Effective Non-Stationary Clutter Suppression Method via Elevation Oblique Subspace Projection for Moving Targets Detection with a Space-Based Surveillance Radar

Abstract

The clutter becomes non-stationary for a space-based surveillance radar (SBSR), which is harmful for the moving targets detection due to the earth’s rotation. The non-stationarity will degrade the accuracy of clutter covariance matrix (CCM) estimation and increase the clutter degree of freedom (DOF), thereby degrading the performance of clutter suppression. To solve this problem, this paper proposes a novel non-stationary clutter suppression method using an elevation oblique subspace projection method. After analyzing the range ambiguity and non-stationarity of the clutter, the proposed method utilized the oblique projection matrix to project the signal onto the subspace spanned by the near-range and far-range clutter components along the subspace spanned by the main lobe clutter component. Then, the projected signal was used to estimate the elevation covariance matrix and calculate the optimal weight vector for the elevation adaptive filter. The proposed method can suppress the non-stationary clutter effectively with a higher improvement factor (IF) and a narrower main lobe width. Finally, the simulation results were given to verify the correctness and effectiveness of the proposed method.

1. Introduction

Compared to an airborne radar, a space-based radar has a further detection range and a larger detection area due to having a much higher flight height. Therefore, the space-based radar has a wide application prospect and has received a great amount of attention in recent times [1,2,3,4,5]. One of the potential application aspects is to equip the surveillance radar onto the satellites and develop the space-based surveillance radar (SBSR). Benefiting from the larger detection area and higher flight height, the SBSR can provide a stronger ground moving target indication (GMTI) [6] and anti-stealth capability compared to the airborne surveillance radar. However, the specific properties of the SBSR will also bring extra problems. First, the extended detection range will cause more serious range ambiguity phenomena, thereby increasing the number of range ambiguity components in the echo. Second, the earth’s rotation will lead to an extra crab angle [7]. The crab angle will make the clutter, which has the same cone angle, is located in different range gates and have different Doppler frequencies. Under the influence of the earth’s rotation, the clutter echo of the SBSR becomes non-stationary. The non-stationarity will then influence the clutter suppression performance at the presence of the range ambiguity. On the one hand, the non-stationarity means that the clutter from different range gates have different space–time distributions, thereby reducing the number of independent identical distribution (i.i.d) samples that can be used for clutter covariance matrix (CCM) estimations. On the other hand, the non-stationarity will increase the clutter degree of freedom (DOF), which will increase the number of i.i.d samples that need to be used for clutter suppression according to the RMB rule [8]. Both of these problems will lead to the degradation of non-stationary clutter suppression.

As different range ambiguity clutter components have different elevation angles and levels of non-stationarity [9], a feasible idea that has been proposed to suppress the non-stationary clutter is to use elevation-dimension data to perform an elevation adaptive filter before CCM estimation in azimuth. The elevation filter method was initially proposed by the authors of [10] and was then utilized for non-stationary clutter suppression by the authors of [11]. The main problem of the elevation filter is that the clutter echo contains the main lobe clutter component, which will cause main lobe distortion and increase the sidelobe level for the elevation response pattern. To solve this problem, an elevation robust Capon beamforming (ERCB) method was proposed by the authors of [12] with diagonal loading (DL). However, the value of DL is typically hard to control. If the value is too small, the effect will not be good enough to solve the problem. The notch will become shallower and biased while the value obtained will be too big. An elevation sum and differ beamforming method was proposed by the authors of [13], which uses the echo of the first pulse to avoid the influence of the main lobe clutter component. The main problem of this method is that it cannot retrieve enough training samples. Meanwhile, a subspace projection preprocessing (SPP) method was proposed by the authors of [14] to project the echo onto the orthogonal complement space of the subspace spanned by the near-range clutter component. However, with this method, residual main lobe clutter will be present following orthogonal projection, thereby degrading the performance on far-range clutter suppression.

To solve the problem mentioned above, this study proposed a novel near-range clutter suppression method via elevation oblique subspace projection. First, the range ambiguity and non-stationarity were analyzed for the clutter of the SBSR. The clutter was divided into three types, which include the near-range clutter component, the main lobe clutter component, and the far-range clutter component. Then, using the oblique projection matrix, the echo was projected onto the subspace spanned by both the near-range clutter and the far-range clutter component, while the main lobe clutter component was cancelled. Finally, the projected signal was used for the elevation adaptive filter and azimuth STAP.

2. Signal Model and Clutter Characteristic Analysis

2.1. Signal Model of SBSR

The observation geometry of the SBSR is shown in Figure 1, where $O$ is the geocenter, $R_e$ is radius of the earth, and ${\theta }_e$ is the geocentric angle. The satellite travels at the height of $H$ with the speed of $V_p$. Point $A$ is the nadir point while point $B$ is the observation point with the slant range $R_s$. The elevation angle and azimuth angle are denoted by $\phi$ and $\theta$, respectively. Assuming that the antenna array is a uniform planar array with interval $d$, each elevation sub-array encompasses $M$ elements distributed along the direction of the z-axis, while each azimuth sub-array has $N$ elements distributed along the direction of the x-axis, respectively. The crab angle caused by the earth’s rotation between the x-axis and $V_p$ is denoted as ${\varphi }_c$. In the case where the number of pulses is $K$, then the clutter echo of the array element with coordinate $\left(m,n\right)$ for the $k^{th}$ pulse, and $l^{th}$ range gate can be formulated as follows:

$$X_l\left(m,n,k\right)={\displaystyle \sum _{p=1}^{N_r}{\displaystyle \sum _{i=1}^{N_c}{\sigma }_{l,p,i}\cdot a_s\left(m,n,l,p\right)a_t\left(k,{\phi }_{l,p},{\theta }_i\right)}}$$

(1)

where $m=1~M$, $n=1~N$, and $l=1~L$. $L$ and $N_r$ denote the number of the range gate and range ambiguity, respectively. $N_c$ is the number of the statistically independent clutter patch with an amplitude of $\sigma$ in each range cell. The steering vector in the space and time domains can be expressed respectively as follows:

$$a_s\left(m,n,{\phi }_{l,p},{\theta }_i\right)=\exp \left(j2\pi \left(m-1\right)\frac{d}{\lambda }\cos {\phi }_{l,p}\right)\cdot \exp \left(j2\pi \left(n-1\right)\frac{d}{\lambda }\sin {\phi }_{l,p}\cos {\theta }_i\right),$$

(2)

$$a_t\left(k,{\phi }_{l,p},{\theta }_i\right)=\exp \left(j2\pi \left(k-1\right)\frac{2V_p}{\lambda f_r}{\rho }_c\sin {\phi }_{l,p}\cos \left({\theta }_i+{\varphi }_c\right)\right).$$

(3)

where $\lambda$ and $f_r$ denote the wavelength and the pulse repetition frequency, respectively. ${\rho }_c$ and ${\varphi }_c$ are the crab magnitude and crab angle caused by the earth rotation, which are derived from [6].

2.2. Analysis of the Range Ambiguity Characteristics for the SBSR

Due to the limitations in the earth’s curvature, the maximum of the slant range for the SBSR can be calculated as follows:

$$R_{s\max }=\sqrt{H\left(2R_e+H\right)}.$$

(4)

Based on the above equation, this indicates that the slant range of the SBSR is ranged from $H$ to $R_{s\max }$. Following this step, all the range ambiguity components of the observation point $B$ can be calculated as follows:

$$R_{s,p}=R_s+p\frac{c}{2f_r}$$

(5)

where $p$ is an integer which can satisfy $H\le R_{s,p}\le R_{s\max }$. Then, the elevation angle of each range ambiguity component can be calculated as follows:

$${\phi }_p=\arcsin \left(R_e/R_{s,p}\cdot \sin {\theta }_{e,p}\right)$$

(6)

where ${\theta }_{e,p}$ is the geocentric angle corresponding to $R_{s,p}$.

A schematic diagram to show all the clutter range ambiguity components in a range gate for SBSR is displayed in Figure 2, with the main parameters listed in

Table 1. Main parameters.

Parameter	Value
Orbit height	500 km
Carrier frequency	0.5 GHz
Crab angle	3.77 deg
Azimuth angle	90 deg
Elevation angle	30 deg
Number of pulses	128
Array number in azimuth	256
Array number in elevation	16
Pulse repetition frequency	5000 Hz

. It is important to note that there are two main differences regarding the parameters between the AEW (airborne early warning) radar and SBSR that are worthy to mention. First, the elevation angle of the SBSR is much smaller as the SBSR is much higher than the targets compared to the AEW radar. Second, the SBSR should choose a medium pulse repetition frequency (MPRF) rather than a high repetition frequency (HPRF) like the AEW radar due to having a much larger doppler bandwidth caused by the speed of the satellites. Using Equations (4)–(6) with the parameters in Table 1, we can obtain the number of range ambiguities and the elevation angle of each range ambiguity component. In total, there are 68 range ambiguity components which are denoted by 68 circles with different colors. All these components can be divided into three main types, which are the near-range clutter, main lobe clutter, and far-range clutter denoted by blue, red, and green, respectively. It can be seen how the near-range clutter is distributed most sparsely with a high intensity as it is located quite close to the main lobe with a smaller slant range. As the slant range expands, the distributions of the main lobe clutter and far-range clutter become denser, especially for the far-range clutter.

2.3. Analysis of Non-Stationarity for the Clutter of the SBSR

As according to Equation (3), the theoretical range–doppler curves of the clutter in the main lobe of the cone angle are displayed in Figure 3 with the parameters listed in Table 1. The curved spectrum lines show that the clutter components from different range gates have different distributions, which indicates the non-stationarity. Imperatively, Figure 3 also illustrates that these different clutter components have different levels of non-stationarity. The doppler frequencies of the near-range clutter components from different range gates change most greatly, indicative of the highest level of non-stationarity. As the slant range becomes larger, the level of non-stationarity becomes lower, especially for the far-range clutter.

3. The Proposed Method

Based on the analysis regarding the characteristics of non-stationary clutter for the SBSR in the previous section, we can see that the near-range clutter component has the highest level of non-stationarity, which impacts the most on CCM estimation. Therefore, the elevation adaptive pattern should point to the main beam steering and form notches in the direction of the near-range clutter components. This indicates that the elevation covariance matrix should contain the near-range clutter component. Additionally, although the far-range component has quite a lower level of non-stationarity, it has the greatest number of range ambiguities and distributes the most densely, thereby providing great contributions to the high DOF of clutter. Therefore, the sidelobe level of elevation adaptive patterns in the direction of the far-range clutter component should also be quite low in terms of improving the performance of further space-time adaptive processing (STAP) [15] in azimuth, meaning that the elevation covariance matrix should contain the far-range clutter component as well. Finally, the presence of the main lobe clutter component in the elevation covariance matrix will cause main lobe distortions of the elevation adaptive pattern and increase the sidelobe level. Moreover, the targets are assumed to be located in the main beam, which will degrade the effects on estimating the elevation covariance matrix of the near-range clutter. Therefore, both the main lobe clutter component and the target need to be eliminated in the final elevation covariance matrix.

To achieve the goal mentioned above, this study proposes an elevation oblique subspace projection (EOSP) method. A schematic diagram is displayed in Figure 4, where ${\Omega }_m$ denotes the subspace spanned by the main lobe clutter component and ${\Omega }_{nf}$ denotes the subspace spanned by both the near-range clutter and far-range clutter components, respectively. ${\Omega }_m^{\perp }$ represents the orthogonal complementary subspace of ${\Omega }_m$. The oblique projection matrix $P_l$ for the $l^{th}$ range gate is expressed in Equation (7) [16,17], can project the signal onto ${\Omega }_{nf}$ along the ${\Omega }_m$, thereby keeping the near-range clutter and far-range clutter components and eliminating the main lobe clutter component at the same time.

$$P_l=H_{nf}\cdot {\left(H_{nf}^H\cdot H_m^{\perp }\cdot H_{nf}\right)}^{-1}\cdot H_{nf}^H\cdot H_m^{\perp }\in {ℂ}^{M\times M}.$$

(7)

As displayed in Equation (7), ${\left(\cdot \right)}^H$ and ${\left(\cdot \right)}^{-1}$ denote the conjugate transposed and the inverse of a matrix, respectively. $H_{nf}$ and $H_m^{\perp }$ can be formulated as follows:

$$H_{nf}=\left[a_{el}\left(\left\{{\phi }_n\right\}\right),a_{el}\left({\phi }_f\right)\right]$$

(8)

$$H_m^{\perp }=I-a_{el}\left(\left\{{\phi }_m\right\}\right){\left(a_{el}^H\left({\phi }_m\right)a_{el}\left(\left\{{\phi }_m\right\}\right)\right)}^{-1}{a}_{el}^H\left(\left\{{\phi }_m\right\}\right)$$

(9)

where $I$ denotes the identity matrix with size of $M\times M$ and $a_{el}$ is the elevation steering vector, which can be further expressed as follows:

$$a_{el}\left(\phi \right)={\left[1,\exp \left(j2\pi \frac{d}{\lambda }\cos \phi \right),\dots ,\exp \left(j2\pi \frac{\left(M-1\right)d}{\lambda }\cos \phi \right)\right]}^T$$

(10)

where ${\left(\cdot \right)}^T$ denotes the transposed of a matrix, and $\left\{{\phi }_n\right\}$, $\left\{{\phi }_m\right\}$, and $\left\{{\phi }_f\right\}$ denote the row vectors composed by the elevation angle of each near-range clutter, main lobe clutter, and far-range clutter component, respectively. The three row vectors are dependent on the exact calculation results of range ambiguity under different range gates. It is worth mentioning that the column-rank of the matrix $\left[H_{nf},a_{el}\left(\left\{{\phi }_m\right\}\right)\right]$ must be smaller than the number of its row [17]. This condition means that the components we use to span the clutter subspace should be smaller than the value of M exactly. That is to say, it is hard to use all the range ambiguity components to span the clutter subspace as the number of antenna elements in the elevation plane is limited. Considering that the system degree of freedom (DOF) equals to M, we can always select no more than M components to span the clutter subspace. Just as the analysis above, the far-range clutter components distribute densely, which means that the adjacent far-range clutter components have similar elevation angles. Therefore, in practice, we can choose part of the far-range clutter components to form $\left\{{\phi }_f\right\}$, as shown in Equation (8), where we can choose the far-range clutter components by their elevation angle with equal interval criteria.

After elevation oblique subspace projection, the projected signal is used for elevation covariance matrix estimation, which can be formulated as follows:

$$R_{Xl}={\displaystyle \sum _{n=1}^N{\displaystyle \sum _{k=1}^K\left(P_l\cdot X_l\left(:,n,k\right)\right){\left(P_l\cdot X_l\left(:,n,k\right)\right)}^H}}/NK$$

(11)

where $X_l\left(:,n,k\right)$ can be represented as:

$$X_l\left(:,n,k\right)={\left[X_l\left(1,n,k\right),X_l\left(2,n,k\right),\cdots ,X_l\left(M,n,k\right)\right]}^T.$$

(12)

Following this, the elevation optimal weight vector for the $l^{th}$ range gate can be expressed as:

$${\omega }_l=\frac{R_{Xl}^{-1}\cdot a_{el}\left({\phi }_0\right)}{a_{el}^H\left({\phi }_0\right)R_{Xl}^{-1}{a}_{el}\left({\phi }_0\right)}$$

(13)

where ${\phi }_0$ denotes the main beam steering in elevation. The signal after elevation processing can be formulated as:

$$Y_l\left(n,k\right)={\omega }_l^H\cdot X_l\left(:,n,k\right).$$

(14)

Finally, the block diagram of the equipment involved in the implementation of the proposed method, as well as the software tools dedicated to the simulations and processing of the radar signals are shown in Figure 5.

4. Experiment Results

To verify the correctness and effectiveness of the proposed method, a simulation experiment was made with the results given in this chapter. The main parameters are listed in Table 1. The maximum of the elevation angle was $68{.02}^{\mathrm{o}}$. As a comparison, the results of elevation common beamforming (ECBF) with 30 dB Chebyshev weight, ERCB [10], and SPP [12] were also provided, and the data used here was simulated.

Here, we chose the 138th range gate with 68 range ambiguity components (corresponding to Figure 2) to analyze the elevation response patterns of the four methods. The results obtained are shown in Figure 6, which illustrates how ECBF has no ability to form a notch in each near-range and far-range ambiguity component. For ERCB, DL was able to make the elevation response pattern free of main lobe distortion. However, the notch in the region of near-range clutter component became biased at the same time. The performance of ERCB on non-stationary clutter suppression was thereby limited. SPP can form two idea notches in the region of the near-range clutter component. However, the residual main lobe clutter component after orthogonal projection made the beam-steering biased and the sidelobe level higher as a result. The proposed method was also able to form notches in the direction of the near-range component effectively, with the number of range ambiguities also being bigger than the number of array elements in elevation. To satisfy the condition mentioned below Equation (10), we chose eight far-range ambiguity components to form using Equation (8). Figure 6 illustrates that the elevation response pattern has quite a low sidelobe level, corresponding to the far-range component in this manner than that of the three other methods. Therefore, the proposed method has the ability to suppress both the near-range and far-range clutter components.

Next, we used the elevation weight vectors of each method to perform elevation filter for all the range gates. The range doppler spectrums in the main lobe of the cone angle for the four methods are shown in Figure 7a–d, respectively. The elevation response pattern of ECBF cannot form the notch in the position of the near-range and far-range clutter. Therefore, there is still a curved spectrum line in the region of the near-range clutter, while the residual power of the far-range clutter was high. Then, for ERCB, it can be seen that a part of the curved spectrum line was suppressed compared to ECBF. However, as the notch was biased, the other part of the near-range clutter was not well suppressed with the residual curved spectrum still shown in Figure 7b. Figure 7c illustrates that the near-range clutter was well-suppressed by SPP. However, the power of the far-range clutter was still relatively high, which can be proved by the spectrum’s width and the color bar. For our proposed method, even though its notches in the near-range clutter was a bit higher than that of SPP, its level was determined to be low enough to suppress the near-range clutter. And benefiting from the low sidelobe level corresponding to the far-range clutter, the result of the proposed method shown in Figure 7d has a narrower spectrum width and a lower residual power compared to the three other methods.

The output signals after elevation filter by the four methods were used to perform the same STAP method in azimuth as well. Four improvement factor (IF) curves of the 138th range gate are displayed in Figure 8. It illustrates that the IF curves of ECBF and ERCB degrade severely in the normalized doppler region from −0.12 to 0.10, respectively, due to not only the residual power of the near-range clutter but also the performance reduction in CCM estimation caused by non-stationarity. The degradation phenomenon did not appear for SPP due to its idea notches in the region of the near-range clutter component. However, as the elevation sidelobe level of SPP corresponding to the far-range clutter was high, the IF curve of SPP in the normalized doppler region from about −0.38 to −0.25, respectively, was determined to lower. Finally, as both the near-range clutter and the far-range clutter were well suppressed, the IF curve of the proposed method was higher than the three other curves with a narrower main lobe width.

Finally, to verify the proposed method in a higher carrier frequency, another experimental result was analyzed as displayed in Figure 9, with the carrier frequency increased to 1.25 GHz. The clutter suppression results of ECBF, ERCB, SPP, and the proposed EOSP for all range gates and Doppler bins are given in Figure 9a–d, respectively. Though the Doppler bandwidth of the main lobe clutter expanded a lot due to the increase in the carrier frequency, the proposed EOSP was still able to suppress the near-range clutter effectively, which ECBF and ERCB were not able to achieve. Additionally, compared to the other three methods, the range doppler spectral of the proposed EOSP had the widest clear zone due to its best suppression performance on the far-range clutter.

5. Conclusions

Aiming at non-stationary clutter suppression for moving targets detection with the SBSR, a novel non-stationary clutter suppression method based on elevation oblique subspace projection has been proposed in this study. After analyzing the non-stationarity and dividing the clutter into three types, the proposed method used an oblique projection matrix to project the signal onto the subspace spanned by the near-range and far-range clutter components along the subspace spanned by the main lobe clutter component for further elevation adaptive filter and azimuth STAP. Finally, simulation results were provided to verify the correctness and effectiveness of the proposed method.

It is crucial to note that the number of array elements in elevation is an important but limited factor. Future research will focus on improving the robustness of the proposed method with less array elements in elevation at the presence of the array error.