Adaptive clutter suppression based on iterative adaptive approach for airborne radar

Signal Processing ] (]]]]) ]]]–]]]

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Contents lists available at SciVerse ScienceDirect

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Signal Processing

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journal homepage: www.elsevier.com/locate/sigpro

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Adaptive clutter suppression based on iterative adaptive approach for airborne radar

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Zhaocheng Yang n, Xiang Li, Hongqiang Wang, Weidong Jiang

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Electronics Science and Engineering School, National University of Defense Technology, Changsha 410073, China

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a r t i c l e i n f o

abstract

Article history: Received 1 August 2012 Received in revised form 22 February 2013 Accepted 19 March 2013

To improve the performance of the recently developed weighted least-squares-based iterative adaptive approach (IAA) in space–time adaptive processing (STAP) for weak or slow targets detection, we propose a novel IAA scheme to adaptively suppress the ground clutter by using the secondary training data (STD). Especially, we use the IAA to estimate the clutter plus noise covariance matrix from a very small number of STD. The resulting clutter plus noise covariance matrix can be utilized to form the STAP filter and then suppress the clutter. To reduce the computational complexity of the IAA, we exploit the sparsity of large clutter components in the angle-Doppler image and develop a modified IAA algorithm employing a soft-thresholding to adaptively determine the entries of each iteration that should be updated. Simulation results show that our proposed scheme outperforms the conventional IAA scheme over weak or slow targets detection and the modified IAA algorithm exhibits a comparable or even a better performance than the IAA algorithm but a lower computational complexity. & 2013 Elsevier B.V. All rights reserved.

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Keywords: Space–time adaptive processing Adaptive clutter suppression Iterative adaptive approach Airborne radar

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65

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1. Introduction

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Since the early 1970s space–time adaptive processing (STAP) methods have been actively considered for look down airborne radar where target signals have to compete with strong ground clutter returns [1–3]. However, there are many practical limitations preventing the use of the optimum full-rank STAP processor. One of them is the requirement of a large number of independent and identically distributed (IID) training samples to estimate the interference covariance matrix, which becomes even more serious in practical situations because of the non-stationary and nonhomogeneous interference environment. Reduced-dimension and reduced-rank methods have been considered to counteract the slow convergence of the

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57 59 61

n

Corresponding author. Tel.: +86 73184575760. E-mail addresses: [email protected] (Z. Yang), [email protected] (X. Li), [email protected] (H. Wang), [email protected] (W. Jiang).

full-rank STAP [4–9]. These methods can reduce the number of training snapshots to twice of the reduceddimension, or twice of the clutter rank. The parametric adaptive matched filter (PAMF) based on a multichannel autoregressive model [10] provides another alternative solution to the slow convergence of the full-rank STAP. Furthermore, the sparsity of the received data and filter weights is exploited to improve the convergence for a generalized sidelobe canceler architecture and a direct filter processor in [11,12]. However, it still needs to improve the convergence or reduce the sample support when employing these approaches because the number of required snapshots is large relative to those needed in IID scenarios. Direct data domain (D3) STAP approaches only use the snapshot in the test range cell but no training data, which can avoid the nonhomogeneity in the secondary training data (STD) and radically eliminate the impacts of nonhomogeneous environments [13]. However, this benefit comes at the cost of reduced system degrees of freedom (DOFs) resulting in a decreased performance.

0165-1684/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.03.033

Please cite this article as: Z. Yang, et al., Adaptive clutter suppression based on iterative adaptive approach for airborne radar, Signal Processing (2013), http://dx.doi.org/10.1016/j.sigpro.2013.03.033i

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Z. Yang et al. / Signal Processing ] (]]]]) ]]]–]]]

Knowledge-aided (KA) STAP aims at exploiting environmental knowledge and has been developed to enhance the detection performance especially in the case of lacking training support [14–22]. However, the exact form of prior knowledge is problem-dependent and hard to be derived. Recently developed STAP algorithms based on sparse representation/sparse recovery (SR) try to formulate the STAP problem by employing a sparse regularization that the target signal or the clutter spectrum is sparse in the whole angle-Doppler plane [23–28]. This kind of approaches can not only work on multiple training samples, but also on D3 case, which exhibit significant better performance than conventional algorithms in very short training samples or avoid performance degradation caused by system DOFs reduction in conventional D3 STAP. But the performance of this kind of approaches depends on the SR algorithms, which always require some parameters to be set. To reduce the need for the STD or the accurate prior knowledge of the clutter statistics in STAP, a weighted least-squares-based iterative adaptive approach (IAA) is presented to form angle-Doppler images of both clutter and targets for each range bin of interest in [30]. Then the resulting angle-Doppler images can be used with localized detection approaches for moving target indication (MTI) [30]. However, there are mainly two problems about IAA algorithm for MTI applications. One is the high computational complexity especially processing data with large sizes. Several authors have developed fast implementations of the IAA using the Gohberg–Semencul (G–S)-type factorization of the IAA covariance matrices [31–33] resulting in great computational savings. But this approach requires the IAA covariance matrices to be a Toeplitz block Toeplitz matrix. As for STAP, it is only suitable for the case of uniformly spaced linear array (ULA) and a constant pulse repetition frequency (PRF) with uniformly sampled spatial frequencies and Doppler frequencies. The other problem is that the IAA algorithm directly detects the targets from the estimated angle-Doppler images of each range bin of interest without clutter suppression, which easily leads to weak or slow targets missing [34]. In order to overcome this problem, a novel IAA scheme is proposed in the paper. The proposed IAA scheme firstly estimates the clutter plus noise covariance matrix using multiple snapshots from the STD. Then the STAP filter is designed by utilizing the estimated clutter plus noise covariance matrix followed with the moving targets detection. Furthermore, in order to reduce the computational complexity of the IAA algorithm, we exploit the fact that the number of large clutter components is much smaller than that of the whole angle-Doppler bins and develop a modified IAA algorithm employing a softthresholding to adaptively determine the entries of each iteration that should be updated resulting in a reduced dimension to compute the clutter power in each iteration. A similar algorithm, called the IAA-RC, has been developed in the context of the 2-D direction-of-arrival (DOA) in [38], which can be also applied to the STAP problem with the proposed scheme. The differences between the proposed algorithm and the IAA-RC lie in three aspects. First, the IAARC algorithm initializes the noise power region with some iterations of the conventional IAA algorithm, while our

proposed algorithm provides a regularized diagonal matrix (which is adaptively estimated from the samples) for the noise power. Second, the IAA-RC algorithm fixes the reduced region in the iterations after the initialization step, while our proposed algorithm determines the entries that should be updated for each iteration thereby a varying reduced region. Third, the IAA-RC algorithm employs a threshold relative to the minimum estimated signal power, while our proposed algorithm sets the threshold relative to the mean of the estimated noise power. Simulation results illustrate the effectiveness of our proposed algorithm. The main contributions of this paper can be highlighted as follows.

63

(a) A novel IAA scheme using the STD is proposed to improve the detection performance of the conventional IAA scheme for weak or slow targets. (b) A modified IAA algorithm employing a soft-thresholding operation is proposed to reduce the computational complexity of the conventional IAA algorithm. (c) A detailed comparison is presented to show the computational complexity of the proposed and conventional IAA algorithms. (d) A study and comparative analysis of our proposed algorithm including the SINR performance, the convergence speed and the detection performance with conventional IAA algorithms is carried out.

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The work is organized as follows. Section 2 introduces the signal model in airborne radar applications. Section 3 details the approach of the proposed IAA scheme and the modified IAA algorithm and also discusses the computational complexity. The simulated airborne radar data are used to evaluate the performance of the proposed algorithms in Section 4. Section 5 provides the summary and conclusions.

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2. Signal model

101

The system under consideration is a side-looking pulsed Doppler radar with a ULA consisting of M elements on the airborne radar platform, as shown in Fig. 1. The

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Fig. 1. Airborne radar geometry with a ULA antenna.

Please cite this article as: Z. Yang, et al., Adaptive clutter suppression based on iterative adaptive approach for airborne radar, Signal Processing (2013), http://dx.doi.org/10.1016/j.sigpro.2013.03.033i

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Z. Yang et al. / Signal Processing ] (]]]]) ]]]–]]]

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platform is at altitude hp and moving with constant velocity vp . ϕ refers to the angle-of-arrival (AOA) relative to the normal of the array. The radar transmits a coherent burst of pulses at a constant PRF f r ¼ 1=T r , where T r is the pulse repetition interval (PRI). The transmitter carrier frequency is f c ¼ c=λc , where c is the propagation velocity and λc is the wavelength. The number of pulses in a coherent processing interval (CPI) is N. The received signal from the iso-range of interest is represented by a space– time NM  1 data vector x. Ignoring the impact of range ambiguities, a general model for the space–time clutter plus noise snapshot x is given by [1] Nc

x ¼ ∑ sc;n vðf d;n ; f s;n Þ þ n;

ð1Þ

n¼1

17 19 21 23 25 27 29 31 33 35 37

where n is the Gaussian white thermal noise vector with the noise power s2n on each channel and pulse; Nc is the number of independent clutter patches over the iso-range of interest; sc;n , f s;n and f d;n are the random complex amplitude, the spatial frequency and the Doppler frequency of the nth clutter patch, respectively; vðf d ; f s Þ is the NM  1 space–time steering vector with the spatial frequency f s and the Doppler frequency f d . Let us discretize the whole angle-Doppler plane uniformly into sufficiently small grid points, e.g., N d ¼ ρd N Doppler bins and Ns ¼ ρs M angle bins, where ρd ; ρs 4 1 determine the smoothness of the angle-Doppler images. For each grid point, the corresponding space–time steering vector is formulated as vðf d;k ; f s;i Þ ¼ vt ðf d;k Þ⊗vs ðf s;i Þ;

ð2Þ

3

63

covariance matrix can be written as H

R ¼ ΦΣΦ þ

s2n I;

ð7Þ

where ðÞ denotes the conjugate transposition operation, I denotes the identity matrix, Σ ¼ diagðaÞ with diagonal elements representing the power of the target signal and clutter patches, a ¼ ½a1;1 ; a1;2 ; …; aNd ;Ns T and ak;i ¼ E½jγ k;i j2 , k ¼ 1; 2; …; Nd , i ¼ 1; 2; …; N s . Here, note that it is assumed that the amplitudes of target signal, clutter patches and thermal noise are statistical independent each other [1–3]. It should be also noted that although here it is a sidelooking ULA and a constant PRF, the following proposed algorithm can be easily extend to arbitrary array geometry and random slow-time samples, which is as well as pointed out by the original IAA work applying to the STAP [30]. In this extension, we just need to reconsider the angle-Doppler bins discretization and also regenerate the space–time steering vectors according to the array geometry and the PRF. The procedure of the proposed algorithm does not require any modification.

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3. Proposed IAA-STAP scheme 85 In this section, we will introduce a new STAP scheme based on IAA using STD to suppress the clutter and detect the target as well as a modified IAA algorithm by employing a soft-thresholding to reduce the computational complexity. In addition, we will detail the computational complexity analysis of the proposed algorithm and compare it with other IAA algorithms.

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3.1. IAA using STD 95

where vt ðf d;k Þ and vs ðf s;i Þ are the temporal and spatial steering vectors of the kth Doppler bin and ith angle bin, given by

The basic idea of the recently developed IAA algorithm tries to solve (5) by minimizing the following weighted least-squares (LS) cost function [30–38]

vt ðf d;k Þ ¼ ½1; …; expðj2πðN−1Þf d;k ÞT ;

min∥x−γ k;i vðf d;k ; f s;i Þ∥2Q −1 ;

ð3Þ

65

H

ð8Þ

97 99

k;i

39 41 43 45 47 49

vs ðf s;i Þ ¼ ½1; …; expðj2πðM−1Þf s;i ÞT ;

ð4Þ

k;i

where ðÞT denotes the transposition operation, f d;k ¼ ð2k−Nd −2Þ=N d , f s;i ¼ ðda =λc Þ sinðð2i−Ns −2Þ=Ns πÞ, and da is the inter-sensor spacing of the ULA. If we do not consider the mismatch between the assumed clutter space–time steering vectors and the true clutter space–time steering vectors, the received clutter plus noise snapshot x can be represented by x ¼ Φγ þ n;

ð5Þ

53

where γ ¼ ½γ 1;1 ; γ 1;2 ; …; γ Nd ;Ns T is called the N d Ns  1 angleDoppler profile with non-zero elements representing the clutter patches [23], and the NM  N d Ns matrix Φ is the space–time steering dictionary, as given by

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Φ ¼ ½vðf d;1 ; f s;1 Þ; …; vðf d;1 ; f s;Ns Þ; …; vðf d;Nd ; f s;Ns Þ:

51

57 59 61

101

where ∥x∥2Q −1 ≜xH Q −1 k;i x and

ð6Þ

For the target presence in the iso-range of interest, the target return is also can be seen as a random amplitude multiplying one column of the space–time steering dictionary and the received signal in the test range bin can be still formulated by (5). Therefore, the clutter (target absence) or target-clutter (target presence) plus noise

Q k;i ¼ R−ak;i vðf d;k ; f s;i ÞvH ðf d;k ; f s;i Þ;

ð9Þ

is the IAA clutter (target absence) or target-clutter (target presence) plus noise (signals at angle-Doppler grid points other than ðf d;k ; f s;i Þ) covariance matrix. By solving the above optimization problem, the parameter ak;i can be calculated as [30]

2

vH ðf d;k ; f s;i ÞR−1 x

IAA a^ k;i ¼ jγ k;i j2 ¼
ð10Þ
; −1 H
v ðf d;k ; f s;i ÞR vðf d;k ; f s;i Þ
The conventional IAA algorithm in [30] directly computes the angle-Doppler image of the test range bin and does not need the ground clutter cancelation. However, the experimental results in [34] show that the conventional IAA cannot replace the classic STAP techniques because weak or slow targets will be missed without clutter suppression. Therefore, we propose a new STAP scheme based on IAA using STD (snapshots from adjacent target-free range bins) to suppress the clutter first and then detect the targets. We assume L snapshots xl ; l ¼ 1; 2; …; L in the STD, to have a complex multivariate

Please cite this article as: Z. Yang, et al., Adaptive clutter suppression based on iterative adaptive approach for airborne radar, Signal Processing (2013), http://dx.doi.org/10.1016/j.sigpro.2013.03.033i

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Z. Yang et al. / Signal Processing ] (]]]]) ]]]–]]]

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IID Gaussian distribution with zero mean and covariance matrix R, given in (7), so that the likelihood of each snapshot xl has the following form [3]:

5

pxl ðxl jRÞ ¼

7

where jRj denotes the determinant operation on R. The associated likelihood function pðx1 ; …; xL jRÞ (joint density of x1 ; …; xL conditioned on R) is thus given by

9 11

1 H −1 e−xl R xl : π NM jRj

ð11Þ

L
pðx1 ; …; xL jRÞ ¼ ∏ pxl ðxl
RÞ

15 17 19 21 23 25 27 29 31 33 35 37 39

¼

1 L H −1 e−∑l ¼ 1 xl R xl : π NML jRjL

ð12Þ

Letting X ¼ ½x1 ; …; xL ∈CNML , the above equation can be rewritten as H −1 1 pðXjRÞ ¼ e−TrðX R XÞ π NML jRjL

ð13Þ

where TrðÞ denotes the trace operator (sum of the diagonal elements of a square matrix). The maximumlikelihood estimation (MLE) of the unknown term ak;i in R is obtained by maximizing the likelihood function with respect to ak;i , especially ML a^ k;i ¼ arg maxðpðXjRÞÞ

¼ arg minð−ln pðXjRÞÞ ¼ arg minðL lnjRj þ TrðXH R−1 XÞÞ:

ð14Þ

Since R ¼ Q k;i þ ak;i vðf d;k ; f s;i ÞvH ðf d;k ; f s;i Þ, we obtain jRj ¼ jQ k;i þ ak;i vðf d;k ; f s;i ÞvH ðf d;k ; f s;i Þj ¼ jQ k;i jð1 þ ak;i vH ðf d;k ; f s;i ÞQ −1 k;i vðf d;k ; f s;i ÞÞ:

ð15Þ

Using the matrix inversion lemma, we can compute the R −1 as R −1 ¼

−1 H ak;i Q −1 k;i vðf d;k ; f s;i Þv ðf d;k ; f s;i ÞQ k;i Q −1 : k;i − −1 1 þ ak;i vH ðf d;k ; f s;i ÞQ k;i vðf d;k ; f s;i Þ

ð16Þ

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Substituting (15) and (16) into (14), and fixing Q k;i , the minimization of (14) with respect to ak;i is equivalent to minimizing n ML a^ k;i ¼ arg min L lnð1 þ ak;i vH ðf d;k ; f s;i ÞQ −1 k;i vðf d;k ; f s;i ÞÞ !) −1 H ak;i XH Q −1 k;i vðf d;k ; f s;i Þv ðf d;k ; f s;i ÞQ k;i X : ð17Þ −Tr 1 þ ak;i vH ðf d;k ; f s;i ÞQ −1 k;i vðf d;k ; f s;i Þ By computing the gradients of the right-hand side (RHS) in (17), equating that to zero and solving for ak;i , we obtain ML a^ k;i ¼

−1 ^ vH ðf d;k ; f s;i ÞQ −1 k;i ðR ML −Q k;i ÞQ k;i vðf d;k ; f s;i Þ

ðvH ðf

−1 2 d;k ; f s;i ÞQ k;i vðf d;k ; f s;i ÞÞ

ð1

2 ðvH ðf d;k ; f s;i ÞQ −1 k;i vðf d;k ; f s;i ÞÞ ML 2 þ a^ k;i vH ðf d;k ; f s;i ÞQ −1 k;i vðf d;k ; f s;i ÞÞ

63

;

ð19Þ

which is strictly positive. That is to say the estimation of ML a^ k;i in (18) is the global minimizer of (17). Moreover, applying the matrix inversion lemma to (9) and substituting the result into (18), we obtain a simple formulation ML of a^ k;i as ML MIAA a^ k;i ¼ a^ k;i þ ak;i −aCapon ; k;i

l¼1

13

ML second derivative at the point a^ k;i as

;

ð18Þ

where R^ ML ¼ ð1=LÞXXH denotes the ML estimation of the clutter plus noise covariance [1–3]. When L ¼1, the above solution gives a MLE of ak;i only using one snapshot, which has the same formulation as (49) in [37]. Similarly as the derivation in [37], taking the second gradients of RHS in ML (17) and inserting the above estimation of a^ k;i , we get the

ð20Þ

where MIAA a^ k;i

¼

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vH ðf d;k ; f s;i ÞR−1 R^ ML R−1 vðf d;k ; f s;i Þ

ð21Þ

77

and aCapon denotes the estimate for ak;i with a standard k;i Capon beamformer (SCB), given by

79

aCapon ¼ k;i

ðvH ðf d;k ; f s;i ÞR−1 vðf d;k ; f s;i ÞÞ2

1 vH ðf

d;k ; f s;i ÞR

−1

vðf d;k ; f s;i Þ

;

:

ð22Þ

83

If the covariance matrix R is known, the SCB estimate aCapon will be around the true parameter ak;i . In this case, k;i ML the parameter a^ k;i in (20) can be simplified as ML MIAA a^ k;i ≈a^ k;i :

ð23Þ

By inspecting (20) and (23), there are some concerns that should be remarked. First, it is well known that R^ ML -R as L-∞ [1–3], which indicates the MLE ML a^ k;i -ak;i . In other words, the more the number of snapshots for training, the better the estimation accuracy of the ML parameter a^ k;i . Second, since (23) requires R, which depends on the unknown parameter a, the proposed algorithm must operate in an iterative way (like the conventional IAA method in [30]). While the size of a is usually very large (N d is chosen from 5N to 10N and Ns is chosen from 5M to 10M) resulting in a high computational complexity. Fortunately, there is a high degree of sparsity in γ corresponding to a high degree of sparsity in parameter a for the STAP problem, which is also exploited in SR-STAP type algorithms [25–29]. That is to say there are a lot of elements of a equating to zeros or approximating to zeros (when in thermal noise case, there will be a lot of elements with small values in parameter a because the thermal noise power is much lower than the clutter power for STAP problem in airborne radar). Therefore, we do not need to compute every single element of a and only have to update those whose values are large ones or more significant ones than others. Based on this idea, we apply a soft-thresholding operation for each iteration of the proposed algorithm to decide which elements should be ML ^ updated. Let a^ k;i ðqÞ and RðqÞ denote the estimated solution and the estimated clutter plus noise covariance matrix using the proposed algorithm in the qth iteration. For the ML q+1th iteration, a^ k;i ðq þ 1Þ can be updated by ML MIAA a^ k;i ðq þ 1Þ≈a^ k;i

¼

−1 −1 vH ðf d;k ; f s;i ÞR^ ðqÞR^ ML R^ ðqÞvðf d;k ; f s;i Þ −1 ðvH ðf d;k ; f s;i ÞR^ ðqÞvðf d;k ; f s;i ÞÞ2

81

85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115 117 119

;

k; i∈ΓðqÞ;

121 ð24Þ

Please cite this article as: Z. Yang, et al., Adaptive clutter suppression based on iterative adaptive approach for airborne radar, Signal Processing (2013), http://dx.doi.org/10.1016/j.sigpro.2013.03.033i

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Z. Yang et al. / Signal Processing ] (]]]]) ]]]–]]]

1

where

3

ML ^ ^ RðqÞ ¼ ∑ a^ k;i ðqÞvðf d;k ; f s;i ÞvH ðf d;k ; f s;i Þ þ ΔðqÞ;

5

and

7 9

Γðq þ

k ¼ 1; …; N d ;

ð25Þ Initialization:
2


vH ðf d;k ; f s;i Þx

a^ k;i ð1Þ ¼
H
, k ¼ 1; …; N d ; i ¼ 1; …; N s ,
v ðf d;k ; f s;i Þvðf d;k ; f s;i Þ


þ 1Þ≥ηðqÞÞ

i ¼ 1; …; Ns :

R^ ML

ð26Þ

13

The set Γ is the adaptive subspace whose entries are needed to update. The symbols ηðqÞ stand for the soft^ thresholding factor, and ΔðqÞ is the regularized diagonal matrix which employs the structure

15

2 ^ ΔðqÞ ¼ s^ n ðqÞI;

11

17 19 21 23 25 27 29 31 33 35 37 39

2 where s^ n ðqÞ is the mean of the elements s^ 2n;h ðqÞ, h ¼ 1; 2; …; NM.

can be given by [37]

2
H ^ −1

v h R ðqÞx
2

; s^ n;h ðqÞ ¼
−1 ^ ðqÞv h


v H hR

estimated noise power The estimate for s^ 2n;h ðqÞ

ð28Þ

ηðqÞ ¼ βs^ ðqÞ;

43

where β is a positive scalar larger than 1 and denotes the factor relative to the thermal noise power (satisfactory results are obtained in our experiments by considering β ¼ 3∼10 dB). In [40], an approach employing a hardthresholding is developed. Compared with this approach, when the noise power is unknown, the proposed algorithm shows an easier thresholding setting because the parameter β represents the factor relative to the noise power and the noise power is adaptively estimated in each iteration. For clarification, we detail the pseudo-Matlab codes of the proposed IAA algorithms using STD in Tables 1–3. For convenience, we call the approach to directly compute parameter a according to (23) as the MIAA algorithm, the approach with regularized matrix to compute the clutter plus noise covariance as MIAA-R algorithm, and the approach employing a soft-thresholding and adaptively estimating the regularized diagonal matrix as MIAA-TR algorithm.

47 49 51 53 55 57 59 61

While q≤Q R^ ðqÞ ¼ ∑Nd

1

2

k¼1

a^ k;i ðq þ 1Þ ¼

ð29Þ

65 67 69 71

H s ^ ∑N i ¼ 1 a k;i ðqÞvðf d;k ; f s;i Þv ðf d;k ; f s;i Þ,

vH ðf d;k ; f s;i ÞR^

−1

ðqÞR^ ML R^

ðvH ðf d;k ; f s;i ÞR^

−1

−1

ðqÞvðf d;k ; f s;i Þ

ðqÞvðf d;k ; f s;i ÞÞ2

,

3 q ¼ q þ 1, End

73 75 77

Table 2 The proposed MIAA-R algorithm.

79

Initialization:
2


vH ðf d;k ; f s;i Þx

a^ k;i ð1Þ ¼
H
, k ¼ 1; …; N d , i ¼ 1; …; N s ,
v ðf d;k ; f s;i Þvðf d;k ; f s;i Þ
1 R^ ML ¼ ∑Ll ¼ 1 xl xH l , q ¼ 1. L

41

45

1 ¼ ∑Ll ¼ 1 xl xH l , q ¼ 1. L

ð27Þ

where v h , h ¼ 1; 2; …; NM denotes the hth column of the identity matrix I. Note that the proposed regularized scheme is different from the scheme in [37]. In the proposed scheme, we use the mean of the estimated noise power elements instead of directly using those, which is similar to the approach in [39]. This is because the thermal noise elements from different channels and pulses are always independent each other and follow the same distribution, as seen in the last term in (7). Furthermore, the proposed regularized scheme will not destroy the structure of the matrix Rc . Especially, the matrix Rc in (7) is a Toeplitz block Toeplitz under the case of ULA and constant PRF with uniformly sampled spatial frequencies and Doppler frequencies (this property is very useful for complexity reduction). The soft-thresholding ηðqÞ can be set as a higher value than thermal noise level but a lower value than the clutter power, i.e. 2

63

Table 1 The proposed MIAA algorithm.

k;i∈Γ

ML 1Þ ¼ argða^ k;i ðq

5



2
H ^ −1

v R ðqÞx

, h ¼ 1; …; NM, s^ 2n;h ðqÞ ¼

h −1
^
v H ðqÞv h
hR

3

2 2 ^ ΔðqÞ ¼ s^ n ðqÞI, s^ n ðqÞ ¼ meanðs^ 2n;h ðqÞÞ, Nd Ns ^ ∑ a^ k;i ðqÞvðf ; f ÞvH ðf R ðqÞ ¼ ∑

4

a^ k;i ðq þ 1Þ ¼

2

i¼1

k¼1

vH ðf d;k ;f s;i ÞR^

83 85

While q≤Q 1

81

d;k

−1

s;i

87 89 ^ d;k ; f s;i Þ þ ΔðqÞ,

−1

ðqÞR^ ML R^ ðqÞvðf d;k ;f s;i Þ , −1 ðvH ðf d;k ;f s;i ÞR^ ðqÞvðf d;k ;f s;i ÞÞ2

5 q ¼ q þ 1, End

91 93 95 97

Table 3 The proposed MIAA-TR algorithm.

99 Initialization:
2


vH ðf d;k ; f s;i Þx

a^ k;i ð1Þ ¼
H
, k ¼ 1; …; N d , i ¼ 1; …; N s .
v ðf d;k ; f s;i Þvðf d;k ; f s;i Þ
1 R^ ML ¼ ∑Ll ¼ 1 xl xH l , q ¼ 1, L Γð1Þ ¼ fð1; 1Þ; ð1; 2Þ; …; ðk; iÞ; …; ðN d ; N s Þg, ηð1Þ ¼ 0, β

103 105

While q≤Q 1

2 3 4

a^ k;i ðq þ 1Þ ¼

vH ðf d;k ; f s;i ÞR^

−1

ðqÞR^ ML R^

−1

ðqÞvðf d;k ; f s;i Þ

−1

ðvH ðf d;k ; f s;i ÞR^ ðqÞvðf d;k ; f s;i ÞÞ2
2


H ^ −1
v R ðqÞx

, h ¼ 1; …; NM s^ 2n;h ðqÞ ¼

h −1
H ^
v h R ðqÞv h


,

2 2 ^ ΔðqÞ ¼ s^ n ðqÞI, s^ n ðqÞ ¼ meanðs^ 2n;h ðqÞÞ, ^ ^ ^ R ðqÞ ¼ ∑k;i∈Γ a k;i ðqÞvðf d;k ; f s;i ÞvH ðf d;k ; f s;i Þ þ ΔðqÞ, 2

5 ηðqÞ ¼ βs^ n ðqÞ, 5 Γðq þ 1Þ ¼ argða^ k;i ðq þ 1Þ≥ηðqÞÞ 6 q¼qþ1 End

101

k; i∈ΓðqÞ,

107 109 111 113 115 117

3.2. STAP filter

119

After we obtain the estimated clutter plus noise covariance matrix, the idea behind linear constraint minimum variance (LCMV) approach is to minimize the clutter plus

121

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Z. Yang et al. / Signal Processing ] (]]]]) ]]]–]]]

3 5 7 9 11

noise output power while constraining the gain in the direction of the desired signal. This leads to the following minimization with constraints: ^ min wH Rw

s:t: wH s ¼ 1;

w

ð30Þ

where s is the NM  1 space–time steering vector in the target direction. Using the method of Lagrange multipliers, the optimal LCMV STAP filter weight vector is given by ^ ¼ w

−1 R^ s

sH R^

−1

s

:

ð31Þ

13

Thus, the target can be detected by the adaptive matched filter (AMF) detector with the form

15

jsH R^

17

−1

x t j2 H 1 ≷ ξAMF ; sH R^ s H0

ð32Þ

−1

21

where xt denotes the snapshot in the test range bin, H 0 is the null hypothesis (i.e., target absence), H 1 is the alternative hypothesis (i.e., target presence), and ξAMF represents the target detection threshold.

23

3.3. Complexity analysis

25

Here, we detail the computational complexity of the proposed MIAA, MIAA-R, MIAA-TR algorithms, the IAA-RC algorithm in [38] employing the proposed scheme (which is shortened as the MIAA-RC), the conventional IAA algorithm in [30], and the IAA-R algorithm in [37]. The computational cost is measured in terms of the number of complex multiplications and additions. The results are shown in Table 4, where J ¼ NM, J~ ¼ Nd Ns , Q denotes the iteration number of the algorithms (especially, in the IAARC algorithm, Q 1 and Q 2 ¼ Q −Q 1 are the iteration numbers for the initialization step and subsequential step respectively), J RC is the size of reduced region for the MIAA-RC algorithm, and J ΓðqÞ stands for the number of entries in the adaptive subspace indices ΓðqÞ for the MIAATR algorithm. From the table, since the number of snapshots in STD L is very small (as shown in the simulations, it usually equates to 6, and satisfies L⪡Nd N s ), the computational complexity of the conventional IAA, the IAA-R, the MIAA, and the MIAA-R algorithms has approximately the same order. On the other hand, the MIAA-RC and the MIAA-TR algorithms all include approaches to reduce the scanning region resulting in a lower computational complexity than other algorithms. Furthermore, the sizes of J RC and J ΓðqÞ vary with the threshold and impact the computational complexity significantly. Fig. 2 provides a

19

27 29 31 33 35 37 39 41 43 45 47 49 51

more direct way to illustrate the complexity requirements for the algorithms compared. It shows the complexity in terms of complex multiplications and additions against the number of the DOFs J ¼ NM. It is clear that the MIAA-TR algorithm provides an even lower complexity than the MIAA-RC algorithm when employing a proper threshold. It should be noted that the above-described algorithms do not have any special requirement for the STAP problem and thereby can be applied to arbitrary array geometries and random slow-time samples. In particular, for the case of ULA and constant PRF, if the spatial frequencies of the discretized angle bins and the Doppler frequencies are uniformly sampled, the clutter (target absence) or targetclutter (target presence) plus noise covariance matrix R described by (7) is a Toeplitz block Toeplitz matrix. Thus, the efficient approaches based on G–S representations and 2-D FFT [31–33] can be directly used to greatly reduce the complexity for the conventional IAA and the MIAA algorithms, where the complexity is Q ð1:5MJ 2 þ J 2 þ 5MJ log2 ð4JÞ þ 3J~ log2 J~ Þ. As for the IAA-R algorithm, the matrix R is not a Toeplitz block Toeplitz matrix due to the impact of the regularized term which leads to the impossibility to use the G–S representations. Although the 2-D FFT can be still exploited to compute the matrix R given a from the previous iteration leading to the complexity of this operation from J~ J 2 to J~ log2 J~ , one should note that the total complexity of the IAA-R algorithm does not decrease much. In the proposed MIAA-R algorithm, the regularized 2 ^ term ΔðqÞ ¼ s^ n ðqÞI is a diagonal matrix with the same 1012

1011

11

57 59 61

65 67 69 71 73 75 77 79 81 83 85 87 89 91

95

10

10

10

109

108

107

106 200

400

600

97

1010

99

109

101

108

103

107

105

106

J=NM

107 200

400

600

J=NM

Fig. 2. The computational complexity in terms of arithmetic operations versus the number of system DOFs J¼NM.

109 111 113

Table 4 Computational complexity comparison.

115

53 55

63

93 1012

Number of Additions

1

Number of Multiplications

6

Algorithms

Multiplications

Additions

IAA in [30]

Q ð2J~ J 2 þ J 3 þ 2J~ JÞ Q ðð2J~ þ 1ÞJ 2 þ J 3 þ 2J~ JÞ

Q ð2ðJ~ −1ÞJ 2 þ J 3 Þ Q ðð2J~ −1ÞJ 2 þ J 3 −JÞ 2ðJ~ Q 1 þ J RC Q 2 ÞJ 2 þ QJ 3 −Q 1 J 2 þ ðLJ þ J−1ÞðJ~ Q 1 þ J RC Q 2 Þ Q ð2J~ J 2 þ J 3 þ ðL−1ÞJ~ J−J 2 −J~ Þ

IAA-R in [37] MIAA-RC in [38] MIAA

2ðJ~ Q 1 þ J RC Q 2 ÞJ 2 þ QJ 3 þ ðL þ 1ÞðJ~ Q 1 þ J RC Q 2 ÞðJ þ 1Þ Q ð2J~ J 2 þ J 3 þ ðL þ 1ÞJ~ J þ LJ~ Þ

MIAA-R

Q ðð2J~ þ 1ÞJ 2 þ J 3 þ ðL þ 1ÞJ~ J þ LJ~ Þ

Q ð2J~ J 2 þ J 3 −J þ ðL−1ÞJ~ J−J~ Þ

MIAA-TR

∑Qq ¼ 1 ðð2J ΓðqÞ þ 1ÞJ 2 þ J 3 þ ðL þ 1ÞJ ΓðqÞ J þ LJ ΓðqÞ Þ

∑Qq ¼ 1 ð2J ΓðqÞ J 2 þ J 3 −J þ ðL−1ÞJ ΓðqÞ J−J ΓðqÞ Þ

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23

elements on the diagonal which will not destroy the Toeplitz block Toeplitz structure of the matrix R^ c . As for the MIAA-RC and MIAA-TR algorithms, although we only compute the powers in the reduced scanning region, the powers in other scanning region can be seen as zeros, where the Toeplitz block Toeplitz structure of the matrix R is also satisfied. Thus, the G–S representations and 2-D FFT can be both exploited to reduce the complexity for the MIAA-R, MIAA-RC and MIAATR algorithms resulting in close complexity as that of the conventional IAA and the MIAA algorithms. Therefore, for the case of ULA and constant PRF with uniformly sampled spatial frequencies and Doppler frequencies, by exploiting the Toeplitz block Toeplitz structure of the matrix R, the complexity of the IAA, MIAA-RC, MIAA, MIAA-R and MIAA-TR algorithms is greatly reduced and has the same order. But the complexity of the IAA-R algorithm is still high. For more general cases with arbitrary array geometries, random slow-time samples or both, the MIAA-RC and MIAA-TR algorithms provide much lower complexity than other algorithms because of the reduced scanning region. Furthermore, the MIAA-TR algorithm has even much lower complexity than the MIAA-RC algorithm.

25

4. Performance assessment

27

In this section, we assess the signal-to-interferencenoise ratio (SINR) performance and the probability of detection (PD) of the proposed MIAA, MIAA-R, MIAA-RC and MIAA-TR algorithms using simulated radar data. The parameters of the simulated radar platform are shown in Table 5. Besides, we set N d ¼ 6N and Ns ¼ 6M throughout the simulations.

3 5 7 9 11 13 15 17 19 21

29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

4.1. Performance of the proposed IAA-STAP scheme In the first example, we evaluate the convergence (the number of snapshots required to estimate the clutter plus noise covariance) of the proposed MIAA-STAP scheme, namely the MIAA-STAP, MIAA-R-STAP, MIAA-RC-STAP and MIAA-TR-STAP algorithms, and compare them with the diagonal loading sample matrix inverse (LSMI) method and the optimum performance which is obtained using the known clutter plus noise covariance. In the simulation, we

assume that the target Doppler frequency is 600 Hz (corresponding to normalized Doppler frequency of 0.3), the diagonal loading factor for the LSMI is 10 dB to the thermal noise power level, the maximum iteration number for the MIAA-STAP, MIAA-R-STAP, MIAA-RC-STAP and MIAA-TR-STAP algorithms is 15, and the threshold parameter β is set to 10 dB for the MIAA-RC-STAP and MIAATR-STAP algorithms. As shown in Fig. 3 (where the results are averaged over 100 independent Monte Carlo runs), we find that the proposed MIAA-STAP, MIAA-R-STAP, MIAARC-STAP and MIAA-TR-STAP algorithms only require 4–6 snapshots to obtain a steady-state performance and exhibit much faster convergence than the LSMI algorithm. It is also should be noted that the performance of MIAA-TRSTAP algorithm is slightly better than that of the MIAASTAP, MIAA-R-STAP and MIAA-RC-STAP algorithms. That may be because the MIAA-TR-STAP algorithm employs a soft-thresholding scheme adaptively dropping small components in the iterations to eliminate the impacts of these unstable values and then feeds the clutter covariance matrix with the estimated thermal noise covariance matrix to avoid powers loss by the dropped components in the solution. In the next example, we access the performance of the proposed IAA-STAP scheme against the target Doppler frequency at the main beam look angle, as plotted in Fig. 4. The potential normalized Doppler frequency space from −0.5 to 0.5 is examined and 100 independent Monte Carlo runs are averaged to obtain the results. The number of snapshots used to train the STAP filter is 6 for the MIAASTAP, MIAA-R-STAP, MIAA-RC-STAP and MIAA-TR-STAP algorithms and 80 for the LSMI algorithm. The other parameters are assumed to be the same as those in the first example. The curves show that the proposed scheme outperforms the LSMI algorithm in the most of Doppler bins, but performs slightly worse in the normalized Doppler range of −0.1 to 0.1 (except for the MIAA-TRSTAP algorithm). But we should note that the number of snapshots used for training the STAP filter of our proposed scheme (6) is much smaller than that of the LSMI algorithm (80). In addition, we see that the SINR performance of the MIAA-TR-STAP algorithm is slightly better than that

63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105

18

107

16

Table 5 Radar system parameters.

109

14

Parameter

Value

Antenna array Antenna array spacing Carrier frequency Transmit pattern Bandwidth Mainbeam azimuth PRF Platform velocity Platform height Clutter-to-noise ratio (CNR) Target azimuth Antenna elements number Pulse number in one CPI

Sideway-looking array λc =2 1.24 GHz Omnidirectional 10 MHz 01 1984 Hz 100 m/s 3000 m 30 dB 01 8 10

SINR (dB)

1

7

12

111

10

113

8

LSMI, MIAA−STAP MIAA−R−STAP MIAA−RC−STAP MIAA−TR−STAP Optimum

6 4 2

115 117 119

0 10

20

30

40

50

60

70

80

Doppler frequency (Hz) Fig. 3. SINR performance against the number of snapshots that are used for training the STAP filter.

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8

1

63

18

1

16

3

14

7 9

SINR (dB)

5

0.8

12

0.7

10

2 0 −0.5

−0.4

−0.3

15 17

67 69

0.5

LSMI MIAA−STAP MIAA−R−STAP MIAA−RC−STAP MIAA−TR−STAP Optimum

4

13

65

0.6

8 6

11

LSMI MIAA−STAP MIAA−R−STAP MIAA−RC−STAP MIAA−TR−STAP Optimum

0.9

−0.2

−0.1

0

0.1

0.2

0.3

0.4

71

0.4

73

0.3

75

0.2

0.5

Doppler frequency (Hz) Fig. 4. SINR performance against the target Doppler frequency with the potential normalized Doppler frequency from −0.5 to 0.5.

19

0.1 0 −20

77 −15

−10

−5

0

5

79 81

1 0.9

21 18

83

0.8

23

85

0.7 16 0.6

27 29

SINR (dB)

25 14

0.2 0.1

8

33

0 −20 6 −0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Doppler frequency (Hz) Fig. 5. The SINR performance versus different values of β for the MIAARCSTAP and MIAA-TR-STAP algorithms.

39 41 43 45 47 49 51 53 55 57 59 61

LSMI MIAA−STAP MIAA−R−STAP MIAA−RC−STAP MIAA−TR−STAP Optimum

0.3

31

37

89

0.4

12

10

35

87

0.5

of the MIAA-STAP, MIAA-R-STAP and MIAA-RC-STAP algorithms, which is similar as that in the first example. It should be noted that the threshold parameter β in the MIAA-RC-STAP and MIAA-TR-STAP algorithms provides a tradeoff between the performance and computational complexity. To illustrate this, we plot SINR performance versus different values of β for the MIAA-RC-STAP and MIAA-TR-STAP algorithms in Fig. 5. From the figure, we observe that there is a range of values of β to obtain a good performance. Intuitively, the larger the β, the lower the complexity. But when β is larger than 15dB, a performance degradation for both the MIAA-RC-STAP algorithm and the MIAA-TR-STAP algorithm is presented. This is because too large of β will drop significant components in the solution resulting in a great bias. In the following examples, we set β ¼ 10 dB for the MIAA-RC-STAP and MIAA-TR-STAP algorithms. In the third example, in Fig. 6, we depicted the PD versus signal-to-noise (SNR) performance for the proposed IAA scheme. The false alarm rate (PFA) is set to 10−3 and for simulation purposes the threshold and probability of detection estimates are based on 1000 samples.

−15

−10

−5

0

91 93 5

Fig. 6. PD versus SNR with two different cases: (a) the target normalized Doppler frequency of 0.3 and (b) the target normalized Doppler frequency of 0.1.

We suppose the target is injected in the boresight (01) with normalized Doppler frequency: case (a) 0.3 and case (b) 0.1, and other parameters are the same as those in the second example. Fig. 6(a) illustrates that the proposed scheme provides much higher detection performance than the LSMI algorithm at the SNR level from −15 dB to −3 dB for case (a). While for case (b), our proposed scheme obtains a very close performance as the LSMI algorithm, which uses a much larger number of snapshots (80) for training the STAP filter than the proposed scheme (6), as shown in Fig. 5(b). This is due to the fact that when the number of training snapshots is small, the energy of the estimated clutter spectrum by the proposed scheme spreads to the slow Doppler bins resulting in a degradation of the detection performance in the case of slow targets. As one can expect, if we increase the number of training snapshots, the detection performance will increase.

95 97 99 101 103 105 107 109 111 113 115 117

4.2. Comparison with conventional IAA-STAP scheme

119

To provide a further investigation for the proposed scheme, we evaluate the performance in terms of receive operating characteristic (ROC) (i.e., the PD versus PFA)

121

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9

63

1

5

9 11

Pd

7

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6 IAA−MCFAR: 0dB IAA−R−MCFAR: 0dB MIAA−AMF: 0dB MIAA−R−AMF: 0dB MIAA−RC−AMF: 0dB MIAA−TR−AMF: 0dB

0.5 0.4 0.3

13 15 17

Pd

3

1

0.2

67 69 71

0.5

IAA−MCFAR: −4dB IAA−R−MCFAR: −4dB MIAA−AMF: −4dB MIAA−R−AMF: −4dB MIAA−RC−AMF: −4dB MIAA−TR−AMF: −4dB

0.4 0.3 0.2

73 75 77

0.1

0.1 0 10−3

65

10−2

10−1

0 10−3

100

10−2

10−1

Pfa

19

100

Pfa

1

21

79 81 83

0.9

85

23 0.8

25

87

0.7 0.6

Pd

27 29

89

0.5 IAA−MCFAR: −7dB IAA−R−MCFAR: −7dB MIAA−AMF: −7dB MIAA−R−AMF: −7dB MIAA−RC−AMF: −7dB MIAA−TR−AMF: −7dB

0.4

31

0.3 0.2

33

91 93 95

0.1

35 37

0 10−3

97 10−2

Pfa

10−1

100

Fig. 7. ROC curves with the normalized target Doppler frequency of 0.3 for different SNR levels. (a) SNR 0 dB; (b) SNR −4 dB; (c) SNR −7 dB.

101

39 41 43 45 47 49 51 53 55 57 59 61

99

curves of the proposed IAA scheme and the conventional IAA scheme developed in [30]. In our simulations, a median constant false alarm (MCFAR) detector is used for the conventional IAA scheme, namely the IAA-MCFAR and the IAA-R-MCFAR. The MCFAR detector has the form of

2 −1



vH ðf d;t ; f s;t ÞR^ xt

10 log10

−1 ^ H
v ðf d;t ; f s;t ÞR vðf d;t ; f s;t Þ
H1

−10 log10 ςðf d;t ; f s;t Þ ≷ ξMCFAR ;

ð33Þ

H0

where ðf d;t ; f s;t Þ are the target Doppler frequency and spatial frequency, and ξMCFAR denotes the target detection threshold. Just as done in [30], the background clutter plus noise level ςðf d;t ; f s;t Þ for the test range bin, spatial frequency f s;t and Doppler frequency f d;t is estimated as the median value of the set of power levels from 10 adjacent range bins at ðf d;t ; f s;t Þ. For each threshold ξMCFAR , the number of correct target detections as well as the number of false alarms are recorded to yield the ROC curves. Moreover, the target at ðf d;t ; f s;t Þ is considered to be

detected correctly if there are any number of detections in the test range bin falling within the interval ðf d;t −π=20; f d;t þ π=20Þ and ðf s;t −π=16; f s;t þ π=16Þ. With regard to the proposed IAA scheme, the ROC curves are obtained using the AMF detector described in Section 3.2. In [34], it is concluded that the conventional IAA scheme will miss the weak and slow targets because of without clutter suppression. Thus, we access the performance of the proposed IAA scheme for weak and slow targets in Figs. 7 and 8, respectively. Fig. 7(a)–(c) depicts the ROC curves with different SNR levels, i.e., (a) 0 dB, (b) −4 dB, and (c) −7 dB, for a high target Doppler frequency (0.3). It is illustrated that our proposed IAA scheme outperforms the conventional IAA scheme in all the three cases. The detection performance significantly degrades with the reduction of the target SNR level. While the degradation of our proposed IAA scheme is much less than that of the conventional IAA scheme. This is because the IAA or IAA-R algorithms are not good at estimating a target with low SNR level. But for the ground clutter in the airborne radar systems, the power of the ground clutter is much higher than that of the targets. The IAA or IAA-R

Please cite this article as: Z. Yang, et al., Adaptive clutter suppression based on iterative adaptive approach for airborne radar, Signal Processing (2013), http://dx.doi.org/10.1016/j.sigpro.2013.03.033i

103 105 107 109 111 113 115 117 119 121 123

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10

63

1

5

9 11

Pd

7

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5 IAA−MCFAR: 0dB IAA−R−MCFAR: 0dB MIAA−AMF: 0dB MIAA−R−AMF: 0dB MIAA−RC−AMF: 0dB MIAA−TR−AMF: 0dB

0.4 0.3

13 15 17

0.2

67

71

0.5 IAA−MCFAR: −4dB IAA−R−MCFAR: −4dB MIAA−AMF: −4dB MIAA−R−AMF: −4dB MIAA−RC−AMF: −4dB MIAA−TR−AMF: −4dB

0.4 0.3 0.2

0.1 0 10−3

65

69 Pd

3

1

73 75

0.1 10−2

10−1

77

0 10−3

100

10−2

10−1

Pfa

100

79

Pfa

81

19 1

21

83

0.9

23

85

0.8 0.7

87

25 0.6

Pd

27

89

0.5 0.4

29

IAA−MCFAR: −7dB IAA−R−MCFAR: −7dB MIAA−AMF: −7dB MIAA−R−AMF: −7dB MIAA−RC−AMF: −7dB MIAA−TR−AMF: −7dB

0.3

31

0.2 0.1

33

010−3

35

10−2

10−1

100

91 93 95 97

Pfa

37

Fig. 8. ROC curves with the normalized target Doppler frequency of 0.1 for different SNR levels. (a) SNR 0 dB; (b) SNR −4 dB; (c) SNR −7 dB.

101

39

49

algorithms can easily provide a good estimation of the clutter components with high energy levels. Therefore, we can first estimate the ground clutter, design the STAP filter to suppress the clutter and then detect the targets. Fig. 8(a)–(c) plots the results of ROC curves for a low target Doppler frequency. It is observed that our proposed IAA scheme provides a worse performance than that for a high target Doppler frequency, but remarkably, obtains much better performance than the conventional IAA scheme.

51

5. Conclusions

53

In this paper, a novel IAA scheme using the STD has been proposed to overcome the drawbacks of the missing of weak or slow targets detection in conventional IAA scheme. The proposed IAA scheme utilizes multiple snapshots to estimate the clutter plus noise resulting in a stable and good estimation. Furthermore, the estimated clutter plus noise covariance is used to form the STAP filter and suppress the clutter first. This scheme exploits the fact that the clutter power is much higher than the targets power as

41 43 45 47

55 57 59 61

99

well as easier to compute compared with the calculation of the targets power, especially the weak or slow targets. To reduce the computational complexity of the MIAA, MIAA-R and MIAA-RC algorithms, a MIAA-TR algorithm that employs a soft-thresholding to adaptively determine the entries required to update has also been developed. Simulation results have shown that the proposed MIAA, MIAA-R, MIAA-RC and MIAA-TR algorithms outperform the conventional IAA and IAA-R algorithms over weak or slow targets detection and the MIAA-TR algorithm exhibits a comparable or even a slightly better performance than the MIAA, MIAA-R and MIAA-RC algorithms and a lower computational complexity.

103 105 107 109 111 113 115

References 117 [1] J. Ward, Space–Time Adaptive Processing for Airborne Radar, Technical Report 1015, MIT Lincoln laboratory, Lexington, MAvol, December 1994. [2] R. Klemm, Principles of Space–Time Adaptive Processing, Institute of Electrical Engineering, London, UK, 2006. [3] J.R. Guerci, Space–Time Adaptive Processing for Radar, Artech House, 2003.

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Please cite this article as: Z. Yang, et al., Adaptive clutter suppression based on iterative adaptive approach for airborne radar, Signal Processing (2013), http://dx.doi.org/10.1016/j.sigpro.2013.03.033i

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