as has been shown in previous research. Maresca Salvatore et al. believed that the Weibull distribution had the best fitting statistical results compared to various statistical models [28]. These results were further corroborated by Shang et al., who determined that the ionospheric clutter power and Weibull cumulative probability density function are a good match [29]. Wu Longshan carried out fitting analysis on the amplitude and Weibull distribution of about 5 × 10⁶ background units in the actual RD spectrum of sky wave OTHR, bistatic HFSWR, and HFSWR, respectively, and obtained better fitting results [30]. Based on this literature, this paper detects the target in the Weibull background.

2.2. OS-CFAR Principle

We assume that the background units obey the Weibull distribution and that the clutter samples (I = 1, 2... N) are statistically independent, where N represents the number of reference units. The Weibull distribution has two parameters, scale parameter b is related to the average power, and shape parameter a is related to the distribution slope. The probability density function and cumulative distribution function of this distribution are shown below [31].

(1)$f(x)=b^{-a/2}a/2x^{(a/2)-1}\exp \left[-{(\frac{x}{b})}^{a/2}\right],x\ge 0$

and:

(2)$F(x)=1-\exp [-{(\frac{x}{b})}^{a/2}],x\ge 0$

The reference unit $x_i$(i = 1, 2,..., N) is ranked according to the magnitude, and the k-th ordered sample $x_k$ is selected as the estimation of the background clutter power level $Z$, namely:

(3)$Z=x_k$

PDF of $x_k$ is:

(4)$f_k(x)=k\left(\begin{array}{l}N\\ k\end{array}\right){[1-F(x)]}^{N-k}{[F(x)]}^{k-1}f(x)$

The detection threshold V of OS-CFAR is obtained by multiplying Z with the threshold factor T. The unit to be detected L is compared with V. If L exceeds the threshold value V, it is determined that the target exists, otherwise the target does not exist. The T value in the detection threshold is determined by the set false alarm probability, and the false alarm probability formula is:

(5)${\mathrm{P}}_{\mathrm{fa}}={\displaystyle {\int }_0^{\infty }\exp [-{(\frac{Tx}{b})}^{a/2}]}{f}_k(x)dx=\frac{N!\Gamma (N-k+1+T^{a/2})}{(N-k)!\Gamma (N+1+T^{a/2})}$

According to Equation (5), under the given false alarm probability, the value of threshold factor T is related to the shape parameter of the Weibull distribution. Therefore, the solution of threshold V in the OS-CFAR detector requires prior information on the shape parameter.

2.3. Parameter Estimation Method Based on Jamming Target Regularization

2.3.1. Regularization of the Jamming Target

Suppressing the interference of outliers to improve the robustness of the algorithm is a widely used technique in various fields, such as numerical analysis, instrument measurement, channel estimation, etc. Here, we apply this idea to the target detection of the high-frequency over-the-horizon radar. In data clustering, Ref. [32] proposes a robust clustering method based on a Gaussian mixture model, which controls the number of outliers by regularizing the negative log-likelihood function of complete data $\left(X,U\right)$ by the weighted norm. The equation is as follows:

(6)$Q\left(\Theta \right)=-L(X,U;\Theta )+\lambda {\displaystyle \sum _{i=1}^N\parallel o_i{\parallel }_{{\Sigma }^{-1}}}$

where $L(X,U;\Theta )$ is the logarithmic likelihood function of the Gaussian mixture model, $\lambda$ is the regularization parameter, and $\parallel o_i{\parallel }_{{\Sigma }^{-1}}$ is weighted norm of the outlier. The parameters and outliers of the Gaussian mixture model are estimated by minimizing $Q\left(\Theta \right)$.

2.3.2. Shape Parameter Estimation

A Gumbel distribution is the logarithmic distribution of the Weibull distribution. The logarithmic transformation leads to small range changes of the detection unit in the measured radar RD spectrum, facilitating the estimation of the jamming target in the background, and the subsequent regularization processing. Therefore, the shape parameter estimation is carried out in the Gumbel background. Given the logarithm of the Weibull background unit:

(7)$f(x)=\frac{1}{\mathrm{b}}\exp (\frac{x-\mathrm{a}}{\mathrm{b}}-\exp \frac{x-\mathrm{a}}{\mathrm{b}})$

where $a$ is the shape parameter of the Weibull distribution, $b$ is the scale parameter, and $x$ is the logarithmic amplitude of the background unit; the $x$ logarithmic likelihood function can be obtained as:

(8)$\ln f(x)=-\ln \mathrm{b}+\frac{x-\mathrm{a}}{\mathrm{b}}-\exp \frac{x-a}{b}$

An estimation of the shape parameters of the background clutter, can be obtained by maximizing the log-likelihood function, that is, solving the equations $\partial \ln f(x)/\partial b=0$, $\partial \ln f(x)/\partial a=0$. The solution of the equation is:

(9)$\widehat{b}=\frac{{\displaystyle {\sum }_{i=1}^N{x}_i\exp (\frac{x_i}{\widehat{b}})}}{{\displaystyle {\sum }_{i=1}^N\exp (\frac{x_i}{\widehat{b}})}}-\frac{1}{N}{\displaystyle {\sum }_{i=1}^N{x}_i}$

(10)$\widehat{a}=\widehat{b}\ln \left(\frac{1}{N}{\displaystyle {\sum }_{i=1}^N\exp (\frac{x_i}{\widehat{b}})}\right)$

As can be seen in Equations (9) and (10), all detection units are involved in the calculation of the parameter estimation, which establishes the jamming target’s influence on the parameter estimation—an influence that increases when the clutter background is in the multi-target scene. Here, we introduce the regularization process for reducing the interference of multiple targets, leading to a robust target detection. Regularizing the negative log-likelihood function of the detection unit to:

(11)$H(x,o)=-{\displaystyle \sum _{i=1}^N\ln (f(x-o))}+\lambda \phi (o)$

where $o$ is the interference target vector, $\lambda$ is the regularization parameter, $\phi (o)$ is the regularization term and the separable penalty function, that is $\phi (o)={\displaystyle {\sum }_{i=1}^N\phi (o_i)}$, $H(x,o)$ and is the target function. It can be seen from the formula that when the $\lambda \phi (o_i)$ increasing speed of the penalty item value is less than the increasing speed of the value $-\ln f(x_i)$, $o_i$> 0 can be obtained to identify the interference target in the background unit; otherwise $o_i$ = 0.

Upon determining the objective function, the non-convex penalty function is introduced as the regularization term of the interference target. The property of this function is specifically analyzed in [33,34]. The function expression is as follows:

(12)$\phi (t;\alpha )=\frac{1-\exp (-\alpha \left|\mathrm{t}\right|)}{\alpha }$

In this paper, the value of $\alpha$ is set to 1, so the expression of the objective function can be re-represented as:

(13)$H(x,o)=-{\displaystyle \sum _{i=1}^N\ln (f(x_i-o_i))}+\frac{\lambda }{\mathrm{b}}{\displaystyle \sum _{i=1}^N\left(1-\exp (-o_i)\right)}$

However, it can be seen from the expression that even if the jamming target is effectively identified and the partial derivative is calculated during parameter estimation, the calculated jamming target $o_i$ will still affect the accuracy of parameter estimation. To solve this problem, the target function is modified and an indicator function is introduced to identify whether the detection unit is a jamming target. If so, the elimination operation is performed. The modified objective function is:

(14)$H(x,o)={\displaystyle \sum _{i=1}^{N}I(o_i^k)}\left(-\ln f(x-o_i)+\frac{\lambda }{b}(1-\exp (-o_i))\right)$

where, $o_i^k$ is the jamming target estimation obtained from the last iteration and $I(o_i^k)$ is the indicator function. Following the iteration, if $o_i$ > 0, then $I(o_i^k)=0$, otherwise $I(o_i^k)=1$; the jamming target elimination operation is completed.

The first partial derivative of the modified objective function with respect to $o_i$ is:

(15)$\begin{array}{l}\frac{\partial H}{\partial o_i}=\frac{I(o_i^k)\partial (\ln b-\frac{x_i-a-o_i}{b}+\exp (\frac{x_i-a-o_i}{b})+\frac{\lambda }{b}(1-\exp (-o_i))}{\partial o_i}\end{array}$

Upon completion of an iteration, to prevent the identified target from participating in the estimation, the estimation of $o_i$ is revised, and the first order partial derivative expression is set to 0 to obtain:

(16)$o_i={\left[b\log (\frac{\exp (\frac{x_i-a}{b})}{1+\lambda \exp (-o_i{}^k)})\right]}_{+}$

where, $y={[x]}_{+}$ is equivalent to $y=\max (0,x)$. In addition, $o_i\ge 0$, due to:

(17)$\frac{\exp \left(\frac{x_i{}^{up}-a}{b}\right)}{1+\lambda \exp (-o_i^k)}=1$

Among them, $x_i^{up}$ is the upper boundary of $x_i$. According to the cumulative distribution function of the Gumbel distribution:

(18)$\exp \left(\frac{x^{up}-a}{b}\right)=1+\lambda =-\log (P_{fa})$

Sorted out:

(19)$\lambda =-1-\log (P_{fa})$

where $P_{fa}$ is the false alarm probability of the target recognition.

Once the estimation of jamming target $o_i$ is completed, the partial derivatives of parameters a and b can be obtained from Equation (14) and the solution of the scale parameters can be written as:

(20)$b=\frac{{\displaystyle \sum _{i=1}^{N}I(o_i)(x_i-o_i)\exp (\frac{x_i-o_i}{b})}}{{\displaystyle \sum _{i=1}^{N}I(o_i)\exp (\frac{x_i-o_i}{b})}}-\frac{{\displaystyle \sum _{i=1}^{N}I(o_i)(x_i-o_i)-\lambda {\displaystyle \sum _{i=1}^{N}I(o_i)(1-\exp (-o_i))}}}{{\displaystyle \sum _{i=1}^{N}I(o_i)}}$

The solution of the shape parameter can be written as:

(21)$a=b\log \left(\frac{1}{{\displaystyle \sum _{i=1}^{N}I(o_i)}}{\displaystyle {\sum }_{i=1}^{N}I(o_i)\exp \left(\frac{x_i-o_i}{b}\right)}\right)$

At the beginning of the iteration $I(o_i^0)=1$, all background units are involved in jamming target estimation and parameter estimation. When the objective function converges, the iterative operation ends. The algorithm estimation process is as Table 1.

2.4. Correction of the k Value

Once the background clutter is sorted, if the value of k is small, the estimation of the average power level of the background clutter is small, which increases the false alarm probability; If the value of k is larger or the jamming target is selected as the estimation of the background clutter, the estimation of the average power level of the background clutter is too large, the detection threshold is increased, and the detection of the target to be measured will be missed, reducing the detection probability. Therefore, the selection of the k value of ordered statistics plays a decisive role in the threshold generation of OS-CFAR. As has been previously reported, when the radar echo signal passes through the pulse matching filter, the output fits the bell envelope of the Sa function [35]. In this instance, the target echo energy may occupy multiple resolution units. It is no longer a point target, but an extended target occupying multiple units.

Assuming that in the N length of the background clutter data, there are R targets with protection units positioned to the left and right sides, the number of unilateral protection units is set as R1 and the number of bilateral protection units is set as R2; then R1 + R2 = R. If the background clutter unit is sorted in ascending order, the target will exist in the tail (R + R1 + 2R2). Therefore, the following equation can be derived about the selection of the k value when all targets and protection units are fully considered:

(22)$k\le N-\left(R+R1+2R2\right)$

According to [36], when the value of k is about 3N/4, the detection performance is better in the multi-target environment. Therefore, the equation is modified as:

(23)$k=\frac{3}{4}\left(N-\left(R+R1+2R2\right)\right)$

In the discussion in the previous three sections, we deduced the detection process of OS-CFAR in the Weibull clutter background. The algorithm flowchart is shown in Figure 2. In the simulation, Weibull clutter is first generated under set parameters, and a fixed number of targets are added to it. Then, we introduce regularization processing, and stop updating the shape parameter when the cost function H converges. Given the false alarm probability, we obtain the nominal factor T under this shape parameter. This value is combined with the indicator function during regularization processing to modify the value of k, thus obtaining the final detection threshold. We name the algorithm as the regularization processing algorithm of the modified k value ordered class detector (BRACOS-CFAR).

3. Simulation Results and Actual Measurement Verification

3.1. Convergence Analysis of the Shape Parameter Estimation Algorithm

In the simulation, we use the Weibull background with data length of 64, amplitude subject to scale parameter of 1.5, and shape parameter of 2. The false alarm probability is 1 × 10⁻⁴. There is a protection unit to the left and right sides of all of the units to be tested. However, the first and last units only have one protection unit.

Twenty simulation targets with a signal-to-noise ratio of 20 dB are randomly injected into the Weibull background, and the convergence simulation result obtained after $1\times {10}^4$ Monte Carlo simulations is shown in Figure 3. The algorithm undergoes about 45 iterations, and the cost function converges to a constant. The convergence to different values is due to the randomness of the data in each simulation.

3.2. Error Analysis of the Shape Parameter Estimation

In order to evaluate the effectiveness of the shape parameter estimation algorithm, we define the relative error:

(24)$e=\frac{x}{x_{real}}-1$

where $x$ is the estimated value of the parameter and $x_{real}$ is the actual value of the simulation preset parameter.

The simulation conditions are the same as those in Section 1. The relative error histogram of shape parameter estimation after $1\times {10}^4$ Monte Carlo simulations is shown in Figure 4. The relative error obtained by the algorithm deviates from zero only within a short range, indicating that the shape parameters estimated by the proposed algorithm are effective in the case of multiple targets.

3.3. Jamming Target Recognition Performance

The proposed algorithm aims to identify jamming targets in the clutter in order to increase the number of background units involved in parameter estimation. Therefore, the recognition of the jamming target has a significant impact on the final detection performance. Figure 5 shows the performance of the proposed method for jamming target recognition. The length of the simulation data is 64. The left column of the Figure 5 shows the recognition of the simulation target, and the right Figure 5 shows the recognition performance surface. According to the pictures, the recognition performance of the algorithm is related to the SCR of the target. With the improvement of the SCR of the jamming target, the recognition probability gradually increases, and can be applied to different Weibull backgrounds. The performance and quantity of target recognition are also related. When the target SCR is large enough, the algorithm can recognize jamming targets within about 20% of the background unit ratio.

3.4. Value Analysis of K

The performance of the BRACOS-CFAR detector at different k values under the Weibull distribution background is analyzed through 10,000 Monte Carlo simulations. The process is as follows:

Step1: Generate the background clutter of the Weibull distribution obeying specific scale parameters and shape parameters;

Step2: Add targets with a specific signal-to-clutter ratio to the background clutter;

Step3: Eliminate the jamming targets using the regularization algorithm and estimate the shape parameter;

Step4: Determine the threshold factor T of the shape parameter under the specific false alarm probability. Then, obtain the adaptive detection threshold V by multiplying the threshold factor T with the corresponding Z under the k value;

Step5: Perform 10,000 cycles of detection, calculate the total number of times the target to be tested is greater than the adaptive threshold V, and divide the number of times by the total number of detections to obtain the detection probability under a specific SCR. Change the SCR size, repeat the previous steps, and draw the detection probability change curve.

The K values are 26, 29, 32, 35, 38, and 41. When the target is not injected, the background clutter follows the Weibull distribution with a scale parameter of 1.5 and shape parameter of 2. The number of reference window units is 64. Five interference targets are injected, and the detection performance curve is as shown in the Figure 6.

When the reference unit injects five interference targets, the selection of the k value has a direct impact on the detection performance of the detector (Figure 6). According to Formula (23), the correct k value should be 32. When the k value is selected as 26 and 29, the detector’s performance decreases as the k value decreases. It can be known from Formula (5) that when the k value decreases, in order to maintain a constant false alarm probability, the nominal factor T will increase accordingly, resulting in a larger threshold and affecting the detection performance. With the increase of the k value, the detection performance of the detector decreases gradually. When a larger k value is selected, the selected background clutter is estimated to be the jamming target or the extended target of the adjacent unit of the jamming target, resulting in a large Z value that increases the threshold selection. In turn, a larger threshold selection causes misdetections of the target and reduces the probability of accurate detection. This verifies the feasibility of the modified k value method and makes up for the inaccurate selection of the k value.

3.5. Algorithm Performance Evaluation

Having confirmed the accuracy of the parameter estimation and corrected the k value selection, the performance of the detector was evaluated through a simulation with preset Weibull distribution parameters that are known prior to the target injection.

False alarm probabilities are set as 5 × 10⁻³ (Figure 7a) and 1 × 10⁻⁴ (Figure 7b). The blue, red, and green lines represent different Weibull clutter backgrounds with shape parameters of 2, 1.5 and 0.5, respectively. The number of background units is 64, including five targets. When the given probability of a false alarm is 5 × 10⁻³, the actual probability of a false alarm obtained does not exceed 6.1 × 10⁻³. When the given probability of a false alarm is 1 × 10⁻⁴, the actual probability of a false alarm obtained is no more than 1.4 × 10⁻⁴. Although in some cases, when the actual probability of a false alarm is slightly higher than the predetermined probability of a false alarm, it is acceptable. In general, the detector has a good characteristic of a constant false alarm.

In the simulated experiment shown in Figure 8, the false alarm probability is set as 10⁻⁴ with a total number of detection units of 100. In the Figure 8, black represents the proposed detector, blue is the contrast detector, and different detectors are marked with different symbols. This simulated experiment compares the proposed detector with other detectors by injecting a different number of simulation targets into different test backgrounds. We started by injecting 1, 5, and 10 simulation targets into the Weibull clutter background with the shape parameter of 0.5. The data represented in Figure 8a–c show that the detection performance of log-t, CA, and MLH detectors drops sharply with the increase of the number of simulation targets. Meanwhile OSGO and OS detectors maintain a good detection probability, but are outperformed by the proposed detector performance due to the modified k value selection. In addition, both OSGO and OS detectors require prior knowledge of the shape parameters, which is not necessary to obtain when operating the new proposed detector. Increasing the number of targets by injecting 10 simulation targets into the Weibull clutter background with shape parameters of 1.5 and 2, respectively (Figure 8d,e), confirms that the proposed detector can maintain an accurate detection performance in multi-target backgrounds. In contrast, the detection probability of log-t, CA, and MLH detectors is significantly reduced as a result of multi-target interference. From Figure 8f, it can be seen that for the same detection probability, the larger the shape parameter, the smaller the required SCR. Figure 8g shows the performance curves for different target numbers when the shape parameter is set to 1.5. We can see that as the number of targets increases, the performance of the detector slightly decreases, but overall it is acceptable.

These simulations show that BRACOS-CFAR can adapt to different detection backgrounds and resist a higher range of target interference. In addition, compared to the background of the Weibull clutter with different shape parameters, with the increase of the shape parameters of the Weibull clutter, the detection probability of the target will also increase under the specific signal-to-clutter ratio. In general, the proposed BRACOS-CFAR can identify targets well in the Weibull clutter background of multiple targets. Compared with some detectors, it can effectively resist the interference of multiple targets and can be applied to different Weibull detection backgrounds.

3.6. Actual Measurement Verification

Data collected by the bistatic high-frequency Surface wave radar system was used to further test the performance of BRACOS-CFAR on the actual measurement verification. Following the baseband signal processing, the received signal is converted into the ARD spectrum. In the RD spectrum obtained, the detection background is mainly composed of atmospheric noise and sea clutter. The Weibull distribution is used to describe the statistical characteristics of the amplitude of the detection unit.

As shown in Figure 9, the measured RD spectrum is mainly composed of atmospheric noise, sea clutter, and targets, and contains a large number of targets in the close-range unit. The RD spectrum shows no significant difference between the high-order sea clutter and atmospheric noise. Ignoring the first order sea clutter and the strong targets, the power of other background units is close. For a given RD spectral range profile, the first-order sea clutter is treated as a target. Therefore, it is considered that the background contains only sea clutter and targets, and the algorithm is only used for the amplitude of the background unit.

In this experiment, the specified distance unit in the RD spectrum is detected within a frequency range of (−1, 1) Doppler (Hz). There are 212 background units and the false alarm probability is set to 10⁻⁴. As shown in Figure 10, by testing multiple batches of data of the middle-distance unit of RD spectrum, the BRACOS-CFAR detection results show that the detector can identify strong targets well. Additionally, the weak targets in the strong target shelter area can also be recognized. There are often one or two more detection points compared with other detectors. Classic detectors that are used for comparison often fail to detect weak targets located in strong target occlusion areas. The main reason behind this is that when detecting weak target units, the presence of strong targets causes an increase in the detection threshold, making it hard to detect weak targets. This detection result demonstrates the excellent capability of BRACOS-CFAR to detect in the background of multi-target Weibull clutter.

4. Conclusions

This paper aims to address the problem of multi-target detection in high-frequency over-the-horizon radar background. We derived the detection process of OS-CFAR under a Weibull clutter background. In the case of unknown distribution parameters, we use regularization processing to eliminate interference targets in background clutter by utilizing the target’s sparse characteristics and the amplitude information of background cells, and accurately estimate the shape parameter of the background clutter. Considering the extension characteristics of actual engineering targets, the k value is modified to obtain the final detection threshold. Simulation and measurement data show that the proposed BRACOS-CFAR can effectively resist interference from multiple targets and maintain good detection performance. Applying the amplitude information of all background cells to the algorithm is not only suitable for multi-target scenarios in the actual RD spectrum of a high-frequency radar, but also provides a reliable method for target detection in other radar systems. Although we have carried out a lot of research in multi-target scenarios, our method still has limitations in clutter-edge scenes, which is an area for future research.

Author Contributions

Conceptualization, Y.L. and Z.W.; methodology, Z.W.; software, Y.L.; validation, Z.Z. and L.X.; formal analysis, Y.L.; investigation, Y.L.; resources, Z.W.; data curation, W.Z.; writing—original draft preparation, L.X.; writing—review and editing, L.X.; visualization, L.X.; supervision, Z.W.; project administration, Y.L. and Z.W.; funding acquisition, Z.W. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

High Frequency	HF
Over-the-Horizon Radar	OTHR
Range–Doppler	RD
Azimuth–Range–Doppler	ARD
Ordered Statistics	OS
Constant False Alarm Rate Detector	CFAR
Cell Under Test	CUT
Cell-Averaging	CA
Censored Cell-Averaging	CCA
Ordered Data Variability	ODV
Gamma	GM
Weighted Amplitude Iteration	WAI
Generalized Likelihood Ratio	GLR
Linear Discriminant Analysis	LDA
Matrix Information Geometry	MIG
High Frequency Surface Wave Radar	HFSWR
Signal-to-Noise Ratio	SCR
Detection Probability	Pd
False Alarm Probability	Pfa
Based Regularization and Correction Ordered Statistics	BRACOS
Ordered Statistic with Greatest Option	OSGO
Maximum Likelihood	MLH

Footnotes

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Figures and Table

Figure 1. Measured data of the HF radar. (a) RD spectrum of the actual HFSWR. (b) Distance profile of the RD spectrum.

Figure 2. Algorithm flow chart of the new detection method.

Figure 3. Convergence interval and mean curve of the objective function.

Figure 4. Relative error histogram of the shape parameter estimation.

Figure 5. Simulation target recognition and recognition performance surface under different backgrounds.

Figure 5. Simulation target recognition and recognition performance surface under different backgrounds.

Figure 6. Detection performance of different k values when the reference unit contains five jamming targets.

Figure 7. Actual false alarm probability.

Figure 8. Comparison of the detection performance under different simulation targets and shape parameters.

Figure 8. Comparison of the detection performance under different simulation targets and shape parameters.

Figure 9. RD spectrum and distance profile of the measured data.

Figure 10. Four consecutive data detection results.

Figure 10. Four consecutive data detection results.

Table 1

Estimation process of the shape parameter algorithm based on jamming target elimination.

Shape Parameter Estimation Algorithm Based on Jamming Target Elimination
Input: $x$, $P_{fa}$
Output: $a$
Initialization: $o_i=0,a=2,b=median(x)$
$I(o_i)=1$
2: do
$\lambda$ is Calculated by (19) 4: $o_i$ is Calculated by (16)
$b$ is Calculated by (20)
6: $a$ is Calculated by (21)
While (14) does not converge

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