Signal Processing 91 (2011) 2268–2276
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Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Registration-based compensation using sparse representation in conformal-array STAP Ke Sun a, Huadong Meng a,n, Fabian D. Lapierre b, Xiqin Wang a a b
Department of Electronic Engineering, Room 901A, Main building, Tsinghua University, Beijing 100084, China Royal Military Academy, Department of Electrical Engineering, Brussels 1000, Belgium
a r t i c l e in f o
abstract
Article history: Received 16 November 2010 Received in revised form 25 February 2011 Accepted 5 April 2011 Available online 12 April 2011
Space–time adaptive processing (STAP) is a well-known technique in detecting slow-moving targets in the clutter-spreading environment. When considering the STAP system with conformal radar array (CFA), the training data are range-dependent, which results in poor detection performance of traditional statistical-based algorithms. Current registration-based compensation (RBC) is implemented based on a sub-snapshot spectrum using temporal smoothing. In this case, the estimation accuracy of the configuration parameters and the clutter power distribution is limited. In this paper, the technique of sparse representation is introduced into the spectral estimation, and a new compensation method is proposed, namely RBC with sparse representation (SR-RBC). This method first establishes the relationship between the clutter covariance matrix (CCM) and the clutter spectral distribution. Based on this, it avoids the problem of lacking stationary training data and converts the CCM estimation into the solving of the underdetermined equation only with the test cell. Then sparse representation method, like iterative reweighted least square (IRLS) is used to guide the solution of the underdetermined equation towards the actual clutter distribution. Finally, the transform matrix is designed using the CCM estimation so that the processed training data behaves nearly stationary with the test cell. Because the configuration parameters and the clutter spectral response are obtained with full-snapshot using sparse representation, SR-RBC provides more accurate clutter spectral estimation, and the transformed training data are more stationary so that better signal-clutter-ratio (SCR) improvement is achieved. & 2011 Elsevier B.V. All rights reserved.
Keywords: STAP Conformal array Sparse representation Iterative reweighted least square
1. Introduction Airborne/spaceborne space–time adaptive processing (STAP) is the technique of choice in detecting slowmoving targets in the presence of a strong clutter background. Conventional STAP processors using a side-looking uniform linear array (ULA) have the desirable property that the relationship between the clutter spatial and Doppler frequencies is range-independent. Therefore the training data from adjacent range cells behave stationary and can
n
Corresponding author. Tel.: þ 8662794095; fax: þ8662781378. E-mail address:
[email protected] (H. Meng).
0165-1684/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2011.04.008
be utilized to estimate the clutter covariance matrix (CCM) so that the adaptive filter can be effectively generated to improve the output signal-clutter-ratio (SCR) in the test cell [1,2]. However in many radar and sonar applications, achieving perfectly ULA geometries may not always be practical. In addition, the complex configuration, e.g., conformal radar array (CFA), does provide certain advantages including minimal payload weight, the potential for increased aperture, and increased field of view [3,4]. However, the relationship between the clutter spatial-Doppler frequencies becomes non-linear and range-dependent in the case of CFA. Therefore the sample covariance matrix computed from the training data mismatches with the test cell and results in degraded performance in canceling clutter.
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array to the ground clutter scatterers is discussed. Suppose the conformal array is deployed onto a particular surface of the radar platform such as the fuselage, wing, or nose cone. The array response is generally non-linear from element to element comprising the multichannel receiving array. To determine the response of a point clutter scatterer, the coordinate system in Fig. 1 is used, with the x-axis aligned to true north, the y-axis pointing to west, and the z-axis perpendicularly directed away from the Earth’s surface. Here, the cylindrical array with M rings is adopted, each of which is composed of N isometry array elements. The parallel rings are perpendicular to the y-axis with a spacing of d. The array elements within each ring are isometrically placed with a circle radius r. Although the discussion is carried out with the cylindrical array in this paper, this method can be effectively implemented in other CFA configurations. In this instance, the angle vector is defined as w ¼[f,y]T, where symbols f,y stand for the azimuth and elevation angles of certain clutter scatterer Q, respectively.
Various methods have been proposed to deal with the range-dependent clutter. Angle–Doppler compensation (ADC) and adaptive angle–Doppler compensation (A2DC) [5,6] attempt to align the peaks of the clutter ridge of the training data with that of the test cell. However, they only accomplish partial compensation and thus are effective only for highly directive antenna beampatterns. The derivative-based updating (DBU) method assumes that the clutter central Doppler frequency is linear with range. However, this assumption is rarely satisfied in the short-range case, and thus, DBU also does not work effectively with all configurations such as CFA and/or bistatic radar [7]. Recently, the registration-based compensation (RBC) [8–10] method is proposed using a mathematical description of the clutter ridge in the angle–Doppler domain. This method estimates both the configuration parameters and the clutter power distribution. Then, the transform matrix is designed so that the training data is nearly stationary with the test cell. RBC implements both the mainlobe and sidelobe clutter compensation. However, because the peak extraction is implemented in the subsnapshot spectrum, the estimation accuracy of both the configuration parameters and clutter power distribution is limited, which results in degraded performance. In this paper, the technique of sparse representation is introduced into the problem of clutter spectral estimation, and a novel registration-based compensation called SR-RBC is proposed to further improve the SCR performance. The remainder of this paper is organized as follows. Section 2 describes the basic signal model deployed with the CFA configuration. Section 3 introduces the theory of the sparse representation and illustrates the details of the SR-RBC algorithm. Section 4 uses the simulated data to illustrate the advantages of the proposed method. Section 5 gives a conclusion of the proposed method and describes the future work.
*
The unit vector k ðwÞ points orthogonal to the propagating planar wavefront and is given by *
*
*
*
k ðwÞ ¼ cos ycos fUe x þcos ysin fUe y sin yUe z :
ð1Þ
Additionally, the direction vector to the nth element on the mth ring is defined as * * * * 2p 2p ðn1Þ Ue x þ dðm1ÞUe y r cos ðn1Þ Ue z : s m,n ¼ r sin N N ð2Þ From the illustration shown in [3], the element-level spatial response at the kth range cell is given by * * * * pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi xs=k ¼ Bvk as,1,1 g1,1 ejð2p=lÞk ðwÞds 1,1 ,. . .,as,1,N g1,N ejð2p=lÞk ðwÞds 1,N , * * * * pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi . . .,as,M,1 gM,N ejð2p=lÞk ðwÞds M,1 ,. . .,as,M,N gM,N ejð2p=lÞk ðwÞds M,N
#T
2. CFA signal model ¼ Bvk ½as g ss ðwÞ, A conformal array assumes the shape of the radarbearing platform and generally belongs to the class of non-linear array. The specific advantages of conformal array include better aerodynamic shape compatibility with the airframe, potentially greater effective apertures, less payload weight, and so on. Therefore, STAP deployed with the conformal array has great potential in the future airborne radar system, especially, unmanned aerial vehicles [3,4]. In this section, the space–time response of the conformal
where z CN(0,1) (if the response is non-fluctuating, then B is a complex scalar with unity magnitude and uniformly distributed phase), and vk is a normalized voltage term following from the radar range equation. The spatial covariance matrix taper (CMT) as reflects the inter-array amplitude-phase inconsistency with E as aH s ¼ As . g ¼ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi T pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi g1,1 ,. . ., g1,N ,. . ., gM,1 ,. . ., gM,N is the antenna gain vector, and is the Hadamard product. Therefore the
z v O s
y
r
x H
ð3Þ
R k
Q Fig. 1. Geometry of the CFA configuration.
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space steering vector is defined as * * * * * * ss ðwÞ ¼ ejð2p=lÞk ðwÞds 1,1 ,. . .,ejð2p=lÞk ðwÞds 1,N ,. . .,ejð2p=lÞk ðwÞds M,1 ,. . .,
kth range cell follows as [4]: Rk ¼
Na X Nc X
s2s ðwp,q ,kÞAst ðwp,q ,kÞ gst ðwp,q ,kÞgHst ðwp,q ,kÞ
p ¼ 1q ¼ 1 *
*
ejð2p=lÞk ðwÞds M,N
T :
ð4Þ
Simultaneously, the Doppler steering vector describes the pulse-to-pulse phase change due to the platform and target motion. Considering a stationary clutter scatterer Q at angle w, the corresponding Doppler steering vector is given by T * * * * * * st ðwÞ ¼ ej2pð2k ðwÞdv p =lÞ1T ,ej2pð2k ðwÞdv p =lÞ2T ,. . .,ej2pð2k ðwÞdv p =lÞPT ,
ð5Þ *
where v p represents the platform velocity vector and is given by *
*
*
*
*
v p ¼ 9v p 9sin sUe x þ 9v p 9cos sUe y ,
ð6Þ
where s denotes the crab angle between the flight direction and the central axis of the cylindrical array. The symbol T is the pulse repetition interval and P is the total number of pulses comprising the temporal aperture. A simple modification to (5) is needed if the scatterer is moving, i.e., the moving target. Therefore, for a temporal snapshot of a certain scatterer Q at the (m,n)th array element, the kth range cell is given by pffiffiffiffiffiffiffiffiffi xt=k ðm,nÞ ¼ Bvk gm,n ðat st ðwÞÞ, ð7Þ where at is a temporal CMT with E at aH t ¼ At , which is caused by the intrinsic clutter motion and system jitter, etc. Both spatial and temporal decorrelation expands the clutter rank and spreads the clutter ridge in the angle– Doppler domain. Therefore, the space–time response of CFA to a stationary scatterer at angle w conveniently follows as: xk ¼ Bvk ðat st ðwÞÞ ðas g ss ðwÞÞ,
ð8Þ
where indicates the Kronecker product. The space–time correlation taper as t ¼ asat satisfies Ast ¼ E ast aH st ¼ At As . A realistic model for the ground clutter return results from the coherent summation of many clutter scatterers within the bounds of each iso-range [4]. Therefore, the clutter space–time snapshot at the kth range cell takes the form of ck ¼
Na X Nc X
Bðwp,q ,kÞvðwp,q ,kÞðat st ðwp,q ,kÞÞ
p ¼ 1q ¼ 1
ðas gðwp,q ,kÞ ss ðwp,q ,kÞÞ,
ð9Þ
where Na indicates the number of ambiguous range cells, Nc is the number of statistically independent clutter scatterers at each iso-range, and wp,q indicates the certain angle vector of the qth clutter scatterer at the pth iso-range. Moreover, by virtue of the statistical independence of each clutter scatterer, the clutter space–time covariance matrix of the
sst ðwp,q ,kÞsH st ðwp,q ,kÞ:
ð10Þ
In this case, the clutter is substantially range-dependent so that the traditional CCM estimation such as loaded sample matrix inversion (LSMI) tends to have the average behavior of the training data. Therefore the corresponding STAP filter response exhibits mismatch for the test cell with insufficient null depth and excessive clutter spread, which causes performance degradation. A series of methods have been proposed to deal with the range-dependent clutter. Methods such as angle– Doppler compensation and adaptive angle–Doppler compensation [5,6] accomplish the peak response, but the sidelobe clutter suppression is limited. RBC implements both the mainlobe and sidelobe compensation [8–10]. However, because the configuration parameters and the clutter power distribution are obtained using the sub-snapshot spectrum, the estimation is not accurate enough. Therefore, the stationarity of the processed training data is destroyed and causes performance degradation of the STAP filter. In the following part, the rangedependent compensation is proposed using the technique of sparse representation, which has the capability of obtaining more accurate clutter response with the fullsnapshot spectrum so that the compensation performance is further improved.
3. Range-dependent compensation using sparse representation The key requirement for STAP with any geometry configuration is the accurate knowledge of the clutter spectral response (i.e., the shape of the clutter ridge in the angle–Doppler domain) [1,2]. In the common case of ULA, the training data behaves stationary and can be utilized to estimate the accurate clutter response (termed as CCM) of the test cell. In the non-side-looking and/or CFA STAP cases, the clutter behaves range-dependent. To solve this problem, a series of preprocessings is proposed. RBC utilizes the technique of temporal smoothing to obtain a sub-snapshot spectrum and then generates the transform matrix at each range cell so that the processed training data behaves nearly stationary. Therefore, the sub-snapshot spectrum is the key to guarantee the desirable performance in the subsequent processings such as the estimation of configuration parameters and clutter power distribution. In this paper, an accurate clutter spectral response is also required, which is similar to that in RBC. However, the difference is that SR-RBC can obtain a more accurate clutter spectrum with full-snapshot and the estimation of the configuration parameters is avoided. In the following part, the technique of sparse representation is first introduced and then utilized in our problem of the clutter spectral estimation. Then, the procedures of the overall algorithm are elaborated.
K. Sun et al. / Signal Processing 91 (2011) 2268–2276
As shown in Fig. 2, after the discretization in angle– Doppler domain, each cell in this plane corresponds to a certain space–time steering vector and all of these vectors compose the overcomplete basis Uk. Because the STAP clutter scenario usually has a high CNR [1,2], the distribution in the angle–Doppler plane is mainly determined by the clutter distribution. Due to the angle–Doppler dependence of the clutter scatterers, the significant elements of the spectral distribution only focuses along the clutter ridge in the angle–Doppler domain, whose slope is determined by the radar configuration parameters and behaves range-dependent in the case of CFA. Therefore the clutter spectrum is sparse, i.e., only a small number of elements is significant and the others are quite small. This statement is even valid in the case of an omnidirectional antenna, where the clutter scatterers come from all the directions but the number of cells occupied by the clutter ridge is still small compared with the whole angle–Doppler plane.
3.1. Clutter spectral estimation using sparse representation First discretize the angle and Doppler frequency axes into Ns ¼ rsNM,Nt ¼ rtP grids in the angle–Doppler domain. The parameters rs,rt are the zoom scales along the angle and Doppler axes, respectively. Let fi ¼2pi/Ns,1 rN, and fd,j ¼j/Nd,1rj rNd denote the uniformly discretized azimuth angles and Doppler frequencies, respectively. The corresponding angle vectors for the kth range cell are given as
wi,k ¼ ½fi , yk T ,1 r i rNs :
ð11Þ
Thus, the received data of the kth range cell can be written in matrix form as xk ¼
N s Nt X
ai UUi,k þ nk ¼ Uk ak þ nk ,
ð12Þ
i¼1
where the NMP NsNt matrix Uk is the overcomplete basis composed with all the possible space–time steering vectors as
Uk ¼ ½sst ðw1,k ,fd,1 Þ,. . .,sst ðwNs ,k ,fd,1 Þ,. . ., sst ðw1,k ,fd,Nd Þ,. . .,sst ðwNs ,k ,fd,Nd Þ:
3.2. Sparse-induced compensation for range-dependent clutter
ð13Þ
According to the theory of sparse representation [11,12], when the actual distribution is sparse in a domain, the illposed problem in (12) can be efficiently solved. The basic form of sparse representation is defined as
The vector ak stands for the spectral distribution of the kth range cell in the basis, (i.e., the space–time spectrum in the angle–Doppler domain), and nk is the observation noise. Eq. (12) is the fundamental equation in this paper and has two important characteristics. First, estimating the space–time spectrum ak is equivalent to solving the linear Eq. (12) with the snapshot xk. Second, the basis Uk is overcomplete and the problem is ill-posed because the zoom scales rs,rt are greater than one to obtain the highresolution spectrum. Generally, when the positions of the actual clutter scatterers are known in advance, this illposed problem can be simplified into an overdetermined equation, which can be effectively solved by least square (LS) [6]. However, this prior knowledge is hard to guarantee in the actual clutter scenario. On the other hand, the theory of sparse representation has proven that: even when the actual positions are unknown, the ill-posed problem can be effectively solved provided that the actual clutter spectral distribution a0 is sparse [11,12]. Next the sparsity of the clutter spectral response is illustrated.
a^ ¼ arg min99a990 subject to 99xCa992 r e,
ð14Þ
where e is the data fitting allowance, 99.99p stands for the Lp norm and thus 99.990 denotes the number of the nonzero elements of a vector. However, this optimization is a combinatorial problem and NP-hard. To address this difficulty, a number of practical algorithms have been proposed to approximate this sparse solution. One effective way is to replace the objective function with the L1 norm [12]. It has been proven that this approximation can achieve quite desirable performance but demands a high computational effort especially for large-scale problems. In addition, a series of fast and greedy approximations are proposed, among which iterative reweighted least square (IRLS) appears to be both effective and time-saving [13–15]. IRLS uses the reweighted L2 norm minimization to make
Range cell #k Range cell #k+1
fd
2271
Mainlobe Range cell #k+2
Sidelobe
fs Fig. 2. Clutter spectral response of CFA STAP.
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recursive adjustments to the weightings until most of the elements in the solution are close to zero and generate a sparse solution. The applications in medical imaging and direction of arrival (DOA) estimation have also illustrated that IRLS may provide better spectral estimation than the l1 norm minimization when given with an appropriate initial value [16,17]. Therefore, considering both the estimation performance and computational load, SR-RBC adopts IRLS in the part of spectral estimation. In this paper, a novel method called SR-RBC is proposed to make registration-based compensation (the initial idea was developed in our earlier work [18]). This method has the advantage of serving two purposes simultaneously. First, the estimation of the configuration parameters and the clutter power distribution is integrated into the solution of ill-posed problem with the constraint of sparsity so that the estimation is data-based and no prior knowledge is needed. Second, the sparse representation such as IRLS can effectively solve the estimation problem and obtain more accurate clutter spectral response with full-snapshot. Based on these, the transform matrix is generated so that the processed training data behaves more stationary with the test cell and the SCR performance is further improved. The details of the algorithm are given below.
3.2.1. Clutter spectral estimation using IRLS To better understand why the sparse signal can be achieved, some basic illustration of IRLS is given. As an iterative algorithm, an appropriate initial value is necessary for IRLS to assure the final convergence [13,14]. Here we adopt the Fourier spectrum as
að0Þ ¼ UH i,k xk ,1 r ir Ns Nt , i,k
ð15Þ
where the initial vector akð0Þ is unbiased but low-resolution. The initialization does not have to be sparse, otherwise, some potential elements may be lost and not recovered in the subsequent iterations. The initial weighting matrix and the adaptive subspace are given as Wð0Þ ¼ diag ½abs að0Þ , k
G ¼ fi,1 ri r Ns Nt g,
ð16Þ
where diag(U) is the diagonalization. The estimation updating at the lth iteration is given accordingly as
aðlÞ 9 ¼ WðlÞ ðUk 9G WðlÞ Þy xk , k G
ð17Þ
ðlÞ 9 k G
stands for the G subset of the vector, where a H y aðlÞ and ðAÞ ¼ ðA AÞ1 AH denotes the pseudoinverse k operation of matrix A. As the iterations are implemented, some of the elements in the estimation aðlÞ become close k to zero. Thus the procedure of the reweighted least square in (17) can be only carried on a subspace Uk9G. Additionally, the dimension of the subspace should also be adjusted during the iteration as G ¼ arg 9aðlÞ 9 Z Th , 1 ri rNs Nt , ð18Þ i,k ðlÞ denotes the ith where Th stands for the threshold and ai,k element of the solution aðlÞ . Then, the weighting matrix k
can be updated as ðlÞ Wðl þ 1Þ ¼ diag fai,k ,i 2 Gg : Finally, the convergence judgment is made with ðlÞ ðl1Þ a a k k r x, ðlÞ a
ð19Þ
ð20Þ
k
where x stands for a small constant. Otherwise, the recursive process in (17)–(19) is repeated. After obtaining the high-resolution clutter spectral estimation, the CCM estimation can be given as X 2 ^ c, k ¼ 9a^ k ðiÞ9 Ui,k Ui,k H þ bL I, ð21Þ R i2C
where a^ k ðiÞ is the ith element of the final spectral ^ k at the kth range cell, and bL is a small estimation a loading to match the noise level. 3.2.2. Range-dependent compensation using SR-RBC The idea of designing the transform matrix using CCM so that the processed training data is stationary with the test cell was first proposed in RBC [9]. In this paper, SR-RBC follows the similar idea with RBC, but because the clutter spectrum and the corresponding CCM estimation is obtained with higher accuracy, the stationarity of the processed training data can be further improved. In other words, the matrix Tk is generated such that the processed training data x~ k ¼ Tk xk is stationary with the test cell xt CN(0,Rcj). Substituting the CCM of the processed data into the stationarity definition leads to h H R~ c, k ¼ E x~ k x~ H ð22Þ k ¼ Tk Rc,k Tk ¼ Rc,t : Using the eigenvalues decomposition, Rc,k and Rc,t can be expressed as 1=2 1=2 H V k Kk Rc,k ¼ Vk Kk VH , ð23Þ k ¼ V k Kk 1=2 1=2 H Rc,t ¼ Vt Kt Vt Kt ,
ð24Þ
where Kk and Kt are the diagonal matrixes containing the eigenvalues for the kth training data and the test cell, respectively, and Vk,Vt are the corresponding matrixes containing the eigenvectors as columns. Thus, the processed CCM at the kth range cell R~ c, k is further expressed as 1=2 1=2 H R~ c, k ¼ Tk Vk Kk Tk Vk Kk : ð25Þ Thus, to generate the stationary training data as R~ c, k ¼ Rc,t , we have 1=2
Tk Vk Kk
1=2
¼ Vt Kt
,
ð26Þ
then the transform matrix is correspondingly given as 1=2
Tk ¼ V t K t
K1=2 VH k: k
ð27Þ
In this way, the processed training data x~ k behaves nearly stationary with the test cell and can be utilized to estimate the CCM of the test cell using statistical-based methods like LSMI.
K. Sun et al. / Signal Processing 91 (2011) 2268–2276
(DOF) loss is generated, and more accurate spectral estimation is expected. Besides, because the prior knowledge is not required in the spectral estimation, SR-RBC is also an adaptive compensation method.
In summary, SR-RBC seeks to generate the transform matrix using the clutter spectral estimation, which follows the basic idea of RBC. However, there exists some critical difference between them, which is thought to be the reason behind the improvement in performance. RBC generates the sub-snapshot spectral response in the angle–Doppler domain and selects peaks to estimate the configuration parameters using the curve-fitting. Thus, the clutter power distribution and the CCM estimation is obtained at each range cell. On the contrary, SR-RBC combines the procedures of estimating both the configuration parameters and the clutter power distribution into the solution of ill-posed problem with the constraint of sparsity. Because the spectral estimation in SR-RBC is carried out with full-snapshot, no degree of freedom
4. Simulations In this section, the airborne radar is deployed with cylindrical arrays. The problem of range ambiguity is not considered and the configuration parameters are given in Table 1. The spectrum estimations using different methods are first given to verify the advantages of sparse representation. Then the performance such as SCR improvement is tested. Fig. 3(a)–(d) gives the actual clutter response, Fourier, RBC, and SR-RBC spectral estimations, respectively, where the clutter scenario is side-looking CFA. Due to insufficient space–time samples of the STAP processor, the Fourier spectrum has a high sidelobe and the resolution is limited. On the other hand, both RBC and SR-RBC can achieve high-resolution estimation. However, because RBC requires the temporal smoothing to obtain a CCM estimate with sufficient rank [10], the dimension of the CCM estimate is reduced so that the RBC spectrum has a limited accuracy, i.e., the clutter power spread and some missing of the actual scatterers along the clutter ridge. To avoid this problem, RBC performs the peak extraction in this sub-snapshot spectrum and then fits it with the mathematical model to estimate the configuration
Table 1 Configuration parameters. Parameter
Symbol
Value
Number of rings Number of arrays on each ring Number of pluses Platform velocity Pulse repetition interval Range sample rate Radar wavelength Inter-ring spacing Radius of the ring Platform height Clutter-to-noise ratio
M N P v PRI fs
4 4 16 300 m/s 0.25 ms 5 Mhz 0.3 m 0.15 m 0.15 m 3000 m 30 dB
l d r H CNR
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Fig. 3. Side-looking CFA (a) actual clutter response (b) Fourier estimation (c) RBC estimation, and (d) SR-RBC estimation.
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Fig. 5(a) gives the IFLoss performance of different STAP algorithms with side-looking CFA. Because the clutter behaves range-dependent, a direct statistical method like LSMI shows a degraded performance and the clutter notch is mismatch with the actual scenario. ADC aligns the peaks of the clutter ridge of the training data with that of the test cell. However, it only accomplishes partial compensation and the performance is still not desirable. On the other hand, RBC and SR-RBC lead to both mainlobe and sidelobe compensation. However, because the estimation of both the configuration parameters and clutter power distribution is based on the accuracy of the clutter spectrum, the RBC performance is limited. However, since SR-RBC can obtain high-accurate spectral estimation with full-snapshot, the corresponding training data after the transformation behaves more stationary, which brings desirable SCR improvement. Moreover, because SR-RBC directly estimate the Doppler response in the spectral domain, the estimation of configuration parameters, such as velocity and crab angle, is avoided. Thus, as shown in Fig. 5(b), SR-RBC still preserves desirable performance with the non-side-looking CFA.
parameters [9,10]. However, this extraction is partly sensitive to the spurious peaks and other artifacts, which appear in the sub-snapshot spectrum. Moreover, even if the estimation of configuration parameters is obtained with high accuracy, the clutter power estimation using LS might lose some actual clutter scatterers because the estimation is only carried out in the locations of the extracted peaks. Thus, the accuracy in the sub-snapshot spectrum limits the performance of RBC. On the other hand, as shown in Fig. 3(d), because SR-RBC can obtain a desirable sparse solution at each range cell using IRLS, there is no sub-snapshot tradeoff and high-accurate spectrum estimation is expected. A parallel simulation was carried out with non-side-looking CFA, where a similar conclusion was obtained in Fig. 4(a)–(d). Next the SCR improvement by different STAP algorithms is given to illustrate the advantage of SR-RBC. Conventionally, the efficiency of the STAP filter is evaluated by the normalized SCR improvement, which is defined as [1] IFLoss ¼
^ H s92 9w SCRout =SCRin ¼ ðSCRout =SCRin Þopt ^ H RwUs ^ H R1 s, w
ð28Þ 5. Conclusions
^ -1 s, ^ ¼ mR where the estimated adaptive filter is given as w ^ is the CCM estimation using a given technique (such as R ADC, RBC, or SR-RBC), s denotes the steering vector of the moving target, R is the actual CCM, and tr(R) is the input clutter power. The slant range of the test cell is Rs ¼1.5H, which is a typical short-range case. The training data are the adjacent range cells with the amount of L¼40.
In this paper, the sparsity of the clutter spectral response in CFA was analyzed, and a new compensation strategy called SR-RBC was proposed to deal with range-dependent clutter. The key advantage of SR-RBC is its capability of obtaining high-accurate spectral estimation with full-snapshot due to the technique of sparse representation. In this
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Fig. 4. Non-side-looking CFA (a) actual clutter response (b) Fourier estimation (c) RBC estimation, and (d) SR-RBC estimation.
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way, SR-RBC can acquire a better transform matrix at each range cell so that the stationarity of the processed training data is further improved. The following are some considerations for further research. First, the current overcomplete basis Uk is fixed in the sparse representation. However, due to the practical nonideal factors such as clutter internal motion and/or channel mismatch, this predefined overcomplete basis does not always match with the actual data, and therefore, the corresponding sparsity might decrease. Therefore, solving the sparse representation problem where both an overcomplete basis and actual sources are unknown seems to be quite important. Second, because IRLS is an iterative algorithm seeking to approximate the actual clutter spectrum, an appropriate initial value, and more adaptive mechanisms are necessary to approach the overall sparse solution.
for his fruitful suggestions and discussions. In addition, the authors also want to give respects to Professor Y.L. Wang at Wuhan Radar Academy, China, for his guidance and selfless helps on the research. This work was supported in part by the National Natural Science Foundation of China (No. 40901157) and in part by the National Basic Research Program of China (973 Program, No. 2010CB731901). References [1] R. Klemm, Principles of Space–Time Adaptive Processing, (IEE Press, 1999. [2] R. Klemm, The Applications of Space–Time Adaptive Processing, second ed., IEE Press, 2004. [3] R.K. Hersey, W.L. Melvin, J.H. McClellan, Clutter-limited detection performance of multi-channel conformal arrays, Signal Processing 84 (9) (2004) 1481–1500.
Acknowledgment
[4] R.K. Hersey, W.L. Melvin, J.H. McClellan, E. Culpepper, Adaptive ground clutter suppression for conformal array radar systems, IET Radar, Sonar and Navigation 3 (4) (2009) 357–372.
The authors would like to express their great gratitude to Professor Fabian D. Lapierre at Royal Military Academy
[5] B. Himed, Y. Zhang, A. Hajjari, STAP with angle–Doppler compensation for bistatic airborne radars, in: Proceedings of IEEE National Radar Conference, Long Beach, California, May 2002, pp. 22–25.
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[6] A.G. Jaffer, P.T. Ho, B. Himed, Adaptive compensation for conformal array STAP by configuration parameter estimation, in: Proceedings of IEEE National Radar Conference, Verona, NY, April 2006, pp. 24–27. [7] M.A. Zatman, Performance analysis of the derivative based updating method, Adaptive Sensor Array Processing Workshop, MIT Lincoln Laboratory, Lexington, MA, March 2001, pp. 13–14. [8] F.D. Lapierre, J.G. Verly, Registration-based range dependence compensation for bistatic STAP radar, Eurasip Journal on Advances in Signal Processing (2005) 85–981 (2005) 85–98. [9] F.D. Lapierre, P. Ries, J.G. Verly, Foundation for mitigating range dependence in radar space–time adaptive processing, IEE Proceedings on Radar, Sonar and Navigation 3 (1) (2009) 18–29. [10] P. Ries, F.D. Lapierre, J.G. Verly, Geometry-induced range dependence compensation for bistatic STAP with conformal arrays, IEEE Transactions on Aerospace and Electronic Systems, accepted for publication. [11] D.L. Donoho, M. Elad, V.N. Temlyakov, Stable recovery of sparse overcomplete representations in the presence of noise, IEEE Transactions on Information Theory 52 (1) (2006) 6–18. [12] J.A. Tropp, Just relax: convex programming methods for identifying sparse signals in noise, IEEE Transactions on Information Theory 1 (1) (2006) 6–18.
[13] I.F. Gorodnitsky, B.D. Rao, Sparse signal reconstruction from limited data using FOCUSS: a re-reweighted minimum norm algorithm, IEEE Transactions on Signal Processing 45 (3) (1997) 600–616. [14] B.D. Rao, K. Engan, S.F. Cotter, J. Palmer, K. Kreutz-Delgado, Subset selection in noise based on diversity measure minimization, IEEE Transactions on Signal Processing 51 (3) (2003) 760–770. [15] N. Mourad, J.P. Reilly, Minimizing nonconvex functions for sparse vector reconstruction, IEEE Transactions on Signal Processing 58 (7) (2010) 3485–3496. [16] P.H. Schimpf, H.S. Liu, C. Ramon, J. Haueisen, Efficient electromagnetic source imaging with adaptive standardized LORETA/FOCUSS, IEEE Transactions on Biomedical Engineering 52 (5) (2005) 901–908. [17] D. Malioutov, M. Cetin, A.S. Willsky, A sparse signal reconstruction perspective for source localization with sensor arrays, IEEE Transactions on Signal Processing 53 (8) (2005) 3010–3022. [18] S. Ke, Z. Hao, L. Gang, M. Huadong, W. Xiqin, A novel STAP algorithm using sparse recovery technique, in: Proceedings of IEEE International Geoscience and Remote Sensing Symposium, South Africa, July 2009, pp. 336–339.