Robust clutter suppression in heterogeneous environments based on multi frames and similarities

A method of robust clutter suppression with space–time adaptive processing (STAP) for airborne radar in heterogeneous environments is proposed, which is based on multi frames and the similarity between the cell under test and each training sample. The proposed method deals with the problem of covariance matrix estimation for STAP in heterogeneous clutter. Firstly, the method expands the set of training samples by selecting similar training frames from past frames. Secondly, initial training samples are selected from the expanded training samples set, that are composed of the samples of the current frame and past frames. Thirdly, initial training samples which may be contaminated by target signal are discarded. Fourthly, the similarities between the cell under test and the remaining training samples are estimated, and training samples which are more similar to the cell under test are assigned higher weights in the estimation of the clutter covariance matrix. The proposed method overcomes the problems of training samples’ heterogeneity and insufficiency in the estimation of the clutter covariance matrix. The accuracy of the estimated clutter character is improved significantly, and thus the performance of clutter suppression is improved. Experimental results based on measured data demonstrate the performance of the proposed method.


Introduction
Airborne phased array radar is widely used to detect moving targets, while its performance degrades in clutter. Space-time adaptive processing (STAP) is adopted to suppress clutter (Li et al., 2019;Rangaswamy et al., 2004a;Wang et al., 2018a). It forms notches at the locations of clutter to suppress clutter. Generally, it is capable to suppress clutter efficiently in homogeneous environments, while its capability of clutter suppression degrades severely in heterogeneous environments (b; Lan et al., 2020a;Sun et al., 2011;Wang et al., 2017a). The clutter covariance matrix of the cell under test (CUT) is necessary for STAP, while it is unknown in the application. Normally, independent and identically distributed (IID) training samples are used to estimate the clutter covariance matrix of the CUT. To ensure its performance of clutter suppression, STAP needs enough IID training samples to estimate the clutter covariance matrix, and its performance of clutter suppression degrades severely when the number of training samples is less than two times the system's degree of freedom (DOF) (Riedl & Potter, 2018;Yifeng et al., 2015;Zhang et al., 2018).
In heterogeneous environments, some of the selected training samples cannot represent the property of the clutter in the CUT, and the estimated clutter covariance matrix is not exact, which degrades the performance of STAP. To overcome the problem of clutter heterogeneity, numerous non-homogeneous detectors (NHD) have been proposed to remove nonhomogeneous training samples from the initial training samples set (Rangaswamy et al., 2004b;Wang et al., 2013;Zhao et al., 2018). However, these NHD methods do not take the property of the CUT into account, and the selected training samples cannot represent the CUT when the CUT is heterogeneous with most of the initial training samples (Tang et al., 2012;Yang et al., 2013). In this situation, the performance of STAP to suppress clutter degrades seriously. On the other hand, to ensure the accuracy of the estimated clutter covariance matrix, the number of training samples should be larger than 2 times the system's DOF, while the number of training samples may not enough after the process of NHD, which degrades the performances of STAP to suppress clutter (Wang et al., 2018b).
To overcome the problems of training samples' heterogeneity and insufficiency in the estimation of the clutter covariance matrix, this paper presents a robust clutter suppression method to improve the performance of STAP in heterogeneous environments. Past frames which share similar properties with the current frame are adopted to expand the initial training samples. Moreover, this paper takes the property of the CUT into account when it selects training samples by the similarities between the CUT and initial training samples, and the similarities are estimated by the correlation coefficients between them. Training samples whose clutter is more similar to the clutter of the CUT are assigned higher weights in the estimation of the clutter covariance matrix. In this way, the proposed method estimates the property of the clutter in the CUT accurately and improves the performance of clutter suppression. Experimental results based on measured data demonstrate that the proposed method improves the performance of clutter suppression in heterogeneous environments effectively.

Signal model for STAP
Denote the observed vector of the CUT as x k0,l0 , where k0 and l0 are the frame index and range gate index of the CUT, respectively. The problem of target detection can be formulated as the following binary hypothesis test problem where c k0,l0 and n k0,l0 are the clutter and noise vectors of the CUT, respectively; and a k0,l0 is the target amplitude, s(θ 0 , f 0 ) is the target steering vector corresponding to the direction of θ 0 and the Doppler frequency of f 0 . To overcome the adverse effects of clutter on target detection, the clutter should be suppressed. The problem of clutter suppression is described as minimizing the output of clutter power subject to a constant gain in the target direction. The optimum weight of STAP is given by the following mathematical problem (Wang et al., 2017b(Wang et al., , 2018b where () H is the conjugate transpose, and || · || calculates the 2-norm of a vector or scalar. According to formula (2), the adaptive weight of STAP is denoted as where R k0,l0 denotes the clutter covariance matrix of the cell under test (Wang et al., 2018c;Zhang et al., 2018). The STAP output of the cell under test is denoted as in which clutter is suppressed and the power of target keeps constant. However, the clutter covariance matrix is unknown in practical application, and it's normally estimated by independent and identical training samples which share similar properties with the CUT,R where x k0,l (l = 1, . . . , L) denotes the sample of the k0th frame at the lth range gate. To guarantee the performance of STAP, the number of training samples should be larger than 2 times the dimension of s(θ 0 , f 0 ). However, in heterogeneous environments, there may be no enough IID training samples that share the same property with the clutter of the CUT. In this situation, the clutter covariance matrix of the CUT cannot be well estimated, which degrades the performance of STAP. Numerous NHD methods have been proposed to select training samples, such as the generalized inner product (GIP) algorithm and loaded GIP algorithm (Rangaswamy, 2005;Rangaswamy et al., 2004b;Tang et al., 2012;Yang et al., 2013;Zhao et al., 2018), while these methods do not take the property of the CUT into account, and the selected training samples cannot represent the clutter of the CUT when the CUT is heterogeneous with most of the initial training samples. As a result, the performance of STAP to suppress clutter in heterogeneous environments suffers severely. On another hand, these NHD methods remove non-homogeneous samples which may result in training samples insufficiency. In this case, the estimated clutter covariance matrix is not accurate, and the performance of clutter suppression degrades.

Proposed method
To overcome the STAP performance degradation resulted from non-homogeneity and training samples limitation, this paper presents a robust covariance matrix estimation algorithm by introducing multi frames and weight factors into the process of clutter covariance matrix estimation, and the weight factors are calculated by the similarities between the training samples and the cell under test. The basic idea of the proposed method is that multi frames have more similar samples than the single frame, and the clutter property of the CUT can be well estimated by the training samples from multi frames, thus, STAP performance of clutter suppression improves. Firstly, the proposed method selects past frames that are similar to the current frame. The samples of the selected past frames and the current frame form the initial training samples set. Secondly, initial training samples which may be contaminated by target signal are discarded. Thirdly, the similarities between the remaining samples and the CUT are estimated, and the weights of training samples in the estimation of the covariance matrix are controlled by the corresponding similarities. Fourthly, the adaptive weight of STAP is calculated by the estimated clutter covariance matrix and applied to suppress clutter. The proposed method expands the initial training samples set and significantly improves the accuracy of clutter covariance matrix estimation, and thus the performance of clutter suppression improves. The detailed description is described as follows.

Multi frames selection
Since the ideal STAP weight is calculated according to the clutter covariance matrix of the clutter in the CUT, and it forms notches at the location of the clutter spectrum. Enough IID training samples which share the same property with the clutter of the CUT are needed. This Section expands training samples set by multi frames. In the process of extended frames selection, the past frames with the same system parameters and similar detection area are firstly selected. After that, the power spectrums of the selected frames areas estimated, and the past frames whose power spectrums are similar to the power spectrum of the current frame are selected.
The power spectrum of the current frame (it's frame index is denoted as k0) is denoted as where s θ i , f j is the scanning steering vector corresponding to the direction of θ i and normalized Doppler frequency of f j , I and J are the number of estimated direction and normalized Doppler frequency, respectively; andR k0 is the covariance matrix estimated by the samples of the current frame,R where x k0,l (l = 1, . . . , L) is the training sample of the frame under test at the lth range gate. The power spectrum of the k2th frame is denoted as whereR k2 is the covariance matrix estimated by the samples of the k2 th frame, where x k2,l (l = 1, . . . , L) is the training sample of the k2th frame at the lth range gate. There are several methods to estimate the similarity between matrixes, such as Euclidean distance and log-Euclidean distance. In this paper, the similarity between the frame under test and the kth frame can be calculated by their Euclidean distance of power spectrum (Wang et al., 2018c), which is denoted as d k2 . (10) The Euclidean distance d k2 reflects the similarity between the k2 th frame and the current frame. Only the current frame and the past frames corresponding to the small d k2 are selected as initial frames. The selected frames should share similar properties with the frame under test, and they are usually the nearest frames to the current frame. The samples of the selected past frames and the current frame form the initial training samples set, which contains much more samples than the current frame. In this way, the proposed method overcomes the problem of training samples limitation to a certain extent.
The sliding window method is adopted to further select training samples in the frame under test (Zhang et al., 2018), and the selected samples are close to the CUT. In the past frames, the training samples are also selected by the sliding window method. In this way, the initial training samples set are extended by samples of the current frame and past frames.

Discarding samples contaminated by target signal
Target signal in the training samples of clutter covariance matrix estimation may result in target-self-null, which must be avoided. Thus, the training samples which might be contaminated by target signal have to be discarded (Zhiqi & Haihong, 2016). Plenty of methods to discard samples contaminated by the target signal have been proposed (Wu et al., 2015), this paper adopts the method of correlated coefficient to discard samples contaminated by target signal (Li et al., 2016). The correlated coefficient of the training sample x k2,l with the target steering vector is As shown in the above formula, the larger c k2,l is, the more similar the training sample with the steering vector is. Thus, to eliminate the effect of target-self-nulling, k samples corresponding to the largest correlated coefficients are discarded from the initial training samples set. Denote the remaining samples set as , and the corresponding training samples are x k2,l ∈ . In the next section, the clutter covariance matrixes are estimated by the remaining samples.

Similarity estimation and covariance matrix estimation
This section firstly estimates the similarities between training samples x k2,l ∈ and the CUT x k0,l0 . Then the clutter covariance matrix of the CUT is estimated based on the similarities. Plenty of methods to estimate the similarities have been proposed, this paper adopts the method of the correlated coefficient which is similar to the method in the last section. The similarity between the CUT and x k2,l is denoted as wherex k0,l0 andx k2,l are the projection components of x k0,l0 and x k2,l on the subspace orthogonal to the target subspace (which is mainly composed of the clutter subspace and noise subspace), respectively.x k0,l0 = Px k0,l0 , x k2,l = Px k2,l , x k2,l ∈ , where P is the orthogonal projection matrix of the target signal. P can be denoted as where I is the identity matrix.
According to the definition of s k2,l , s k2,l is normally larger than 0 and smaller than 1. x k2,l is not similar to x k0,l0 when s k2,l → 0, and x k2,l is similar to x k0,l0 when s k2,l → 1. To better estimate the covariance matrix, training samples that are similar to the CUT are needed. Therefore, the training samples corresponding to large s k2,l are assigned heavy weights and training samples corresponding to small s k2,l are assigned light weights in the estimation of the covariance matrix. The estimated clutter covariance matrix of the CUT by the proposed method isR k0,l0 = 1 k2 l s k2,l k2 l s k2,l x k2,l x H k2,l , x k2,l ∈ .
In the proposed estimation method, the proposed method considers the property of each training sample and takes the training sample property into consideration when it estimates the covariance matrix of the CUT. Thus, the proposed method improves the accuracy of the clutter covariance matrix estimation, and the robustness of clutter suppression in heterogeneous environments improves.
The flowchart of the proposed algorithm is shown in Fig. 1, and the algorithm can be summarized as follows.
• Step 1: select the extended frames according to the waveform, interested region, and spectrum property et.al, and only frames whose property are similar to the frame under test are selected. • Step 2: select initial training samples which are nearby the CUT, excluding the CUT and guard samples to prevent the target self-nulling effect, and the training samples of the frame under test and extended frames form the initial training samples set. • Step 3: discard samples that might be contaminated by the target signal to avoid the targetself-nulling effect. • Step 4: estimate the correlated coefficients between the selected training samples and the CUT, and the correlated coefficients are adopted to measure the similarities between the training samples and the CUT. • Step 5: estimate the clutter covariance matrix of the CUT, and the weight of each training sample in the estimation is calculated according to the similarity between the training sample and the CUT. • Step 6: calculate the adaptive space-time adaptive weight and process the CUT with the STAP weight.

Experiment results
To demonstrate the performance of the proposed method, the proposed method and the classical method based on single frame and GIP are applied to measured data which was collected in a heterogeneous clutter environment. One coherent process interval (CPI) of the measured data contains 64 pulses, and a cooperated target was injected at the 200th range gate with 0.3 normalized Doppler frequency. The range-Doppler plot of the measured data is shown in Fig. 2, which shows the non-homogeneity of the environment. Figure 2a shows the clutter of the past frame and Fig. 2b shows the clutter of the current frame. Comparing the clutter in Fig. 2a, b, it shows that the clutter in the two frames looks the same.
The STAP based on a single frame adopts the classical GIP method to discard nohomogeneous training samples. The STAP result of the classical method based on a single frame is shown in Fig. 3. The result of the proposed method is shown in Fig. 4. Comparing Fig. 4 with Fig. 3, the clutter residue of the classical method in the black rectangle is 49 dB, and the clutter residue of the proposed method in the same black rectangle is 44 dB. At the same time, the power of the injected target remains the same. It shows that the proposed method improves the performance of clutter suppression.
A modified sample matrix inversion (MSMI) test statistic is plotted versus range bin for each of the results obtained at the normalized Doppler of 0.3 in Fig. 5. The value of range averaged statistic value was one of our performance measures in this Paper, which demonstrates the superiority of clutter suppression. STAP results of a classical method based on a single frame are plotted at the normalized Doppler of 0.3, and the range averaged statistic value is 53 dB. The result of the proposed method at the same normalized Doppler is also plotted, it illustrates that the range averaged statistic value is 49 dB, which is 4 dB lower than the classical method. Experimental results show that the residential power of the proposed method is lower than that of the classical method, which demonstrates the performance of the proposed method. Figure 6 shows the statistical output of the signal-to-clutter-plus-noise ratio (SCNR) against the input SCNR. The result of each input SCNR is statistically averaged by 200 (a) past frame (b) current frame  Fig. 6 shows that the output SCNR of the proposed method is 3~4 dB higher than that of the classical method, which illustrates the performance improvements of the proposed method.

Discussion
Experimental results show that the capability of clutter suppression of the proposed method is better than that of the classical method. This is due to the classical method may not have enough IID training samples in the estimation of the covariance matrix, which degrades the performance of clutter suppression. On another hand, the classical method does not consider the property of the CUT, thus, the selected training samples might not share the same property with the CUT in heterogeneous environments, and the estimated covariance matrix is not correct when the clutter of the CUT is heterogeneous with most of the initial training samples. In this case, the ability of STAP to suppress clutter degrades. The proposed method expanded the initial training samples set and considers the property of the CUT. It selects enough training samples which are IID with the CUT, and improves the estimated performance of the clutter property, thus, it improves the performance of STAP in heterogeneous environments.
The proposed method overcomes the performance degradation of STAP resulted from training samples insufficiency and heterogeneous, which adopts past frames and training samples similarities. The selected past frames should be measured from the same zone with the frame under test, otherwise, the proposed method cannot select past frames to extend the training samples set. On another hand, the proposed method estimates the similarity between the CUT and each training sample, and its calculation is larger than the classical method.

Conclusions
Based on the traditional STAP method, a weighted method is proposed, which is based on multi frames. After comparing with a traditional single frame and NHD method, it has been proven that the proposed method can achieve a good clutter suppression performance. Firstly, it uses past frames to expand the set of training samples. Secondly, training samples that are heterogeneous with most of the initial training samples are discarded by the classical NHD method. Thirdly, the similarity between the cell under test and each selected training sample is estimated. Then, the clutter covariance matrix is estimated according to the similarities. Since the proposed method expands the set of initial training samples and takes the property of each sample into account, it improves the performance of the clutter covariance matrix estimation. Thus, the proposed method can effectively suppress clutter in heterogeneous environments. Experimental results based on real data demonstrate the effectiveness of the proposed method.
Yifeng Wu was born in Anhui Province, China, in 1988. He received the Ph.D. degree from Xidian University in 2016, He is currently a associate professor of Sun Yat-Sen University. His research interests include target detection, array signal processing.
Xiaobo Deng was born in Hunan Province, China, in 1982. His research interests include radar signal processing, waveform design, and array signal processing.
Yufeng Cheng was born in Jiangsu Province, China, in 1973. His research interests include radar system and airborne radar. Jun Tang received the Ph.D. degree in electrical engineering from Tsinghua University, China, in 2000. He is currently a Professor with the Department of Electronic Engineering, Tsinghua University. His research interests include array signal processing, information theory and MIMO radar, and compressive sensing in radar application.