Adaptive detection of distributed targets in compound-Gaussian clutter with inverse gamma texture

Digital Signal Processing 22 (2012) 1024–1030

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Digital Signal Processing www.elsevier.com/locate/dsp

Adaptive detection of distributed targets in compound-Gaussian clutter with inverse gamma texture ✩ Xiuqin Shang a , Hongjun Song a , Yu Wang a , Chengpeng Hao b,∗ , Chuan Lei c a b c

Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, 100190, Beijing, China School of Electronics and Information Engineering, Beihang University, Beijing 100191, China

a r t i c l e

i n f o

Article history: Available online 23 May 2012 Keywords: Adaptive radar detection Nonhomogeneous environments Rao test Wald test

a b s t r a c t The problem of coherent detection for distributed target in compound-Gaussian clutter with inverse gamma texture is studied and three detectors. One-step generalized likelihood ratio test (GLRT), maximum a-posteriori GLRT and two-step GLRT, are proposed respectively in a Bayesian architecture. Resultantly, these detectors have similar detection structures with their test statistics modulated by the shape and scale parameters of the texture. Alternatively, they can be reformulated into another form with their test statistics associated with the scale parameter and detection thresholds related with the shape parameter. And this detection structure can be seen as a matched filter form with a shape-parameter-dependent threshold like the detectors for point target. Subsequently, the proposed detectors are compared with two-step GLRT based on compound-Gaussian clutter without considering texture model, their detection performances are evaluated, and their robustness are analyzed via Monte Carlo simulations. Results enlighten us that: (1) the three Bayesian detectors bear pretty much the same detection performances; (2) the detection performances fluctuate more intensely when the shape parameter or the scale parameter is smaller; (3) the shape parameter has more influences on the detection performances than the scale parameter, as it is an indication of the clutter impulsiveness. © 2012 Elsevier Inc. All rights reserved.

1. Introduction The coherent detection of point or distributed target against a background of Gaussian or non-Gaussian clutter has gained importance in radar community, and there are two fundamental aspects, the target and the clutter, to be considered. On one hand, whether a target can be seen as a point one depends on the range resolution of radar and the range extent of target. When the range extent of target is much larger than the range resolution of radar, the target energy will scattered and spread more than one range bins. This case happens frequently in high-range resolution radars (HRRs), which gives significant enhancement of detection performances due to two aspects. (1) Increasing the range resolution of radar reduces the energy per cell backscattered by distributed clutter; (2) Resolved scatterers introduce less fluctuation than an unresolved point target [1]. Such target is called range-distributed target or range-spread target therefore, and its detection problem has been involved in homogeneous and partially homogeneous clutter [1] and compound-Gaussian clutter [2–9].

✩ This work was supported by the National Natural Science Foundation of China under Grant Nos. 60802072 and 61172166. Corresponding author. Fax: +86 010 8254 7705. E-mail address: [email protected] (C. Hao).

*

1051-2004/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.dsp.2012.05.002

On the other hand, choosing a suitable clutter model is of fundamental in designing radar detector and different clutter model leads to different performances. A suitable clutter model must be reasonable and mathematically efficient for real environments. Firstly, it should represent the single pulse amplitude statistics accurately and also the correlation between pulses within a multiple pulse burst accurately [10,11]. Secondly, the clutter model should lead to mathematical tractability as much as possible, which also determines the implementation and complexity of the resulting detection structure. In particular, the clutter statistics can be modeled as Gaussian distribution in radar systems with relatively low resolution capabilities. Detection problem under Gaussian assumptions is considered extensively and some classic detectors are given in literature. The non-Bayesian one-step is addressed by Kelly, where the test and training signals are utilized and a generalized likelihood ratio test (GLRT) with constant false alarm rate (CFAR) is derived by maximizing the two likelihood functions over a set of unknown parameters, target amplitude associated with covariance matrix [12]. And the non-Bayesian two-step one, termed AMF, is derived in [13,14]. In recently years, detection problem is reconsidered under Gaussian assumptions with random covariance matrix [15], and corresponding detectors are derived by integrating conditional probability density functions (PDFs) under both H 0 and H 1 hypotheses over random covariance matrix.

X. Shang et al. / Digital Signal Processing 22 (2012) 1024–1030

The compound-Gaussian model is suitable one for representing clutter echoes from those areas covered by forests, vegetations and oceans and is widely used in high resolution radar with low grazing angle [16–18], the main problem of which involves texture model mainly involving gamma or inverse gamma distribution, parameters estimation and evaluation and target detection [6,11,19]. The compound-Gaussian clutter with gamma and inverse gamma texture are consistent with k-distributed and t-distributed clutter, respectively. The compound-Gaussian model with inverse gamma texture is fit for see-clutter data [17], where the maximum likelihood (ML) and method of fractional moments (MoFM) are proposed for the texture parameters estimation. In [18], the joint estimation for target and clutter parameters is given, including complex target amplitude and spatial covariance matrix of speckle. The PX-EM algorithm is proposed for the computation of the ML estimator and the corresponding Cramer–Rao bounds (CRB) are analyzed. The related detection problem based on compoundGaussian model has been studied with little considering the texture model. The compound-Gaussian model is used in STAP, where the robustness of probability of detection ( P D ) and probability of false alarm ( P FA ) are discussed for different texture parameters [6,7]. In [9,19], the compound-Gaussian model with gamma and inverse gamma texture is applied to multiple input multiple output (MIMO) radar respectively, by which the GLRT-based detectors are derived. It is noted that the likelihood ratio can be recast into another two forms, estimator–correlator form and MF form with data-dependent threshold [11]. And also the suboptimum detectors are achieved, turning out to be CFAR, by either approximating the likelihood ratio directly or utilizing a suboptimal function to model the data-dependent threshold in MF form. The detection for point target has been dealt with based on compound-Gaussian model with inverse gamma texture and three detectors are proposed based on Bayesian one-step GLRT (B1S-GLRT), maximum a posteriori GLRT (MAP-GLRT) and Bayesian two-step GLRT (B2SGLRT) decision rule, referred to [12] and references therein. Additionally, performance evaluations are given by P D and P FA , their CFAR properties are analyzed consequently and their robustness are simulated. Here the objective of this paper is to extend the work of [20] and to analyze detection structures for distributed target in compound-Gaussian model with inverse gamma texture and their performance evaluations. Similarly, three detectors, Bayesian one-step generalized likelihood ratio (B1S-GLRT), maximum aposteriori GLRT (MAP-GLRT) and Bayesian two-step GLRT (B2SGLRT) are proposed, turning out to be closely related to the shape and scale parameters of the texture. As it is difficult for the closed form for the P D and the P FA for distributed target in compoundGaussian model, we resort to Monte Carlo simulations to analyze the detection performances and their robustness. The rest of this paper is organized as follows: the compoundGaussian clutter with inverse gamma texture is briefly introduced in Section 2, followed by three detectors based on this model in Section 3; subsequently, the performance evaluations and simulations are given in Section 4; and discussions and conclusions are included in Section 5. 2. Clutter model It is stated by Sangston in [11], the appropriate multidimensional non-Gaussian model used for radar detection must incorporate the following features: (1) it must account for first-order statistics of the clutter echoes; (2) it must incorporate pulseto-pulse correlation between data samples. Compound-Gaussian model is an attacking one not only for its suitable for modeling sea-clutter in high resolution and/or lower grazing angle application data but also for its consistency with k-distributed clutter

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model, which gives a deep insight into the scattering mechanism [21,22]. It can be represented with a product model [17,18]

c=

√

τ g,

(1)

where g is a multivariate complex Gaussian vector called speckle with covariance Σ , and τ is a nonnegative real random process called texture representing local clutter power. One essential problem of compound-Gaussian model is the choice of the texture model, with gamma or inverse gamma distribution mainly used. The compound-Gaussian model with inverse gamma texture is suitable for sea-clutter validated by real data [17]. Namely, the texture follows inverse gamma distribution with its probability density function (PDF) parameterized by shape parameter α and scale parameter β

f τ (τ ; α , β) =

1

β α (α )

τ −(α +1) exp(−1/β τ ), τ  0,

(2)

where (·) denotes the gamma function. In this case, the model in (1) is a multidimensional t-distributed clutter model [17]. In this case, the statistical expectation and the variance of texture can be easily derived.

E (τ ) =

1

β(α − 1)

Var(τ ) =

α > 1,

, 1

β 2 (α − 1)2 (α − 2)

(3)

,

α > 2,

(4)

and so can the amplitude distribution of the clutter

f r (r ; α , β) =

√

2r β(α + 1) (β r 2 + 1)α +1 (α )

(5)

with r = τ | g | representing the clutter amplitude and its PDF and that of the clutter texture are illustrated in Fig. 1, for different texture parameters. In the following, the detection problem for distributed target is studied based on this model. 3. Detector designs As it has been stated that the one-step, maximum a-posterior and two-step detectors have been derived under Bayesian and non-Bayesian framework and they are related. However, there few studies about detectors against compound-Gaussian clutter. In [2], the two-step GLRT has been derived for point target based on compound-Gaussian clutter with considering texture model and there none ones for range-spread target against compoundGaussian clutter with or without considering texture model. The goal of this chapter is to manifest detectors for range-spread targets under compound-Gaussian clutter with inverse gamma texture and the relations with those for point target given in [25] has been investigated. Assume that the radar transmits a coherent train of N pulses and the received signals can be achieved after demodulating, matched filtering and A/D sampling. Let N-dimensional x = [x(0) · · · x( N − 1)] T vector represents the received N complex samples, where superscript T denotes the transpose operator. And the detection problem for distributed target can be cast into the following binary hypothesis

 ⎧ xl = c l , ⎪ ⎪ ⎪H 0: ⎪ ⎨ zk = c k ,  ⎪ ⎪ ⎪ H : xl = al v + c l , ⎪ ⎩ 1 zk = c k ,

l = 1, . . . , L , k = 1, . . . , K , l = 1, . . . , L ,

(6)

k = 1, . . . , K ,

where v denotes the target steering vector, and al s are unknown deterministic parameters accounting for the target reflectivity and

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where  ·  denotes determinant of a square matrix. The texture is modeled as inverse-gamma-distributed random variable, and the PDF of which is characterizing by the shape parameter α and the scale parameter β . Additionally, we assume that the texture parameters are known or have been estimated from real clutter data by ML method or MoFM [17]. Firstly, we focus on the detection architectures of this problem with known speckle covariance matrix and then the adaptive ones can be easily obtained by utilizing its estimators instead. 3.1. B1S-GLRT According to Neyman–Pearson (NP) theorem, the optimum NP detector is the likelihood ratio test (LRT) under the assumption of independent data [21,27]. Then the detection problem considered here can be formulated as [28]

∞

L max a1: L

H l=1 0 f (xl |al , l ; H 1 ) f τl ( l ) d l 1 ≷ λ.

L  ∞ l=1 0 f (xl | l ; H 0 ) f τl ( l ) d l H 0

τ

τ

τ

τ

τ

(9)

τ

The unconditional PDF of xl under both hypotheses can be achieved by integrating the conditional one with respect to τl . After manipulation, they can be written as

∞ f (xl | H 0 ) =

f (xl |τl ; H 0 ) f τl (τl ) dτl 0

=

1

π

N Σ

(α + N ) (α + N ) , β β α (α ) l0

(10)

∞ f (xl | H 1 ) =

f (xl |al , τl ; H 1 ) f τl (τl ) dτl 0

=

(α + N ) (α + N ) 1 , β π N Σ β α (α ) l1

(11)

where 1/βl0 = xlH Σ −1 xl + 1/β , and 1/βl1 = (xl − al v ) H Σ −1 (xl − al v ) + 1/β . The NP detector given in (9) takes on simpler form when taking (10)–(11) into it. Fig. 1. The inverse gamma texture and the amplitude of the compound-Gaussian clutter with inverse gamma texture for different shape parameters and scale parameters.

the channel propagation effects. x1 , . . . , x L are called as primary data, collected from the cells under test (CUT), and z 1 , . . . , z K are termed as secondary data, standing for the snapshots without any useful target and share the same structure of the covariance matrix as the primary data. The clutters c l s and c k s are modeled by compound-Gaussian model with inverse gamma texture, given by a √ product model c l = τl g with Σ = E ( g g H ), where E (·) and superH denotes the statistical expectation operator and conjugate script transpose operator, respectively. For given τl , the clutter has conditional expectation E (c l c lH |τl ) = τl Σ , and then the conditional PDF of xl under H 0 hypothesis can be written as [26]

f (xl |τl ; H 0 ) =



1

π N ΣτlN

exp −

1

τl

xlH Σ −1 xl





=

1

π N ΣτlN

 exp −

1

τl

a1: L

.

(7)

(xl − al v ) H Σ −1 (xl − al v ) ,

L  H1 (βl1 /βl0 )(α + N ) ≷ λ.

a M L ,l =

v H Σ −1 xl v H Σ −1 x v

l = 1, . . . , L ,

,

(13)

and the corresponding detector L 

1/β + xlH Σ −1 xl

1/β

+ xlH Σ −1 xl

H1

≷ λ1/( N +α ) .

− | v H Σ −1 xl |2 / v H Σ −1 v H 0

(14)

The NP detector termed as B1S-GLRT as well, to be differentiated from the following B2S-GLRT and it can be formulated compactly as

−( N + α ) (8)

(12)

H0

l =1

Differentiating (12) with respect to target amplitudes al s, l = 1, . . . , L, we can obtain the maximum likelihood estimate (MLE) of the target amplitudes

l =1

Similarly, the conditional PDF of xl under the alternative hypothesis can be given as [26]

f (xl |al , τl ; H 1 )

max

 L 

ln 1 −

l =1

| v H Σ −1 xl |2 ( v H Σ −1 v )(1/β + xlH Σ −1 xl )

H 1 ≷ ln λ. H0

(15)

X. Shang et al. / Digital Signal Processing 22 (2012) 1024–1030

3.2. MAP-GLRT The MAP-GLRT obeys the following decision rule [28]

max a1: L

maxτ1:L

L

l =1

maxτ1:L

L

f (xl |al , τl ; H 1 ) f τl (τl )

H1

f (xl |τl ; H 0 ) f τl (τl )

H0

l =1

≷ λ.

(16)

It requires MAP estimates of τl under both H 0 hypothesis and H 1 hypothesis (denoted by τˆ1: L | H 0 and τˆ1: L | H 1 respectively), satisfying

τˆ1:L | H 0 = max τ1:L

L 

f (xl |τl ; H 0 ) f τl (τl )

(17)

f (xl |al , τl ; H 1 ) f τl (τl ).

(18)

l =1

and

τˆ1:L | H 1 = max τ1:L

L  l =1

Making use of the PDF of τl and conditional PDF of the primary data under both hypothesis, the two MAP estimates are finally obtained, i.e.

τˆ1:L | H 1

 

1

τˆ1:L | H 0 =





tr Σ −1 T l0 + 1/β ,

(19)

N +α+1   −1 1 = tr Σ T l1 + 1/β , N +α+1

(20)

where T l0 = xl xlH , and T l1 = (xl − al v )(xl − al v ) H . Inserting them into the MAP-GLRT, we can get

max a1: L

 L   tr(Σ −1 T l0 ) + 1/β H 1 1/( N +α +1) ≷λ . tr(Σ −1 T l1 ) + 1/β H 0 l =1

(21)

The maximum can be achieved at these target amplitudes given in (13), and the detector finally takes on the following form

−( N + α + 1)

L 

 ln 1 −

l =1

| v H Σ −1 xl |2 ( v H Σ −1 v )(1/β

+ xlH Σ −1 xl )

H 1 ≷ ln λ.

B2S-GLRT needs two steps: firstly it assumes the local power τl is known and derives the GLRT for the primary data by maximizing LRT over al s, l = 1, . . . , L; and then it inserts the MAP estimator of τl , into the resulting GLRT. The GLRT for the primary data is given as

a1: L

f (xl |al , τl ; H 1 )

H1

f (xl |τl ; H 0 )

H0

l =1

≷ λ.

(23)

Taking the unconditional PDF of the primary data into the GLRT, maximizing it over the target amplitudes, we can obtain the GLRT after manipulation. L  l =1

 exp

( N + α + 1)| v H Σ −1 xl |2 ( v H Σ −1 v )(1/β + xlH Σ −1 xl )

H 1 ≷ λ.

−N

 L 

ln 1 −

l =1

| v H Σ −1 xl |2 ( v H Σ −1 v )(xlH Σ −1 xl )

H 1 ≷ ln λ.

(26)

H0

Actually, the covariance matrix of the speckle component is unknown, which needs to be estimated from the secondary data. There are several alternative estimators, such as sample covariance matrix (SCM), normalized sample covariance matrix (NSCM) and recursive maximum likelihood estimator. However, SCM is often used in Gaussian clutter while NSCM is more suitable for nonGaussian environment and ML-based estimator is much more operations involved. After trading off the performance and complexity, we consider the NSCM, given by K  zk zkH ˆ =N Σ . H

K

k =1

(27)

zk zk

When it is inserted into the proposed detectors, the adaptive ones can be achieved. When L = 1, i.e., the target only lies in one range bin, the proposed detectors are reduced to these detection structures for point target. If β → ∞ is satisfied at the same time, they eventually reduce to the adaptive coherence estimator (ACE)1 with data-dependent threshold. And if β = 1 meanwhile, the detectors are boiled down into the Kelly GLRT, with data-dependent threshold as well. In addition, the proposed B1S-GLRT can be written in another form

−

 L 

ln 1 −

l =1

| v H Σ −1 xl |2 ( v H Σ −1 v )(1/β + xlH Σ −1 xl )

H 1 ≷ ln λ/( N + α ). H0

(28)

3.3. B2S-GLRT

L

with the shape and the scale parameters of the texture. For comparison, the two-step GLRT (2S-GLRT) is given in (26), which are proposed based on compound-Gaussian clutter without considering texture model. It is enlightened that the texture parameters modulate the test statistics when the text has a prior inverse gamma distribution, which has been proved reasonable and suitable for see clutter data and used in many radar applications.

H0

(22)

l =1 max L

1027

(24)

H0

Then the test statistics of the proposed detectors is related to the scale parameter, while the data-dependent threshold is a function of the shape parameter, which is consistent with the detectors for point target [20]. 4. Performance assessments This section is devoted to a performance assessment of the presented detectors, also in comparison to previously proposed solution. More precisely, we compare our detectors to the 2S-GLRT. We conduct the analysis in two phases: first, we compare the performances of the different detectors; second, we study the robustness of the proposed detectors. We assume that the coherent radar transmits N = 8 pulses in one coherent pulse interval (CPI), and the target under test spreads L = 3 range bins. There are K = 20 training data available for target detection. When the P FA is set, the performance evaluation is carried out via 100/ P FA independent Monte Carlo trials under the following assumptions:

• The clutter follows multivariate compound-Gaussian model

Taking the logarithm of the derived GLRT, yielding

(25)

with its texture component inverse gamma distributed. The PDFs of the texture and its amplitude are given in (2) and (8), respectively;

Comparing the B1S-GLRT, MAP-GLRT and B2S-GLRT, given in (15), (22) and (25), respectively, we can find that the test statistics of the B1S-GLRT, MAP-GLRT and B2S-GLRT have closed relation

1 The ACE is not only the adaptive matched filter in spherically invariant noise and asymptotically optimum detector in compound-Gaussian clutter [23,14], but also the matched subspace detectors (MSDs) with unknown noise covariance [24].

( N + α + 1)

L 

| v H Σ −1 xl |2

H1

l =1

( v H Σ −1 v )(1/β + xlH Σ −1 xl )

H0

≷ ln λ.

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X. Shang et al. / Digital Signal Processing 22 (2012) 1024–1030

Fig. 2. Performances of the B1S-GLRT, MAP-GLRT, B2S-GLRT for different

• The speckle component of the clutter has an exponential correlation structure covariance matrix. So Σ can be given mathematically as

Σ(i , j ) = ρ |i − j | ,

1  i, j  N

(29)

where ρ is the one-lag correlation coefficient and its typical values for sea clutter data are in the range [0.9, 0.99] [19]. • The signal-clutter-ratio (SCR) is defined as

SCR =

L 

|al |2 v H Σ −1 v .

l =1

• The 2S-GLRT, based on compound-Gaussian model without considering texture model, are considered and compared with the proposed Bayesian detectors. And the texture is generated randomly, with its mean and variance the same as those of inverse gamma texture; • The detection performances of the proposed detectors are analyzed, so dose the robustness of the detectors.

α , compared with those of the 2S-GLRT.

4.1. Detection performance In Fig. 2, the detection performances of the B1S-GLRT, MAPGLRT, B2S-GLRT, based on compound-Gaussian model with inverse gamma texture, and those of the 2S-GLRT, based on compoundGaussian model without considering the texture model, are illustrated and compared. The texture of the clutter is generated by inverse gamma distribution with the scale parameter β = 1 and the shape parameter α = 2, 1.6, 1.2. The speckle covariance matrix of the clutter is given by (29) with one-lag correlation coefficient ρ = 0.95. Comparing Fig. 2(a)–(c), we can find that: (1) the 2S-GLRT has slightly and tiny better detection performances than the Bayesian detectors at low SCR. However, when SCR is not very low, its performances are much worse than those of the Bayesian ones; (2) when the shape parameter decreases, the performances of the Bayesian detectors and those of the 2S-GLRT all decreases, however, the former decreases not that fast than the latter; (3) the B1S-GLRT, MAP-GLRT and B2S-GLRT bear pretty much the same performances, however, the former two detectors need the logarithm operations. As the shape and the scale parameters are estimated from the real radar data, it’s essential to analyze

X. Shang et al. / Digital Signal Processing 22 (2012) 1024–1030

Fig. 3. Robustness of the B1S-GLRT with the shape parameter for β = 1 and P FA = 10−3 .

the detection performances vary with the estimation errors of the shape and the scale parameters. 4.2. Robustness performance The robustness of the proposed detectors is illustrated and analyzed in the following. Here, only the robustness of the B1S-GLRT is focused, and those of the MAP-GLRT and the B2S-GLRT can be achieved similarly. Figs. 3 and 4, respectively, analyze the robustness of the B1S-GLRT due to the estimation errors of the shape parameter α and the scale parameter β . Fig. 3 depicts the detection performances of the B1S-GLRT for β = 1, P FA = 10−3 , α = 1.8, 2 and those α s with their errors limited within ±0.1. Fig. 4 plots the detection performances of the B1S-GLRT for α = 2, P FA = 10−3 , β = 2, 10 and those β s with their errors limited within ±0, 1 and ±1, respectively. From Fig. 3, we can find that: (1) the B1S-GLRT has better detection performances for larger α and given β ; (2) the detection performances are much fluctuating for small value of α . This is because the small the shape parameter is, the impulsive the clutter is; (3) the performances are better when α is overestimated and worse when α is underestimated. From Fig. 4, it can be seen that: (1) the B1S-GLRT has better detection performances

Fig. 4. Robustness of the B1S-GLRT with the shape parameter for 10−3 .

1029

α = 2 and P FA =

for larger β and given α ; (2) the detection performances of the B1S-GLRT is inclined to stabilization as β goes to infinite; (3) the detection performances are less fluctuating for large value of β ; (4) the detection performances are better when β is overestimated and worse when β is underestimated. Additionally, when comparing Fig. 3 with Fig. 4, we can penetrate that the shape parameter has much more influences on the detection performances than the scale parameter does, which is consistent with that alpha is the indication of clutter impulsiveness. The larger the parameter is, the more stable the clutter is. It is also needed to point that we have similar conclusions about the robustness of the B1S-GLRT for distributed target as those detectors for point target. 5. Conclusions In this paper, the adaptive detection for distributed target is dealt with, based on compound-Gaussian model with inverse gamma texture, and correspondingly three detectors are proposed according to generalized likelihood ratio test in a Bayesian framework. The detection structure of the proposed detectors can be written as two forms: one has their test statistics related to the shape parameter and the scale parameter; the other has their test statistics related to the scale parameter, while with their detection

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thresholds being a function of the shape parameter. Additionally, the latter detection structure can be reformulated as the matched filter form like the detectors for point target. Subsequently, their detection performances are analyzed via Monte Carlo simulations and compared with those of 2S-GLRT, the detector for distributed target based on compound-Gaussian model without taking texture model into account. Finally, the robustness of those Bayesian detectors is analyzed, showing that the detectors behave well when the shape parameter and the scale parameter suffer from errors. Especially, the scale parameter has less influence on these Bayesian detectors than the shape parameter. This conclusion is also consistent with that α represents impulsiveness and fluctuations of clutter. When α is overestimated, the clutter variance is larger and the performance degrades. References [1] E. Conte, A. De Maio, G. Ricci, GLRT-based adaptive detection algorithms for range-spread targets, IEEE Trans. Signal Process. 49 (2001) 1336–1348. [2] X. Shuai, L. Kong, J. Yang, Performance analysis of GLRT-based adaptive detector for distributed targets in compound-Gaussian clutter, Signal Process. 90 (2010) 16–23. [3] J. Guan, Y. Zhang, Y. Huang, Adaptive subspace detection of range-distributed target in compound-Gaussian clutter, Digital Signal Process. 19 (2009) 66–78. [4] B. Liu, B. Chen, J.H. Michels, A GLRT for multichannel radar detection in the presence of both compound Gaussian clutter and additive white Gaussian noise, Digital Signal Process. 15 (2005) 437–454. [5] E. Conte, A. De Maio, Distributed target detection in compound-Gaussian noise with Rao and Wald tests, IEEE Trans. Aerospace Electron. Syst. 39 (2003) 568– 582. [6] J.H. Michels, B. Himed, M. Rangaswamy, Performance of STAP tests in Gaussian and compound-Gaussian clutter, Digital Signal Process. 10 (2000) 309–324. [7] J.H. Michels, M. Rangaswamy, B. Himed, Performance of parametric and covariance based STAP tests in compound-Gaussian clutter, Digital Signal Process. 12 (2002) 307–328. [8] Y. Zheng, T. Shao, E. Blasch, A fast-converging space–time adaptive processing algorithm for non-Gaussian clutter suppression, Digital Signal Process. 22 (2012) 74–86. [9] G. Cui, L. Kong, X. Yang, J. Yang, Distributed target detection with polarimetric MIMO radar in compound-Gaussian clutter, Digital Signal Process. 22 (2012) 430–438. [10] A. Farina, A. Russo, F. Scannapieco, Theory of radar detection in coherent Weibull clutter, IEE Proc. F: Commun. Radar Signal Process. 134 (1987) 174– 190. [11] K.J. Sangston, F. Gini, M. Greco, Structures for radar detection in compound Gaussian clutter, IEEE Trans. Aerospace Electron. Syst. 35 (1999) 445–458. [12] E.J. Kelly, An adaptive detection algorithm, IEEE Trans. Aerospace Electron. Syst. 22 (1986) 115–127. [13] W. Chen, I.S. Reed, A new CFAR detection test for radar, Digital Signal Process. 1 (1991) 198–214. [14] E. Conte, M. Lops, G. Ricci, Adaptive matched filter detection in spherically invariant noise, IEEE Signal Process. Lett. 3 (1996) 248–250. [15] S. Bidon, O. Besson, J.Y. Tourneret, A Bayesian approach to adaptive detection in nonhomogeneous environments, IEEE Trans. Signal Process. 1 (2008) 205–217. [16] A. Farina, F. Gini, M. Greco, K.J. Sangston, Optimum and sub-optimum coherent radar detection in compound-Gaussian clutter: a data-dependent threshold interpretation, in: Proceedings of the 1996 IEEE National Radar Conference, Ann Arbor, MI, May 1996, pp. 160–165. [17] A. Balleri, A. Nehorai, J. Wang, Maximum likelihood estimation for compoundGaussian clutter with inverse gamma texture, IEEE Trans. Aerospace Electron. Syst. 43 (2007) 775–779.

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Chengpeng Hao received the B.S. and the M.S. degrees in electronic engineering from Beijing Broadcasting Institute, Beijing, China, in 1998 and 2001 respectively. He received the Ph.D. degree in signal and information processing from Institute of Acoustics, Chinese Academy of Sciences, Beijing, China, in 2004. Currently Dr. Hao is an Associate Professor in the State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences. He is an IEEE Member since 2008, and an IEICE Member since 2010. His main research interests are in the field of statistical signal processing with more emphasis on adaptive sonar and radar signal processing.

Xiuqin Shang received the B.S. degree in Electronic Information Science and Technology from Shandong University, Shandong, Chain, in 2002. And she received the Ph.D. degree in Communication and Information System from the Graduate University of Chinese Academy of Sciences, Beijing, China, in 2011. Since 2011, Dr. Shang has joined Huawei Company. Her research interests include space–time adaptive processing, adaptive detection and MIMO detection in wireless communications, especially in Long Term Evolution (LTE).

Hongjun Song received the B.S. degree in Radio Technology from the University of Science and Technology of China, Hefei, China, in 1991. He received the M.S. and the Ph.D. degrees in Communication and Electronics System from Institute of Electronics, Chinese Academy of Sciences (IECAS), Beijing, China, respectively in 1994 and 1998. Dr. Song has been a Research Fellow with the Spaceborne Microwave Remote Sensing System Department, Institute of Electronics, Chinese Academy of Sciences (IECAS) and have been in charge of system simulation, control software design for Chinese Spaceborne SAR system.