Radar False Alarm Suppression Based on Target Spatial Temporal Stationarity for UAV Detecting

Abstract

Due to its ease of implementation without an additional transmitter, radar communication integrated systems using conventional communication signals have become an effective means for monitoring unmanned aerial vehicles and other aircraft. However, their non-ideal radar signal form causes strong signals to mask weak signals, and clutter suppression is required to detect a target. As the energy of a target signal is extremely low and conventional clutter suppression methods have limited performance, the residual clutter persists after the time-varying clutter is suppressed, resulting in many false alarm points on the processed range–Doppler (RD) map. The two-dimensional distribution of these false alarms on the range–Doppler is very similar to that of the target and difficult to discriminate, which seriously affects target detection and tracking. To reduce false alarm points and improve the performance of target detection, the difference in the spatial–temporal stationarity between a target signal and clutter in a short time is discussed in this paper; a radar false alarm suppression method based on a target’s spatial–temporal stationarity is proposed by using the difference in the stationarity between the target signal and false alarm points in the range, Doppler, energy and azimuth. In this algorithm, to ensure the short-time stationarity of the target, the RD map of the short-time interval sub-frames is obtained by using the sliding matching filtering method and the peak points are extracted. Then, the Mahalanobis distance between the peak points of each sub-frame is used to eliminate the false alarm points. Finally, the false alarm points are further eliminated by target tracking and the real target information to improve the radar target detection performance. The simulation experiments show that this method can eliminate more than 90% of the false alarms points while maintaining the target detection performance. The analysis of actual data obtained from the field experiment indicate that the implementation of this algorithm, which effectively suppresses false alarms, leads to improved target detection outcomes. These enhancements can facilitate the tracking of UAVs and other aircraft.

1. Introduction

In recent years, the exponential increase in the utilization of UAVs and the corresponding threats posed by their operators have rendered the accurate identification of UAVs and other aerial targets a critical measure for ensuring safety and security [1]. Therefore, it is necessary to detect the UAVs remotely, which has led to many efforts and research projects for UAV detection. Radar technology is widely used in UAV detection due to its universality and reliability, and radar communication integration technology has become a hot research field for monitoring air targets because of its low cost and wide coverage. In this field, the orthogonal frequency-division multiplexing (OFDM) signals have gained significant attention, owing to their extensive utilization across communication systems, including digital broadcasting, mobile communication and WLAN. In addition, OFDM signals typically offer the advantage of a wide bandwidth, making them a focal point of interest within an integrated radar and communication domain [2].

Although OFDM signals show considerable performance in communication, compared to traditional radar signals, weak signals are often masked by strong signals because the signal form is not specifically designed for radar detection. These strong signals include strong target signals and strong clutter, etc. However, in practical applications, the ambiguity floor of the strong clutter is the main factor obscuring targets; to achieve better detection performance, the relevant researches have mainly focused on clutter suppression [3,4,5,6,7,8,9,10,11,12]. These clutter suppression techniques are mainly based on the spectral difference between the moving target echo and the clutter, which suppresses the clutter by a filtering method and then extracts the target signal. To achieve a higher detection probability, it is imperative to suppress the strong clutter, which is characterized by a high ambiguity floor. The Extensive Cancellation Algorithm (ECA) [3] was originally introduced by projecting a signal onto the direction orthogonal to the clutter subspace, a method that is not limited to a specific signal model. However, suppressing a large region on the RD map requires a large clutter dictionary matrix, which in many cases poses significant challenges in terms of memory and computational requirements. Therefore, improvements aimed at reducing the memory footprint have mainly revolved around segmentation [4,5,6], iteration [7,8,9], and approximation [10,11,12].

However, there are some problems with these clutter suppression techniques [13,14]. First of all, when the clutter is time-varying, it exhibits a specific velocity, which constrains the enhancement factor of the moving target indication (MTI). Consequently, it is difficult to fully meet the clutter suppression requirements. In addition, only the amplitude information of the radar echo signal is used in the detection process, and the false alarm rate can be well controlled in a weak clutter environment, but in a complex strong clutter environment, the clutter and the target are not easy to distinguish from the amplitude, and clutter suppression by only controlling the amplitude threshold can easily lead to the loss of the target trace. Therefore, in an integrated radar communication system using OFDM signals, conventional radar clutter suppression technology produces serious clutter residue in the face of complex and strong clutter environments, such as sea surfaces, mountains and cities, resulting in a large number of false alarms [15]. Too many false alarms will seriously affect the performance of radar target detection. On the one hand, a large number of false alarms mixed with the target will make it difficult to accurately detect the target and cause the target to be missed. On the other hand, in the process of target tracking, these false alarms will be included in the track establishment, leading to false tracks, which will not only waste system resources but also increase the computations and complexity of the computer data processing. Therefore, it is of great importance to study the methods for suppressing false alarms [16].

There are numerous methods available to suppress false alarms in the domain of radar technology. For example, the false target detection method based on measurement fusion was adopted to effectively identify true and false targets and reduce the false alarm probability of radar [17]. A block filtering method for ground surveillance radar was proposed [18], which can reduce false alarms by calculating the risk coefficient of the target. Reference [19], using the method of inter-frame data processing, showed that the clutter in a single stagnation area can be processed in time to reduce false alarms. An error detection method was used to suppress false alarms in [20], but the suppression effect was limited. A block filtering method based on clutter feature evaluation can effectively discriminate between radar targets and clutter by comprehensive feature factors [21]; feature extraction in conjunction with a support vector machine to eliminate false alarms. Reference [22] presented an adaptive clutter suppression method for airborne pulse Doppler radar. Based on a comprehensive analysis of the clutter generation process, this method dynamically reduces false alarms by using the specific frequency and range characteristics of the clutter to improve the radar detection performance. Because the false alarms generated by residual clutter after clutter suppression have a two-dimensional distribution similar to that of the target, they are difficult to distinguish. When there are many false alarm points, it is difficult to ensure that a target is not misclassified as a false alarm while maintaining a certain false alarm rate.

Considering that false alarms are mainly generated by clutter residue after clutter suppression, they exhibit non-stationary characteristics in multiple dimensions, such as time and space. Conversely, the target points are generated from the echo reflected by the actual targets and exhibit stable characteristics in multiple dimensions. To suppress false alarms while maintaining targets, this study analyzes the characteristics of the spatial–temporal stationarity of the target signal and clutter in a short time interval, using the stationarity differences between the target signal and the false alarm in range, Doppler, energy and azimuth, and proposes a radar false alarm suppression method based on the spatial–temporal stationarity of the target. Simulations and experiments show that this method can effectively reduce false alarms and significantly improve the efficiency of target detection. The main contributions of this paper are summarized as follows:

Firstly, a time-domain model of the radar signal is introduced and the cause of false alarm points on the RD map of the radar echo signal following clutter suppression is analyzed. The difference in spatial–temporal stationarity between clutter and target in a short time is analyzed from the perspective of signal stationarity, which provides a theoretical basis for the false alarm suppression algorithm proposed later.

Secondly, the principles and specific steps of the radar false alarm suppression method based on target spatial–temporal stationarity are analyzed in detail. To obtain sub-frame data with short time intervals, a multi-frame sliding matched filtering method is proposed, that is, the data of two adjacent frames are spliced and then matched for filtering, and an explanation is provided on how to determine the length of the sliding window. then the peak points with higher energy than the floor energy in each sub-frame are extracted, and the azimuth of each peak point is obtained by using the single-snapshot angle measurement method based on sparse representation. Combining the information of range, velocity and energy, the Mahalanobis distance between the peak points in adjacent sub-frames is calculated, and the target is selected according to the Mahalanobis distance between the peak points. Finally, the target track generation method based on spatial–temporal stationarity is employed to generate the track and further eliminate the remaining false alarms.

Finally, simulations and field experiments were conducted. The results from the simulations indicate that the false alarm suppression algorithm is effective in reducing the number of false alarm points that persist following clutter suppression, while simultaneously preserving weak target signals. In addition, by comparing the target tracking results after the field experiment with the actual UAV track, it was found that more ideal target tracking results could be obtained after false alarms were suppressed by the false alarm suppression algorithm.

The organization of this paper is delineated as follows: The time-domain radar signal model is given and the stationarity difference between target signals and false alarms over a short time is analyzed in Section 2. In Section 3, the principle of the algorithm proposed in this paper is analyzed in detail and the specific steps are explained. Section 4 presents the verification of the algorithm using simulation data. Section 5 describes the use of the false alarm suppression algorithm in the field experiment. Finally, Section 6 concludes this study.

2. Theoretical Model

2.1. Radar Echo Model in the Time Domain

Given that the baseband signal transmitted by the transmitter is $s_{ref}(t)$, it is assumed that the radar echo signal contains $N_t$ moving targets, $N_c$ non-time-varying clutters (including direct waves), $N_v$ time-varying clutters and noise. Then the radar received signal can be expressed as

$$\begin{array}{l}{s}_{surv}(t)={\displaystyle \sum _{i=1}^{N_c}{A}_{c,i}{e}^{j2\pi f_{c,i}t}{s}_{ref}(t-{\tau }_{c,i})}+{\displaystyle \sum _{j=1}^{N_v}{A}_{v,j}(t)e^{j2\pi f_{v,i}(t)t}{s}_{ref}(t-{\tau }_{v,j})}\\ +{\displaystyle \sum _{k=1}^{N_t}{A}_{t,k}{e}^{j2\pi f_{t.k}t}{s}_{ref}(t-{\tau }_{t,k})}+w(t)\end{array}$$

(1)

where w(t) is additive noise. Obviously, there are $N_c$ time-varying clutters, $N_v$ time-varying clutters and $N_t$ targets in Equation (1), where $A_{\_}$, ${\tau }_{\_}$, $f_{\_}$ represent complex amplitude, delay and Doppler frequency, respectively. For time-varying clutters, $A_{v,i}(t)$ and $f_{v,i}(t)$ are time-varying non-stationary random signals. Because of the presence of clutter and noise, the target is always hidden and difficult to find in field experiments.

For better detection performance, clutter suppression is needed. The existing time-domain algorithm takes advantage of the characteristic that the direct wave and clutter in the receiving channel are the delay of the transmitted signal, and the suppression of the clutter by an adaptive filter or orthogonal projection. After being suppressed by the time-domain suppression algorithm, the radar echo is represented as

$$s_{surv}(t)={\displaystyle \sum _{k=1}^{N_t}{A}_{t,k}{e}^{j2\pi f_{t.k}t}{s}_{ref}(t-{\tau }_{t,k})}+{\displaystyle \sum _{j=1}^{N_v}{A^{\prime }}_{v,j}(t)e^{j2\pi {f^{\prime }}_{v.j}(t)t}{s}_{ref}(t-{{\tau }^{\prime }}_{v,j})}+w(t)$$

(2)

As indicated by (2), the non-time-varying clutter is effectively filtered out through suppression. In addition to the targets and noise, the echo signal also contains the time-varying clutter residue after suppression. ${A^{\prime }}_{v,i}(t)$, ${f^{\prime }}_{v,i}(t)$ are the complex amplitude and Doppler frequency of the suppressed residual time-varying clutter, respectively. Both of these signals are characterized as time-varying non-stationary signals.

The phenomenon arises from the distinct time-varying characteristics of each clutter component present in the echo signal, which complicates the accurate estimation of these characteristics necessary for achieving ‘perfect’ suppression. If the actual radar data contain a significant amount of time-varying clutter, the clutter templates constructed within the clutter matrix during clutter suppression may not align with the actual clutter. Therefore, the time-domain clutter suppression algorithm is theoretically capable of entirely filtering out non-time-varying clutter, it can only approximate the filtering of time-varying clutter through the clutter template, thus failing to achieve complete suppression. The signal is sampled at a specified sampling rate $f_s$, and the coherent processing length is $N=T{f}_s$, where T represents the duration of the base-band signal $s_{ref}(t)$, and its vector form is

$$Y={\displaystyle \sum _{k=1}^{N_t}{A}_{t,k}x({\tau }_{t,k},f_{t,k})}+{\displaystyle \sum _{j=1}^{N_v}{A^{\prime }}_{v,j}x({\tau }_{v,j},f_{v,j})}+w$$

(3)

where ${A^{\prime }}_{v,j}$ is the residual of the suppressed time-varying clutter. Although the floor energy of this residual clutter may be significantly low, following the application of the range correlation–Doppler transform, it manifests as peaks that resemble those of the target on the RD map. This phenomenon adversely affects target detection due to the emergence of these false target peaks.

2.2. Stationarity Analysis of Target and Clutter

In contrast to non-stationary random time-varying clutter, the echo state of a target is primarily influenced by its motion state. For an aerial target, let us denote the maximum velocity as $v_m$, and the maximum turning radius as r, the distance corresponding to a range cell in the RD map is $dr$, and the Doppler speed corresponding to a Doppler cell is $dv$. When the Doppler velocity of the target experiences the most significant change, the target executes a turning maneuver at its maximum speed, as shown in Figure 1.

The X-axis represents the radial direction that extends from the target to the receiving station. The range R from the target to the receiving station satisfies $R>>r$. When the target initiates a turn at point A along the Y-axis, and subsequently rotates 90 degrees to reach point C, the change in Doppler velocity is $\mathrm{\Delta }v=v_m$($R>>r$), and it takes $T=\pi r/2v_m$ for the target to traverse from point A to C. Assuming that the change in Doppler velocity from the target to point B corresponds to a single Doppler cell dv, the change in angle $\theta$ satisfies $\theta =\arcsin (dv/v_m)$, which leads to a time duration $T_1$ for the target to move from A to B.

$$T_1={\displaystyle \frac{2\theta }{\pi }}T={\displaystyle \frac{\theta r}{v_m}}$$

(4)

In field experiments, the time $T_{frame}$ of a single frame typically satisfies $T_{frame}\ge T_1$, Consequently, the change in Doppler velocity between adjacent frames serves as the primary factor influencing the positional variation of the target on the RD map. However, when the time interval satisfies $\mathrm{\Delta }t\le T_1$, the Doppler velocity change in the target is constrained to not exceed one Doppler cell within the time duration $\mathrm{\Delta }t$. This implies that the Doppler velocity v of the target remains relatively stable within $\mathrm{\Delta }t$.

The duration required for the target distance to change by one range cell is $T_2=dr/v_m$. In the context of radar systems, this duration is often set to $T_{frame}\le T_2$. This implies that the variation in target range within a single frame of data is unlikely to exceed one range cell, indicating that the change in target range R over the time duration $\mathrm{\Delta }t$ remains relatively stable. Therefore, by selecting an appropriate time interval $\mathrm{\Delta }t$, the target signal can be considered to exhibit stationary characteristics, thereby satisfying the necessary conditions for analysis.

$$\begin{array}{l}{v}_{t,k}(t)\approx v_{t,k}(t+\mathrm{\Delta }t)\\ R_{t,k}(t)\approx R_{t,k}(t+\mathrm{\Delta }t)\end{array}$$

(5)

The alteration in the target’s azimuth $\mathrm{\Delta }\theta$ is associated with the target’s velocity, range and duration of the movement, which in time $\mathrm{\Delta }t$ it is satisfied

$$\mathrm{\Delta }\theta \approx {\displaystyle \frac{v\mathrm{\Delta }t}{R}}={\displaystyle \frac{\mathrm{\Delta }R}{R}}$$

(6)

According to Equation (5), the range changes $\mathrm{\Delta }R\approx 0$ and $\mathrm{\Delta }\theta \approx 0$ within the time interval $\mathrm{\Delta }t$, it follows that ${\theta }_{t,k}(t)\approx {\theta }_{t,k}(t+\mathrm{\Delta }t)$, so the change in target azimuth $\theta$ over time $\mathrm{\Delta }t$ is relatively stable. Thus, within a short time interval $\mathrm{\Delta }t$, the target can be approximated as a static scatterer, and its scattered energy E remains relatively stable during $\mathrm{\Delta }t$.

The false alarms that occur following clutter suppression primarily arise from the time-varying clutter residue that remains after the suppression process. As indicated in Equation (2), the signal can be expressed as

$$x_{resdual}(t)={\displaystyle \sum _{j=1}^{N_v}{A^{\prime }}_{v,j}(t)e^{j2\pi {f^{\prime }}_{v.j}(t)t}{s}_{ref}(t-{{\tau }^{\prime }}_{v,j})}$$

(7)

The working environment is often suboptimal, various types of clutter, such as sea clutter and ground clutter, which are mixed in the target echoes. Sea clutter is characterized as a non-stationary, time-varying phenomenon, under adverse sea conditions, the velocity of wave can exceed 10 m/s. This velocity is significantly lower than that of aerial targets, resulting in sea clutter typically being distributed near both sides of the 0 Doppler frequency. Since it is difficult to eliminate the influence of time-varying clutter through time-domain clutter suppression, the false alarm points in RD map after clutter suppression are generated by the residual time-varying clutter.

Residual time-varying clutter $x_{resdual}(t)$ is characterized by complex amplitude ${A^{\prime }}_{v,j}(t)$ and Doppler frequency shift ${f^{\prime }}_{v,j}(t)$ are non-stationary random signals. Within time interval $\mathrm{\Delta }t$, ${A^{\prime }}_{v,j}(t)$ and ${A^{\prime }}_{v,j}(t+\mathrm{\Delta }t)$, ${f^{\prime }}_{v,j}(t)$ and ${f^{\prime }}_{v,j}(t+\mathrm{\Delta }t)$ are randomly varying and uncorrelated. Therefore, the energy, range, Doppler velocity and azimuth information of false alarm signals generated by time-varying clutter residues also manifest as non-stationary signals that evolve over time. This indicates that the locations of false alarm points are non-stationary in both temporal and spatial dimensions.

3. False Alarm Suppression Algorithm Based on Target Spatial–Temporal Stationarity

The distinction between the target signals and the clutter signals in spatial–temporal stationarity has been previously discussed. Over a short time interval $\mathrm{\Delta }t$, the target can be considered a stationary signal, whereas the clutter is characterized as a non-stationary signal, so the target state is more stable than the false alarm point. Therefore, the target and false alarm point can be distinguished according to their amplitude fluctuation, Doppler velocity change, range change and azimuth stationarity characteristics, achieving the reduction in false alarms.

3.1. Time-Domain Signal Characterization Based on Multi-Frame Sliding

The target is considered stationary over a short time. To ensure the short-time stationarity of the target, a multi-frame sliding matching filtering method is proposed. The radar target echo signal can be regarded as a sample of the reference signal after time delay and Doppler frequency shift, and it is a linear superposition of multiple target echoes, clutters and noise. The echo signal is processed through a range–Doppler two-dimensional matched filter, and the received signal is sampled at a specified sampling rate $f_s$. Assuming that the baseband signal transmitted by the radar is $s_{ref}(n),n=0,1,\cdot \cdot \cdot ,N-1$, the echo signal after clutter suppression is $s_{surv}(n)$, the two-dimensional matched filtering process of the radar echo signal can be expressed as

$$\psi (\tau ,v)={\displaystyle \sum _{n=0}^{N-1}{s}_{surv}(n)s_{ref}{s_{ref}}^{*}(n-\tau )}{e}^{-j2\pi v{\displaystyle \frac{n}{N}}}$$

(8)

Using the method of range correlation-Doppler transform to realize two-dimensional matched filtering, then Equation (8) can be rewritten as

$$\psi (\tau ,v)={\displaystyle \sum _{k=0}^{n_{b-1}}[}{\displaystyle \sum _{n=0}^{N_b-1}{s}_{surv}(n+k{N}_b)}\times s_{sref}^{\ast }(n+k{N}_b-\tau )e^{-j2\pi v{\displaystyle \frac{n}{N}}}]e^{-j2\pi v{\displaystyle \frac{k{N}_b}{N}}}$$

(9)

where $N_b$ is the length of sub-segments (in the OFDM radar communication integrated system is an OFDM symbol length), ${\mathrm{n}}_b$ is the number of sub-segments, and $N=n_b{N}_b$. If $v{\displaystyle \frac{n}{N}}\approx 0$, Equation (9) is simplified as follows:

$$\psi (\tau ,v)={\displaystyle \sum _{k=0}^{n_{b-1}}[}{\displaystyle \sum _{n=0}^{N_b-1}{s}_{surv}(n+k{N}_b)}\times s_{sref}^{\ast }(n+k{N}_b-\tau )]e^{-j2\pi v{\displaystyle \frac{k{N}_b}{N}}}$$

(10)

Bringing Equation (2) into Equation (10) will form a peak in the form of sinc function at the corresponding $({\tau }_{t,k},f_{t,k})$ and $({{\tau }^{\prime }}_{v,j},{f^{\prime }}_{v,j})$ positions. The number of false alarm points caused by clutter is much larger than the target peak due to $N_v\rangle \rangle N_t$.

To reduce false alarms by utilizing the difference in stationarity between the target signals and the clutters, a sliding matching filtering technique is used to ensure the notion of short time intervals. This involves splicing adjacent data frames and using a smaller time window for sliding matching filtering to generate RD information, as shown in Figure 2.

Splicing adjacent frames of suppressed data and sliding with $\mathrm{\Delta }t$ as the time interval to perform matched filtering can obtain several sub-frames RD_i(i = 1, 2,.., m) as shown in Figure 3.

where $m=T_{frame}/\mathrm{\Delta }t$ and RD_i can be expressed as

$${\psi }_i(\tau ,v)={\displaystyle \sum _{k=0}^{n_{b-1}}[}{\displaystyle \sum _{n=(i-1)\mathrm{\Delta }{N}_b}^{N_b+(i-1)\mathrm{\Delta }{N}_b-1}{s}_{surv}(n+k{N}_b)}\times s_{sref}^{\ast }(n+k{N}_b-\tau )]e^{-j2\pi v{\displaystyle \frac{k{N}_b}{N}}}$$

(11)

where $\mathrm{\Delta }{N}_b=\mathrm{\Delta }t\times f_s$, compared to the matched filtering applied to each frame, there is a shorter time interval between sub-frames RD_i to ensure the stationarity of the target signal.

When determining the appropriate length of the sliding window $\mathrm{\Delta }{N}_b$, the primary consideration to address is the stationarity of the target state. This ensures that the target’s position on the RD map remains constant and is not influenced by target motion during the time interval of the sliding window. The range dr associated with each range cell in the RD map and the Doppler velocity dv corresponding to each Doppler cell are define as follows:

$$\begin{array}{c}dr=c/(2*f_s)\\ dv=c/2*f_0*(n_b\times T_b)\end{array}$$

(12)

where $T_b=N_b/f_s$. Bringing Equation (12) into Equation (8), it can be seen that the sliding window time should meet $\mathrm{\Delta }{T}_b\le \min (T_1,T_2)$, $\mathrm{\Delta }{N}_b=\mathrm{\Delta }{T}_b{f}_s$. Since the smaller $\mathrm{\Delta }{N}_b$ is, the more sub-frames RD_i will be, which will increase the computational and memory load. Therefore, the specific value of $\mathrm{\Delta }{N}_b$ should be selected according to the actual situation.

Following the application of multi-frame sliding matching filtering, extract all the peaks whose energy is higher than the floor energy on the RD map of the sub-frame. This process yields the information regarding the range R, Doppler velocity v and energy E of these peaks. While it is assured that the target peak is present among the numerous peaks, distinguishing the target peak remains challenging due to the prevalence of false alarms.

3.2. Spatial Signal Characterization Based on Single-Snapshot Measurement Angle

Upon acquiring the range R, Doppler velocity v and energy E on the sub-frame RD map, it is essential to determine the azimuth information of these peaks in order to effectively differentiate the target from false alarms. Given the data utilized in this study are all single-snapshot data, the DOA estimation method based on sparse representation has been selected for azimuth estimation. This method is characterized by its high precision and resolution, and it does not necessitate decoherence pre-processing nor impose any requirements on the number of snapshots of the data [23].

Consider partitioning the airspace into $\{{\theta }_1,{\theta }_2,\cdot \cdot \cdot ,{\theta }_L\}$ with equal angles, and assume that each possible angle ${\theta }_n$(n = 1,2,...,L) corresponds to a potential target signal $s({\theta }_n)$. Then the array manifold matrix’s $\Phi =\{a({\theta }_1),a({\theta }_1),\cdot \cdot \cdot ,a({\theta }_L)\}$ each column corresponds to the azimuth information of a potential target signal. To accurately reflect sparsity, the number of potential targets L should significantly exceed the actual number of targets N. so a $L\times 1$ dimensional sparse signal $S={[s_1,s_2,\cdot \cdot \cdot ,s_L]}^T$ is constructed, where $s_n=s({\theta }_n)$ (n = 1,2,...,L), only the N positions ${\theta }_n$ corresponding to actual targets possess non-zero value of $s_n$, while the other L-N positions are all zero values. The model for the sparse representation of the signal is

$$Y=\Phi S+E$$

(13)

where $Y=Y_j$ is the snapshot data corresponding to the j-th range gate after pulse compression; E is the noise component within the data; The matrix Φ serves as a redundant dictionary, comprising steering vectors for all potential angles in space, namely

$$\Phi =\{a({\theta }_1),a({\theta }_1),\cdot \cdot \cdot ,a({\theta }_L)\}$$

(14)

As can be seen from Equation (13), the single-snapshot data Y can be represented sparsely using the redundant dictionary Φ, and its representation coefficient is the sparse signal S. By solving the sparse signal S and identifying the locations of its non-zero elements, it becomes possible to estimate the direction of arrival (DOA) angle of the target. To address this issue, a constraint criterion that minimizes the $l_p$ norm (0 < p < 1) is employed, and the concept of regularization is introduced into the algorithm to mitigate the influence of noise on the reconstruction algorithm, thereby enhancing the accuracy of angle estimation [23]. The DOA estimation problem of (13) can be equivalent to solving the following problem.

$$\underset{s}{\min }{‖S‖}_p\hspace{1em}\mathrm{s}.\mathrm{t}\hspace{1em}{‖Y-\Phi S‖}_2^2\le {\beta }^2$$

(15)

where ${‖S‖}_p={\displaystyle \sum _{i=1}^L{\left|S(i)\right|}^p}$, 0 < p < 1 is $l_p$ norm.

Under the maximum a posteriori probability (MAP) criterion, the solution of (15) can be expressed as

$${\stackrel{⌢}{S}}_{MAP}=\underset{S}{\arg \min }J(S)=\underset{S}{\arg \min }{‖Y-\Phi S‖}_2^2+\eta {‖S‖}_p$$

(16)

where the cost function is $J(S)={‖Y-\Phi S‖}_2^2+\eta {‖S‖}_p$, the regularization coefficient is $\eta ={\sigma }^2/{\beta }^p$, ${\sigma }^2$ is noise variance and $\beta =\sqrt{{\displaystyle \frac{1}{2^{(2/p)}}}{\displaystyle \frac{\Gamma (1/p)}{\Gamma (3/p)}}}$ [24].

To obtain the optimal solution of (16), it is necessary to meet its necessary conditions.

$${\displaystyle \frac{\partial J(S)}{\partial S}}=2{\Phi }^H\Phi S_{*}-2{\Phi }^{H}Y+2\lambda \Sigma S_{*}=0$$

(17)

where matrix $\Sigma =diag\{{\left|S(1)\right|}^{P-2},\cdot \cdot \cdot ,{\left|S(L)\right|}^{P-2}\}$, parameter $\lambda ={\displaystyle \frac{\left|p\right|}{2}}\eta ={\displaystyle \frac{\left|p\right|}{2}}{\sigma }^2/{\beta }^p$. It can be obtained from Equation (17).

$$({\Phi }^H\Phi S+2\lambda \Sigma )S_{*}={\Phi }^{H}Y$$

(18)

The weighted matrix is introduced by using the idea of a weighted minimization iterative solution of Focus [25] algorithm.

$$W={\Sigma }^{-{\displaystyle \frac{1}{2}}}=diag\{{\left|S(1)\right|}^{1-(P/2)},\cdot \cdot \cdot ,{\left|S(L)\right|}^{1-(P/2)}\}$$

(19)

Bring into Equation (18)

$$S_{*}=W{({(\Phi W)}^H\Phi W+\lambda I)}^{-1}{(\Phi W)}^{H}Y$$

(20)

Introduce an iterative process, let $W_{k+1}=diag\{{\left|S_k(1)\right|}^{1-(P/2)},\cdot \cdot \cdot ,{\left|S_k(L)\right|}^{1-(P/2)}\}$ and $U_{k+1}={\Phi }_k{W}_k$, the optimal solution can be solved iteratively by

$$S_{k+1}=W_{k+1}{U}_{k+1}^H{(U_{k+1}{U}_{k+1}^H+\lambda I)}^{-1}Y$$

(21)

By solving for the sparse solution $\widehat{S}=S_k$, identify the positions of the non-zero elements or the N largest elements ${\widehat{s}}_n$ in $\widehat{S}$, and then based on the correspondence of $s_n~{\theta }_n$, the estimated DOA value ${\widehat{\theta }}_n$ of the target can be obtained.

$${\widehat{\theta }}_n=\underset{\theta }{\mathrm{argmax} }{P}_{CS}(\theta )=\underset{\theta }{\mathrm{argmax}}{\displaystyle \frac{\left|\widehat{S}\right|}{\max \text{{}\left|\widehat{S}\right|\text{}}}}$$

(22)

where $P_{CS}$ is the normalized space spectrum. From this, the angle estimation of any $(\tau ,f)$ peaks may be obtained.

As illustrated in Figure 4. By estimating the azimuth of each peak point extracted from the RD map of each sub-frame and integrating the range and Doppler information of each peak point, the azimuth spectrum for each peak point within each sub-frame can be obtained, and the azimuth of each peak point are expressed in different colors.

3.3. False Alarm Reduction Method Based on Mahalanobis Distance

The Mahalanobis distance was proposed by the Indian statistician Mahalanobis P C and is a generalization of the Euclidean distance. It calculates the distance between two points by covariance, which is an effective way of calculating the similarity between two unknown sample sets [26].

The peak information matrix $X_i=[x_{i,1},x_{i,2},\cdot \cdot \cdot ,x_{i,N_p^i}]$ in sub-frame RD_i (i = 1,2,…, m) was calculated through the above section. Each sub-frame RD_i has $N_p^i$ peaks, where $x_{i,n}(n=1,2,\cdot \cdot \cdot ,N_p^i)$ is the peak information vector $x_{i,n}=[R_{i,n},v_{i,n},E_{i,n},{\theta }_{i,n}]$ obtained from the RD map of the i-th sub-frame, $R_{i,n},v_{i,n},E_{i,n},{\theta }_{i,n}$ represent the range bin, Doppler bin, energy, and azimuth information corresponding to each peak, respectively. Assuming that there are $N_t$ targets, the peak information matrix $X_i$ can be reformulated as

$$X_i=[X_t,W_d^i]=[x_{i,1},\cdot \cdot \cdot ,x_{i,N_t},{\omega }_{i,1}\cdot \cdot \cdot ,{\omega }_{i,N_d^i}]$$

(23)

where $X_t$, $W_d^i$ are the target peak matrix and the false peak matrix of the sub-frame RD_i, respectively, and $N_p^i=N_t+N_d^i$.

For each peak information vector $x_{i,n}$ in the peak information matrix $X_i$ of the sub-frame RD_i, it can be observed from the range–Doppler information obtained on the RD map that the majority of the peaks in the peak information matrix $X_{i+1}$ of the adjacent sub-frame RD_i+1 do not correspond to $x_{i,n}$. To reduce computational complexity and enhance algorithmic efficiency, an initial pairing can be conducted based on the range–Doppler information. For this purpose, the Euclidean distance is used, range–Doppler ($R_{i,n},v_{i,n}$) position of any peak $x_{i,n}$ in the sub-frame RD_i, a circular region centered on ($R_{i,n},v_{i,n}$) with $r_0$ as the radius is defined as the matching range. The peaks $x_{i+1,k}$$(k=1,2,\cdot \cdot \cdot ,N_p^{i+1})$ in the adjacent sub-frame RD_i+1 satisfies

$$\mathrm{ED}(x_{i,n},x_{i+1,k})=\sqrt{{(R_{i,n}-R_{i+1,k})}^2+{(v_{i,n}-v_{i+1,k})}^2}\le r_0$$

(24)

It can be considered that $x_{i+1,k}$ corresponds to $x_{i,n}$ and should be retained, while any peaks that do not successfully match can be considered false alarms and subsequently eliminated. The matching range is generally set at 2–3 cells, which not only ensures that the target is preserved, but also minimizes the occurrence of false alarms. Consequently, the matrices of matching peak information ${X^{\prime }}_i$ and ${X^{\prime }}_{i+1}$ of adjacent sub-frames are obtained, resulting in a reduction in peak information vectors. This preliminary reduction in the false alarm rate is achieved, thereby decreasing the computational effort required for subsequent steps.

To further decrease the false alarm rate, the Mahalanobis distance between the matched peak ${x^{\prime }}_{i,n}$ in sub-frame RD_i and the matched peak ${x^{\prime }}_{i+1}$ in sub-frame RD_i+1 is computed by utilizing the peak distance, Doppler velocity, energy and azimuth information. The minimum Mahalanobis distance is then designated as the distance $d_{i,n}$ from the peak ${x^{\prime }}_{i,n}$ to sub-frame RD_i+1. This procedure is shown in Figure 5. For $d_{i,n}$ there is

$$d_{i,n}=\min \left(\mathrm{MD}\right({x^{\prime }}_{i,n},{x^{\prime }}_{i+1,k}\left)\right)=\min (\sqrt{{({x^{\prime }}_{i,n}-{x^{\prime }}_{i+1,k})}^{\mathrm{T}}{\Sigma }^{-1}({x^{\prime }}_{i,n}-{x^{\prime }}_{i+1,k})})$$

(25)

$\mathrm{MD}({x^{\prime }}_{i,n},{x^{\prime }}_{i+1,k})=\sqrt{{({x^{\prime }}_{i,n}-{x^{\prime }}_{i+1,k})}^{\mathrm{T}}{\Sigma }^{-1}({x^{\prime }}_{i,n}-{x^{\prime }}_{i+1,k})})$is the Mahalanobis distance from ${x^{\prime }}_{i,n}$ to ${x^{\prime }}_{i+1,k}$. ${\Sigma }^{-1}$ is the inverse of ${x^{\prime }}_{i,n}$ and ${x^{\prime }}_{i+1,k}$ covariance matrix.

$$\Sigma =\left[\begin{array}{cc}\mathrm{cov}({x^{\prime }}_{i,n},{x^{\prime }}_{i,n})& \mathrm{cov}({x^{\prime }}_{i,n},{x^{\prime }}_{i+1,k})\\ \mathrm{cov}({x^{\prime }}_{i+1,k},{x^{\prime }}_{i,n})& \mathrm{cov}({x^{\prime }}_{i+1,k},{x^{\prime }}_{i+1,k})\end{array}\right]$$

(26)

If the covariance matrix is a diagonal matrix, the Mahalanobis distance simplifies to the Euclidean distance. By employing the aforementioned operations, we can obtain the Mahalanobis distance $d_{i,n}=[d_{i,1},d_{i,1},\cdot \cdot \cdot ,d_{i,N_p^i}]$ from all peaks ${x^{\prime }}_{i,n}$ in the sub-frame RD_i to the adjacent sub-frame RD_i+1. Given that the target signal is stationary in relation to false alarms over a short time interval, the Mahalanobis distance associated with the target is less than that of the false alarm. Therefore, to detect $N_t$ targets, it is necessary to identify the peaks corresponding to the $N_t$ smallest Mahalanobis distances in $D_{i,n}$.

The precise number of targets in the field experiment remains indeterminate; therefore, we set a reference threshold denoted as d_thr. When the Mahalanobis distance $D_{i,n}$ from all peaks ${x^{\prime }}_{i,n}$ in sub-frame RD_i to the adjacent sub-frame RD_i+1 is less than the reference threshold d_thr, the corresponding peak points can be classified as suspected targets $X_t^i$. The reference threshold d_thr is defined as the Mahalanobis distance at which the target peak information vector exhibits the greatest variation between adjacent sub-frames. Assuming that the target information in a sub-frame is represented as $T_i=[R_i,v_i,E_i,{\theta }_i]$ and will transition to $T_{i+1}=[R_i+\mathrm{\Delta }R,v_i+\mathrm{\Delta }v,E_i+\mathrm{\Delta }E,{\theta }_i+\mathrm{\Delta }\theta ]$ when it experiences the maximum change in the adjacent sub-frames. Based on practical engineering experience, $\mathrm{\Delta }R$, $\mathrm{\Delta }v$ usually does not exceed 2 cells, $\mathrm{\Delta }E$ does not exceed 5 dB, and $\mathrm{\Delta }\theta$ is 1/2 beam width, so the reference threshold d_thr is.

$$d_{thr}=MD(T_i,T_{i+1})$$

(27)

The final target detection result should be the intersection of multiple sub-frame detection results.

$$X_t=X_t^i\cap X_t^{i+1}\cap \cdot \cdot \cdot X_t^{m-1} i=1,2,\cdot \cdot \cdot ,m-1$$

(28)

Through the above steps, many false alarm points are eliminated.

3.4. Target Track Generation

Following the suppression of false alarms, some residual false alarms remain on the RD map, but these false alarms do not form a stable trace, So the trace can be generated based on the temporal and spatial stability of the target, allowing for the further removal of scattered points. The specific implementation method is as follows: first, all peak points from the first frame of data are taken as the start of the trace; Then the Mahalanobis distance and SNR between points in adjacent frames are employed as the target association criterion; If a target trace is not established in the first frame, newly generated points in each subsequent frame that are not associated with the previous frame are considered as the initiation of the new trace; in order to minimize false alarms, reduce computational load and eliminate multiple frames that lack an associated trace. The process for updating the trace is detailed follows:

(1)

Prediction: Assume that the state of the uncut trace $\left\{T_{k-1}\left(i\right)\right\}$ at time k − 1 is $\left\{x_{k-1}\left(i\right)=\left[R_{k-1}\left(i\right),D_{k-1}\left(i\right),{\theta }_{k-1}(i)\right]\right\}$, with the three parameters representing range, velocity and azimuth, respectively. Assuming that the aircraft travels in a straight line at a constant speed during the subsequent frame time, the state of the point trace at time k can be predicted as follows.

$$p_k=\left[\begin{array}{ccc}1& -{\displaystyle \frac{T\lambda }{2}}& 0\\ 0& w& 0\\ 0& 0& 1\end{array}\right]x_{k-1}$$

(29)

Equation (29) considers that after a duration of time T, the target’s range state undergoes a transformation in accordance with the Doppler state, while the azimuth angle remains constant. The Doppler state is expected to experience minor variations as a result of positional changes relative to the receiving station throughout the flight, so w is defined as the parameter for adjusting the Doppler state.

(2)

Association: $\left\{x_k\left(j\right)=\left[R_k\left(j\right),D_k\left(j\right),{\theta }_k\left(j\right)\right]\right\}$ as the point set detected at time k is associated with the uncut off trace $\left\{T_{k-1}\left(i\right)\right\}$ according to the following criteria:

$\left\{{x_k}^{\prime }\left(m\right)=\left[R_k\left(m\right),D_k\left(m\right),{\theta }_k\left(m\right)\right]\right\}{|}_{\left\{x_k\left(j\right)\right\}\in U_i}$ define as the neighborhood point set in the range of 2 range cells, 4 Doppler cells and 1/2 beam-width (assuming that the range is $U_i$) around the predicted state $p_k\left(i\right)$ of the trace $T_{k-1}\left(i\right)$ is determined through screening; If the neighborhood point set $\left\{x_k^{\prime }\left(m\right)\right\}$ is empty, the trace state is updated directly to $p_{k+1}=\left[\begin{array}{ccc}1& -{\displaystyle \frac{T\lambda }{2}}& 0\\ 0& w& 0\\ 0& 0& 1\end{array}\right]p_k$; If the neighborhood point set $\left\{x_k^{\prime }\left(m\right)\right\}$ contains at least one point, the point exhibiting the strongest SNR is selected as the relevant point.

(3)

Evaluate the quality of the trace $\left\{T_k\left(i\right)\right\}$ and update the trace: A trace that remains unassociated with a new point will be terminated if it is not associated for three consecutive frames, For the associated trace, to reduce the influence of trace offset resulting from measurement errors and process errors, the state of the trace is smoothed as follows:

$$x_k\left(i\right)=x_{k-1}\left(i\right)+p\left(x_k^{\prime }\left(m\right)-x_{k-1}\left(i\right)\right)$$

(30)

p represents the smoothing coefficient, which assumes a value within the range of 0 to 1. For points that lack association with any existing trace, these are designated as new starting point trace. As shown in Figure 6. Through the tracking process, the target points that contribute to a stable trace are preserved, resulting in the formation of a target track. Conversely, scattered points that do not contribute to a coherent track are eliminated, thereby reducing false alarms and enhancing target detection capabilities. The purpose of target tracking is to further eliminate scattered false alarm points that do not contribute to a defined track. However, the current methodology exhibits limitations in robustness when confronted with complex moving targets. Future research should focus on exploring more advanced and robust track correlation technologies to improve the performance of the algorithm.

4. Simulation Experiment Results

Let $s_{sref}$ represent the transmitted signal and $s_{surv}$ denote the radar echo, with a total of N frames of data. Initially, adjacent data frames are concatenated, resulting in the generation of the sub-frame range–Doppler (RD) spectrum through the application of a sliding matched filter with a specified time interval $\mathrm{\Delta }t$. Subsequently, all peak points within the sub-frame RD map are extracted, allowing for the determination of their respective range, Doppler, energy and azimuth information. The Mahalanobis distance is then employed to reduces false alarm points, and finally track the target, further eliminating false alarms that do not contribute to a stable track. The final output y is the target detection results for each frame after false alarm suppression. The whole process is as follows (Algorithm 1):

Algorithm 1: Radar false alarm suppression

Input: $s_{sref}$, $s_{surv}$, ∆t, Output: y 1:   for n = 1, 2, …,N-1 2:       $S_{sref}=\mathrm{cat}(2,S_{sref}^n,S_{sref}^{n+1})$  3:       $S_{surv}=\mathrm{cat}(2,S_{surv}^n,S_{surv}^{n+1})$ 4:       $R{D}_i=\mathrm{makeRD}(\mathrm{\Delta }t,S_{sref},S_{surv})$ 5:       Xi = findpeak(RDi) 6:           for j = 1:size(Xi,1) 7:                 ${\widehat{\theta }}_j=\underset{\theta }{\mathrm{argmax} }{P}_{CS}(\theta )=\underset{\theta }{\mathrm{argmax}}{\displaystyle \frac{\left|\widehat{S}\right|}{\max \text{{}\left|\widehat{S}\right|\text{}}}}$ 8:       end 9:        return Xi = [x1,x2,...], xj= [R,v,e,$\theta$] xj∈ Xi 10:     for i = 1, 2, … 11:           $\mathrm{ED}(x_{i,n},x_{i+1,k})=\sqrt{{(R_{i,n}-R_{i+1,k})}^2+{(v_{i,n}-v_{i+1,k})}^2}\le r_0$ 12:           return  ${X^{\prime }}_i$ ${X^{\prime }}_{i+1}$ 13:           $d_{i,n}=\min \left(\mathrm{MD}\right({x^{\prime }}_{i,n},{x^{\prime }}_{i+1,k}\left)\right)$ 14:           return  $X_t=X_t^i\cap X_t^{i+1}\cap \cdot \cdot \cdot i=1,2,\cdot \cdot \cdot$ 15:       end 16:   end 17:   for k = 1, 2,…,N-1 18:       initialization $\left\{T_{k-1}\left(i\right)\right\}$ 19:       prediction $p_k=\left[\begin{array}{ccc}1& -{\displaystyle \frac{T\lambda }{2}}& 0\\ 0& w& 0\\ 0& 0& 1\end{array}\right]x_{k-1}$ 20:       association $\left\{T_k\left(i\right)\right\}$ 21:       evaluation $x_k\left(i\right)=x_{k-1}\left(i\right)+p\left(x_k^{\prime }\left(m\right)-x_{k-1}\left(i\right)\right)$ 22:   end 23:   return y

4.1. Conventional Time-Domain Suppression

In this section, we conducted a simulation experiment to evaluate the performance of the algorithm. The radar signal utilized is OFDM signal, with a pitch angle set at 15 degrees, primarily aimed at detecting aerial targets within an azimuth range of 100–160 degrees. We randomly introduced 12 simulation targets into the dataset, each characterized by varying ranges, velocities and energy levels. The radar echo comprises both the target echoes and various forms of clutter. The original RD map is generated as shown in Figure 7. Obviously, in the raw data, all targets are obscured by the clutter’s energy.

In order to effectively identify the target, it is essential to suppress clutter present in the original dataset. The time-domain cancellation algorithm is employed for this purpose, as shown in Figure 8. Following the application of clutter suppression, a substantial reduction in clutter is achieved, leading to a decrease in floor energy, thereby allowing for the clear display and identification of the simulation target. The simulation targets highlighted within the red circle in the figure represent high SNR targets with a minimum SNR of 11.2 dB post-suppression. This high SNR is attributable to the high energy levels associated with these targets, rendering them highly distinguishable on the RD map. Conversely, the simulation target located at (128 km, −7.044 Hz), which is encircled in yellow has a low SNR of merely 7.8 dB after clutter suppression, a consequence of its lower energy configuration, resulting in its diminished prominence on the RD map.

4.2. Multi-Frame Sliding Matching Filtering

To achieve short-term stationarity of the target, the two adjacent suppressed frames of data are concatenated, and the sliding matching filter is used to produce sub-frames with uniform accumulation time. The duration of the sliding window is $\mathrm{\Delta }{N}_b=1/3N_b$, it can be inferred that from every two original frames, four sub-frames of data can be generated.

Figure 9 is a Doppler slice of the low SNR simulation target (128 km, −7.044 Hz). This target exhibits a peak that exceeds the floor energy at the corresponding position. Consequently, this target will remain detectable when extracting all peaks that surpass the floor energy on the RD map, thereby facilitating the generation of the peak information matrix.

The comparison of the Doppler slices of the RD map for the low SNR simulation target in the first and second sub-frames is shown in Figure 9. The target exhibits a peak that essentially overlaps on the RD map of both sub-frames, with the primary differences being in energy levels. The peak energy of the target in the first sub-frame is measured at −5.2 dB, while in the second sub-frame it is −2.4 dB, indicating an energy variation of less than 3 dB. In comparison to the majority of other peaks, the behavior of the target peak demonstrates greater stability across the two sub-frames. This observation suggests that, under an appropriately selected sliding window length, the weak target signal maintains relative temporal stability, whereas the false alarms exhibit more randomness. Consequently, this characteristic can be leveraged to reduce the false alarm rate.

4.3. Spatial Signal Processing Based on Single-Snapshot Angle Measurement

Based on the aforementioned operations, it is possible to extract the range, Doppler and energy information of all peaks exhibiting energy levels above a specified threshold within each sub-frame. However, to construct a comprehensive peak information matrix, it is essential to ascertain the azimuth information for each peak in every sub-frame, which is crucial for effectively differentiating between actual targets and false alarms.

Using the weak simulation target located at (128 km, −7.044 Hz) as a case study, we orient it at an angle of 135.5 degrees. Given that the data utilized are derived from a single-snapshot, we employ, we use the spatial signal processing technique based on single-snapshot angle measurement, as discussed in Section 3 to estimate its DOA. The results of the spatial spectrum estimation are shown found at Figure 10. The maximum sparse solution for this target is in 135.5 degrees, which aligns precisely with our initial configuration.

The application of this method for estimating the DOA of the peak points within each sub-frame facilitates the acquisition of their corresponding azimuth information, thereby enabling the generation of the peak azimuth spectrum. The comparison of the peak azimuth spectrum from two adjacent sub-frames is shown in Figure 11. In this figure, the dot represents the peak azimuth spectrum of sub-frame 1, while the cross represents the peak angle spectrum of sub-frame 2. The target under consideration is a weak simulation target, with color coding employed to indicate the values of the azimuth angles. The weak target is successfully detected in both adjacent sub-frames through the application of sliding matching filtering, with detections occurring in the direction of the main lobe. Notably, the azimuth spectrum of the two sub-frames exhibit distinctly different peaks at varying ranges, Doppler shifts and azimuth angles, suggesting that these peaks are spatially unstable in comparison to the target. Consequently, the spatial stationary characteristics allow for the differentiation of the target from false alarms.

4.4. False Alarm Suppression Based on Mahalanobis Distance

Following the acquisition of peak range, Doppler, energy, and azimuth information within each sub-frame, false alarms are reduced in accordance with aforementioned methodology. Peaks exhibiting a Mahalanobis distance that is less than the reference threshold d_thr in each sub-frame are preserved, while those with a Mahalanobis distance exceeding the reference threshold d_thr are attenuated to the floor energy level.

As shown in Figure 12, numerous bright spots are observed to vanish on the RD map following the implementation of false alarm suppression, which means that a substantial number of false alarms have been effectively eliminated. Initially, there were 12,608 detectable peak points on the RD map prior to the application of false alarm suppression; however, only 517 peak points persisted post-suppression. This signifies that over 95% of the false alarm points have been removed. Notably, for the weak target indicated within the yellow circle, the corresponding weak target peak remains detectable on the RD map and its energy level remains unchanged after the suppression of false alarms. This finding suggests that the radar false alarm suppression method, which is predicated on the principles of target spatial–temporal stationarity, is effective in reducing reduce the false alarm rate while simultaneously preserving the weak target signal.

False alarm suppression is essentially a binary classification task aimed at distinguishing between genuine targets and false alarms. To demonstrate the advantages of Mahalanobis distance in this context, we compare it with classical classification algorithms such as SVM and DBSCAN. As shown in Figure 13. prior to the implementation of false alarm suppression, a total of 12,608 peaks were detectable on the RD map. Following the application of various suppression, the application of Mahalanobis distance resulted in the retention of 517 peak points, thereby achieving a false alarm suppression rate of 95.8%. In contrast, the SVM retained 1858 peaks, yielding a suppression rate of 85.2%, while DBSCAN retained 2644 peaks, corresponding to a suppression rate of 79.0%. Thus, Mahalanobis distance exhibited superior efficacy in suppressing false alarms and was subsequently selected for our implementation.

4.5. Target Track Generation Based on Spatial–Temporal Stationarity

Two simulation tracks were positioned in proximity to the range–Doppler unit coordinates (449, −101) and (642, −71), with each track spanning a duration of ten frames of data. The generation of target tracks, predicated on spatial–temporal stationarity is executed subsequent to the processing of ten frames of data utilizing conventional time-domain clutter suppression and false alarm suppression techniques.

The tracking results are shown in Figure 14. The track generation method based on spatial–temporal stationarity effectively tracked two simulated trajectories. This indicates that when a target can establish a stable trajectory, the points along that trajectory can be consistently tracked and preserved throughout the tracking process. Therefore, by using the spatial–temporal stationarity of the target to generate the trajectory, it is possible to further eliminate scattered false alarm points while simultaneously maintaining the target track.

To further investigate the performance of the false alarm suppression algorithm, this study implemented a Monte Carlo experiment. A total of 10 simulated moving targets, positioned variably but maintaining the same SNR, were introduced across 20 consecutive frames of data. This setup effectively resulted in the simulation of 200 simulation targets over the 20 frames. The tracking algorithm was used to track the trajectories of the simulated targets both prior to and following the application of false alarm suppression, with SNR settings were varied from −5 dB to 15 dB, with a total of 25 rounds performed. The percentage of correctly tracked targets relative to the total number of targets in each round of tracking results is calculated as the target detection probability in order to assess the enhancement in target detection capability afforded by the false alarm suppression algorithm, as well as its impact on algorithm performance under varying simulation conditions.

As shown in Figure 15. The probability of detection is low when the target signal is weak, however, it increases as the SNR rises. Once the SNR exceeds 12 dB, the detection probability begins to stabilize at approximately 96%, although it does not reach 100% detection. Furthermore, it can be seen that following the implementation of false alarm suppression, the detection probability for targets significantly improves compared to the pre-suppression state. This enhancement is particularly pronounced for weak targets, which shows that the false alarm suppression algorithm has significantly improved the detection capability for weak targets.

5. Experimental Results

To assess the efficacy of the algorithm in practical applications, we employed a radar signal integration system utilizing OFDM signals to conduct field experiments. The radar operates at a carrier frequency of 11 MHz, with a detection range of 160 km and a pitch angle of approximately 15 degrees. This configuration facilitates the detection of targets within an azimuth angle of 60–120 degrees. Concurrently, several drones were deployed to fly around the test site, recording their tracking information, which serves as a basis for comparing the tracking results produced by the algorithm.

As shown in Figure 16. The clutter present on the original RD map primarily consists of ground object clutter and noise. This clutter base contributes to an increase in the floor energy of the original RD map, which leads to obscuring the target. Therefore, it is imperative to implement clutter suppression techniques to diminish the floor energy and thereby enhance the visibility of the target peaks. The results of traditional time-domain suppression are shown in Figure 17. Following the application of clutter suppression, the floor energy is reduced by approximately 25 dB, which is adequate for revealing the target. However, the presence of residual clutter leads to a significant number false alarms in the suppressed RD map, adversely affecting target detection.

To enhance target detection performance, radar false alarm suppression is conducted based on the spatial–temporal stationarity of the target within the suppressed data, with the results are shown in Figure 18. Prior to the implementation of false alarm suppression, a total of 210,952 peak points were extracted from the entire RD map, following the suppression process, 14,634 peak points were retained. This indicates that Over 93% of the false peaks in the RD map were successfully eliminated. When applying the same methodology to continuous multi-frame, the average effectiveness of the multi-frame data processing can achieve a reduction in false alarms by approximately 91%.

In order to enhance the screening of targets and to evaluate the impact of false alarm suppression on target tracking, a target tracking algorithm based on the spatial–temporal stationarity of the target is employed. This algorithm is applied to track the multi-frame peak points identified prior to and following the suppression of false alarms. The results of this tracking are then compared with the actual trajectory of the UAVs.

Figure 19 and Figure 20 illustrate the tracking results prior to and following the implementation of false alarm suppression, respectively. In these figures, the line connecting the dots represents the tracked trajectory, while the scatter points denote the actual trajectory of the UAVs, with the color of each point indicating the azimuth. The field experiments encountered significant interference without the application of false alarm suppression subsequent to clutter suppression, false alarms would be mixed into the tracking process. Given the instability of these false alarms, their presence can disrupt the tracking continuity, thereby complicating the accurate tracking of the actual UAVs’ trajectory. However, the algorithm proposed in this study, which incorporates false alarm suppression, yields tracking results that closely align with the actual trajectory of the UAV. This outcome effectively demonstrates the efficacy of the radar false alarm suppression algorithm based on the space-time stationarity of the target.

6. Conclusions

This paper examines the reasons for the existence of numerous false alarms on the RD map following conventional clutter suppression. Considering the difference in stationarity between target signals and clutter over short time interval, we propose a radar false alarm suppression algorithm based on the distinctions in stationarity between target signals and clutter in range, Doppler, energy and azimuth. Initially, to ensure the short-time stationarity of the target, the RD map of sub-frames with short time interval is obtained by using a sliding matching filtering method, followed by the extraction of peak points. Subsequently, the Mahalanobis distance between the peak points of each sub-frame is employed to eliminate false alarm points. Finally, further refinement is achieved through target tracking, which retains genuine target information and enhances radar target detection performance. Simulation results indicate that this method can reduce false alarms by over 90% without compromising target detection capabilities. To validate the algorithm's effectiveness in practical applications, field experiments were conducted using the radar communication integration system. The experimental data confirm that the algorithm successfully eliminates false alarm points and enhances target detection performance. A comparison of target tracking results before and after false alarm suppression reveals that the post-suppression tracking results align closely with the actual trajectory of the UAV. Future work will focus on further analyzing the characteristic differences between clutter and targets to achieve improved false alarm removal and enhance target detection capabilities.