Performance analysis of GLRT-based adaptive detector for distributed targets in compound-Gaussian clutter

ARTICLE IN PRESS Signal Processing 90 (2010) 16–23

Contents lists available at ScienceDirect

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

Performance analysis of GLRT-based adaptive detector for distributed targets in compound-Gaussian clutter Xiaofei Shuai , Lingjiang Kong, Jianyu Yang School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China

a r t i c l e i n f o

abstract

Article history: Received 25 November 2008 Received in revised form 9 May 2009 Accepted 13 May 2009 Available online 21 May 2009

The problem of adaptive detection for spatially distributed targets in compoundGaussian clutter is studied. We first derive the optimum NP detector and suboptimum two-step GLRT detector. For the two-step detection strategy, we also introduce three covariance matrix estimation strategies and evaluate their CFAR properties and complexity issues. Next, the numerical results are presented by means of Monte Carlo simulation strategy. In particular, the simulation results highlight that the performance loss due to adaptively estimating the texture is negligible, and that the loss due to adaptively estimating covariance matrix largely depends on the estimation algorithm, the number of the secondary data vectors and the number of the scatterers. & 2009 Elsevier B.V. All rights reserved.

Keywords: Generalized likelihood ratio test (GLRT) Distributed targets Maximum likelihood (ML) estimation Secondary data

1. Introduction It is well-known that high-range resolution (HRR) radar can spatially resolve a target into a number of scatterers depending on the range extent of the target and on the range resolution capability of the radar. This technique brings performance improvement in two facts: (1) increasing the range resolution of the radar reduces the clutter energy in each range cell and (2) resolved scatterers show less fluctuation than an unresolved point target which is contained in only one range cell [1]. However, the clutter becomes very complex and cannot be modeled as the Gaussian one in HRR radar. For these reasons and with the experimental data, the spikiness clutter is usually modeled as a compoundGaussian vector [5,6,8–10,13]. The compound-Gaussian model can be interpreted as a process which is locally Gaussian with local power which is random variable. A successful modern radar detection scheme should be adaptive to the probability density function (PDF) of the interference. Various adaptive detection algorithms

 Corresponding author.

E-mail address: [email protected] (X. Shuai). 0165-1684/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2009.05.008

in non-Gaussian background have been investigated [5,6,8–10,13]. To avoid complicated integral computation of the optimum NP detector, these papers resort to the so called two-step GLRT detection scheme: (1) assuming that the covariance matrix of the speckle component is known and deriving the canonical GLRT detector and (2) estimating the covariance matrix of the speckle component by the secondary data from the range cells adjacent to the cells under test (CUT). In this way, the performance loss due to adaptive detection should be quantified. Recently, in the case of point target, Gini et al. [15,16] discussed the performance differences between the optimum detector, which is a classical whiteningmatched filter (WMF) compared with a data-dependent threshold, and its polynomial approximations. Conte et al. [18–20] also investigated the NP detector versus the GLRT detector. To our knowledge, however, it seems that there is no work discussed the losses due to adaptivity for the case of spatially distributed targets. Motivated by the above cited works, in this paper, we focus on the problem of adaptive detection for spatially distributed targets in compound-Gaussian background. Precisely, we first derive the optimum NP detector and suboptimum two-step GLRT detector. For the two-step detection strategy, we also introduce three estimation

ARTICLE IN PRESS X. Shuai et al. / Signal Processing 90 (2010) 16–23

strategies, i.e. sample covariance matrix (SCM), normalized sample covariance matrix (NSCM) and the recursive ML-estimator, and then evaluate their CFAR properties and complexity issues. Next, we give the numerical results which represent the novel contribution of this paper. In particular, the simulation results of the performances of the NP detector and the two-step GLRT detector with known covariance matrix suggest that the performance loss due to adaptively estimating the texture component should be negligible. Also, we present the detection curves of the twostep GLRT detector with estimated covariance matrix, which show that (1) the recursive ML-estimator outperforms the other two for the realistic scenario and (2) the performance loss is largely influenced by the number of the secondary data vectors and the number of scatterers. The rest of this paper is organized as follows. The statement of the problem and the description of the signal and clutter models are given in Section 2. The NP detector, GLRT detector and GLRT-based two-step detector are derived, respectively, in Section 3. The performance analysis is displayed in Section 4. Lastly, conclusions are given in Section 5. 2. Problem statement Assume that the radar transmits a coherent train of N pulses and that the receiver properly demodulates, filters and samples the incoming waveform. We denote by an N-dimensional vector z ¼ ½ zð0Þ . . . zðN  1Þ T the N complex samples, where T denotes the transpose operator. The binary hypothesis can be written as 8 H : zr ¼ cr ; r ¼ 1; 2; . . . ; L; L þ 1; . . . ; L þ K > > ( < (0 zr ¼ ar p þ cr ; r ¼ 1; 2; . . . ; L > > H1 : zr ¼ cr ; t ¼ L þ 1; . . . ; L þ K : (1) where p denotes the steering vector, and the ars are unknown deterministic parameters which account for the target reflectivity and the channel propagation effects. z1,y,zL are referred as primary data that are collected from cells under test, and zL+1,y,zL+K are secondary data that each of such snapshots does not contain any useful target echo and exhibits the same structure of the covariance matrix as the primary data. Here the received data vectors are assumed to be independent between each range cell. We remark that most of the previous works dealing with adaptive detection in non-Gaussian interference are based on the assumption of independent and identically distributed (IID) samples [1,5,10,13,21]. In fact, such an assumption is proved to be reasonable for real clutter data in [22]. As to the clutter c1 ; . . . ; cLþK in (1), they are modeled as compound-Gaussian vectors, i.e., it is the product of two pffiffiffiffiffi independent random quantities, cr ¼ tr x. Here the speckle x is modeled as a zero mean complex Gaussian vector with covariance matrix EfxxH g ¼ M where Efg is statistical expectation operator. The texture

tr is a positive random variable with PDF ptr ðtr Þ. For a

17

given tr, we obtain the conditional covariance matrix of c as EðccH jtr Þ ¼ tr M. Therefore, the conditional PDF of zr under H0 can be expressed as ! 1 zH zr N r M (2) pzr jtr ;H0 ðzr jtr ; M; H0 Þ ¼ 1ðptr Þ jMj exp 

tr

Then the PDF of zr under H0 is Z 1 pzr jtr ;H0 ðzr jtr ; H0 Þptr ðtr Þ dtr pzr jH0 ðzr jH0 Þ ¼ 0

Similarly, the PDF of zr under H1 can be expressed as Z 1 pzr ar pjar ;tr ;M;H0 ðzr  ar pjar ; tr ; M; H0 Þ pzr jH1 ðzr jH1 Þ ¼ 0

 ptr ðtr Þ dtr 3. GLRT derivation 3.1. Optimal NP detector and canonical GLRT According to the Neyman–Pearson (NP) criterion [2] and in the hypothesis of independent data vectors, the optimum NP detector is the LRT R1 r¼1 0 pzr jar ;tr ;M;H1 ðzr j r ; r ; M; H1 Þptr ð r Þ d r QL R 1 r¼1 0 pzr jtr ;M;H0 ðzr j r ; M; H 0 Þptr ð r Þ d r

QL

LNP ðz1:L Þ ¼

a t t

t

t

t

t

(3) In many practical cases, the parameters M, ar and the priori PDF ptr ðtr Þ are unknown. In addition, the evaluation of the NP detector needs a heavy computational integration along the underlying mixing distribution. For these reasons, we resort to a suboptimal GLRT algorithm where the trs are modeled as unknown deterministic parameters and ML-estimated. The canonical GLRT detection strategy is given by

LGLRT ðz1:L Þ ¼

maxa1:L ;t1:L ;M pz1:L jt1:L ;a1:L ;M;H1 ðz1:L jt1:L ; a1:L ; M; H1 Þ maxt1:L ;M pz1:L jt1:L ;M;H0 ðz1:L jt1:L ; M; H0 Þ (4)

where the conditional joint PDF of the primary data vectors under H1 is pz1:L jt1:L ;a1:L ;M;H ðz1:L jt1:L ; a1:L ; M; H1 Þ ¼

YL r¼1

1 ðptr ÞN jMj

exp 

! ðzr  ar pÞH M1 ðzr  ar pÞ

tr

(5)

and under H0 pz1:L jt1:L ;M;H0 ðz1:L jt1:L ; M; H0 Þ ¼

YL r¼1

1 ðptr ÞN jMj

exp 

1 zH zr r M

tr

! (6)

Maximization in (4) is obtained by replacing the unknown parameters t1:L, a1:L, M with their ML estimators. Unfortunately, joint maximization under the hypothesis H1 is rather a difficult task and a closed-form solution does not exist [5,6,21]. Thus, we resort to two-step GLRT detection scheme.

ARTICLE IN PRESS 18

X. Shuai et al. / Signal Processing 90 (2010) 16–23

t^ k ¼ zHk zk which accounts for the random local power

3.2. The two-step GLRT scheme The two-step GLRT scheme is proposed by Robey [4], and is adopted in many HRR GLRT-based detection algorithms [1,5,8,9,13]. Since the secondary data vectors are target-free, this approach allows to decouple estimation of M and t1:K from estimation of a1:L and t1:L. The adaptive two-step GLRT detection scheme can be described as a block diagram in Fig. 1. Step one: Covariance matrix M is assumed to be known. A GLRT detector using only primary data is derived as ! L X jpH M1 zr j2 (7) LGLRT -1 ðz1:L Þ ¼ N ln 1  1 ðpH M1 pÞðzH zr Þ r¼1 r M If we assume that the radar is not able to resolve individual parts of a target, i.e., L ¼ 1, the detector (7) is identical to that given in [8,9] in the case of a point target in compound-Gaussian clutter. Step two: Covariance matrix M is estimated by using the K secondary data vectors. We will introduce and ^ in the next subsection. analyze the different estimators M ^ and then the two-step GLRT is Replace M in (7) by M recast as 0 1 L ^ 1 zr j2 X jpH M @ A LGLRT -2 ðz1:L Þ ¼ N ln 1  (8) ^ 1 pÞðzH M ^ 1 zr Þ ðpH M r¼1 r It is noted that the main difference between detectors (7) and (8) is that the covariance matrix in (7) is known while it is an estimator in (8). We will study the performance loss due to adaptively estimating the covariance matrix by different estimation algorithm and different number of training data in Section 5. 3.3. The covariance matrix estimation strategies The most natural estimator of M in Gaussian clutter is the sample covariance matrix [3,4] K X

^ SCM ¼ 1 z zH M K k¼1 k k

(9)

while for the non-Gaussian environment most of the previous works use the normalized sample covariance matrix [5,6,8]

change from cell to cell. The third estimator of M is called recursive ML-estimator [11,12,14] K X zk zH k ^ þ 1Þ ¼ N ; Mði ^ 1 zk K k¼1 zH MðiÞ

i ¼ 0; . . . ; N RE

where K zk zH NX k ^ Mð0Þ ¼ K k¼1 zH z k k

^ 1 zk also accounts for and NRE ¼ 3 [6]. The quantity zH k MðiÞ the local power estimator. Now we consider the CFAR property of the two-step GLRT detector (8) with the above different estimators under H0. It is known that detector (8) with SCM is independent of the actual clutter covariance matrix in Gaussian clutter [1,3,4], and that it would induce a dependent relationship with the PDF of the texture component in non-Gaussian clutter. Since all the spiky components of the received clutter echoes factor out from the test statistic, the adaptive detector (8) with NSCM is independent of the texture PDF while it depends on the structure of the clutter covariance matrix [7]. It has been proved that the recursive ML-estimator is independent of both texture and M [6,12,14]. To analyze the computation complexity of these estimators, we evaluate their floating-point operations for real data. Note that Kelly’s SCM involves O(KN2) floating-point operations (flops) [1], the NSCM needs O(2KN2) flops, whereas the recursive ML-estimator requires O(NIT KN3) flops. It follows that the NSCM estimator (10) is slightly more complex than the SCM estimator (9) since the NSCM has to normalize the possible power for each CUT, while the recursive ML-estimator is the most complex estimator since it requires on-line inversion of ^ the matrix MðiÞ for each iteration. 4. Performance assessment In non-Gaussian clutter, the performance analysis is carried out via Monte Carlo simulations and based on 100=pfa independent trials under the following assumptions:

 The clutter is K-distributed, i.e. the PDF is

K X zk zH k ^ NSCM ¼ N M K k¼1 zH z k k

(10)

The difference between estimators (9) and (10) is that the NSCM contains the data-dependent normalization factor

assume M is known z1:K/2 ...

z1:L

sffiffiffiffiffiffi 2n sffiffiffiffiffiffi sffiffiffiffiffiffi ! !n m 2n 2n z K n1 z pz ðzÞ ¼ m m GðnÞ

GLRT detector

... zK/2+1:K

estimation algorithms

(11)

k

two-step GLRT detector

H1 H0

^ M Fig. 1. Block diagram for two-step GLRT scheme.

ARTICLE IN PRESS X. Shuai et al. / Signal Processing 90 (2010) 16–23

and the texture is gamma distributed  n 1 n pt ðtÞ ¼ tn1 en=mt uðtÞ GðnÞ m





Then, the optimum statistic test (12) becomes QL

LNP ðz1:L Þ ¼ Qr¼1 L

1  i; j  N

where r is the one-lag correlation coefficient. It is worth observing that typical values of r for radar sea clutter are in the range ½0:9 0:99 [6]. The SCR has been defined in this section as PL SCR ¼

a

r¼1 ð r pÞ

H

s

hN ðq1 ðzr ÞÞ

r¼1 hN ðq0 ðzr ÞÞ

where GðÞ is the Gamma function, K v ðÞ is the modified second-kind Bessel function of order v, m ¼ E½t is the mean value, and n is a measure of clutter spikiness: the lower the value of n, the spikier the clutter. The clutter data have been generated assuming that the speckle component has an exponential correlation structure covariance matrix. Hence M can be denoted as ½Mij ¼ rjijj ;

19

M1 ðar pÞ

2

In order to limit the computational burden, we assume pfa ¼ 104 throughout the section. Recall that L is related to the range extent of the target and the range resolution of the radar. We consider small values of L(Lr10) and N ¼ 8 to save simulation time. 4.1. Detection performance of the NP detector versus the two-step GLRT detector Due to the unknown parameters ar and M in the NP detector (3), it is impossible to compute the optimal NP detector. In order to evaluate the performance loss due to adaptivity, we assume that the parameter M in the NP detector (3) is already known and that the parameter ar is 1 zr . Then the LRT replaced by its MLE a^ r ¼ pH M1 zr =zH r M can be recast as QL R 1 ^ 0 pzr jar ;tr ;M;H1 ðzr jar ; tr ; M; H 1 Þptr ðtr Þ dtr LNP ðz1:L Þ ¼ r¼1 QL R 1 r¼1 0 pzr jtr ;M;H0 ðzr jtr ; M; H 0 Þptr ðtr Þ dtr

QL

r¼1

¼ QL

r¼1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!! q ðzr Þ q1 ðzr Þ  K Nn 4n 1 m sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!! q ðzr Þ q0 ðzr ÞðnNÞ=2  K Nn 4n 0 ðnNÞ=2

(14)

m

Notice that for the case of L ¼ 1, detector (14) can be expressed as the optimum NP detector (WMF compared with a data dependent threshold) in [15–17,21,22], but the detector in these papers cannot be extended to the distributed targets. In (14), we have assumed that the clutter covariance matrix M and the PDF of the texture are known to the detector. For this reason, we refer to the above detection schemes as optimum NP detector. In a realistic radar scenario, these parameters must be estimated from the data. However, the best approach to this adaptive estimation problem is still an open problem. In the following simulation programs, we will study the performance of the estimation algorithms which have been proposed in Section 4. Precisely, the numerical results describe the performance loss due to adaptively estimating the texture and the covariance matrix by estimation strategies (9)–(11) for distributed targets. In Fig. 2, the detection probabilities of the NP detector, of known-M GLRT detector, and of two-step GLRT detector are plotted versus SCR. The optimum NP detector (solid line) refers to detector (14) with known m, v and M, the known-M GLRT detector (dashed line) refers to detector (7) with known-M, whereas the two-step GLRT detector (dashed-dot line) refers to detector (8) with estimator (10). The curves in Fig. 2 suggest that the known-M GLRT detector outperforms the two-step GLRT detector and that it is very close to the NP detector. The performance loss

(12) Following the papers [15–17], we define   Z 1 q ðzr Þ ptr ðtr Þ dtr hN ðqi ðzr ÞÞ  tN exp  i r 2tr 0  N   1 n q ðzr Þ ðnNÞ=2 ¼ nN1 4n  i m GðnÞ m 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! q ðzr Þ 4n i ; i ¼ 1; 2  K Nn

m

(13)

where q0(zr) and q1(zr) are two quadratic forms under hypotheses H0 and H1, respectively, with 1 q0 ðzr Þ ¼ zH zr r M

and 1 q1 ðzr Þ ¼ ðzr  a^ r pÞH M1 ðzr  a^ r pÞ ¼ zH zr  r M

jpH M1 zr j2 pH M1 p

Fig. 2. Performance loss due to adaptive detection for v ¼ 0.4, L ¼ 4, N ¼ 8 and K ¼ 5N.

ARTICLE IN PRESS 20

X. Shuai et al. / Signal Processing 90 (2010) 16–23

of the known-M GLRT detector with respect to the NP one is due to adaptively estimating the texture. This figure and other simulation results for different L (not shown here for brevity) confirm that the loss is less than 0.5 dB and is negligible. On the other hand, it is observed that the twostep GLRT whose loss is about 2 dB for Pd ¼ 0.9 is poorer than the other two detectors. Precisely, the loss of the two-step GLRT with respect to the NP detector, namely, the one that possesses perfect knowledge of the covariance matrix and texture PDF, is due to adaptive detection. The figure highlights that the loss of adaptive detector (8) is mainly caused by adaptively estimating the covariance matrix. Hence, the further simulation tasks are to study the performance loss caused by estimation algorithms of covariance matrix. 4.2. Detection performance of the two-step GLRT detector with different covariance matrix estimators

^ for L ¼ 6, Fig. 4. Performance of detector (8) with different estimator M r ¼ 0.99, v ¼ 0.5 and K ¼ 5N.

In Section 3, we introduced three different estimators: SCM, NSCM and recursive MLE, and analyzed their computation complexity and CFAR properties. Now we study the performance loss of the two-step GLRT detector with the above different estimation strategies with respect to the known-M GLRT detector. In Figs. 3–5, the detection probabilities of the adaptive detector (8) with the estimators (9)–(11) and known-M are plotted versus SCR. The values of the parameters are r ¼ 0.9, v ¼ 0.5 (Fig. 3), r ¼ 0.99, v ¼ 0.5 (Fig. 4) and r ¼ 0.9, v ¼ 4.5 (Fig. 5). We first analyze the curves in Figs. 3 and 4. It is shown that the adaptive GLRT with NSCM (10) is very close to the GLRT with (11) for r ¼ 0.9, whereas it is poorer than the ^ 1 zk GLRT with (11) for r ¼ 0.99. The reason is that zH k MðiÞ in (11) which can whiten the highly correlated clutter data is more accurate than zH k zk in (10) for large r, although both of them account for the local power estimator of the

^ for L ¼ 6, Fig. 5. Performance of detector (8) with different estimator M r ¼ 0.9, v ¼ 4.5 and K ¼ 5N.

1 0.9 0.8 0.7

Pd

0.6 0.5 0.4 0.3 known-M GLRT

0.2

NSCM GLRT SCM GLRT

0.1 0 -20

recursive ML-M GLRT

-15

-10

-5

0

5

10

SCR (dB) ^ for L ¼ 6, Fig. 3. Performance of detector (8) with different estimator M r ¼ 0.9, v ¼ 0.5 and K ¼ 5N.

range cell k. Next, the curves in Fig. 5 demonstrate that the adaptive detector (8) with the estimator (9)–(11) performs almost the same for the case of homogeneous environment (vZ4.5). This is not surprising since the K distribution converges to the Rayleigh PDF for increasing v. As a consequence, the loss of the GLRT detector with Kelly’s SCM (9) due to model mismatching decreases as n increases. We also observe that the GLRT detector with the recursive MLE outperforms the other two receivers for the realistic clutter rZ0.9. Moreover, the recursive ML-estimator is CFAR with respect to M and texture. Other simulation results not reported here for the sake of brevity have confirmed that the above results are still true for different values of r and v. Thus, the recursive ML-estimator is the most suitable estimator to implement the adaptive detection algorithm for the realistic spiky clutter.

ARTICLE IN PRESS X. Shuai et al. / Signal Processing 90 (2010) 16–23

1

21

1

0.9

v = 0.4, SCR = 2dB

0.9

L = 2, SCR = 0dB

0.8

0.8

0.7

0.7

0.6

0.6

v = 0.5, SCR = 2dB

Pd

Pd

v = 0.8, SCR = 2dB

0.5 0.4

0.4

L = 6, SCR = 2dB

0.3

0.3 L = 10 SCR = -5dB

0.2

0.2 0.1

0.1

0

0 2N

3N 4N 5N 6N K (number of secondary range cells)

2N

7N

Fig. 6. Performance of detector (8) versus K for v ¼ 0.5 and r ¼ 0.8.

3N 4N 5N 6N K (number of secondary range cells)

7N

Fig. 8. Performance of detector (8) versus K for L ¼ 6 and r ¼ 0.9.

1

1

p = 0.95

0.9

0.9

p = 0.9

0.8

0.8

0.7

0.7

0.6

0.6 Pd

pd

0.5

0.5 p = 0.6

0.4

L=6

0.5 0.4

p = 0.8

0.3

L=4

0.3

0.2

0.2

0.1

0.1

0

L=1

known-M GLRT estimated-M GLRT

0

2N

3N 4N 5N 6N K (number of secondary range cells)

7N

Fig. 7. Performance of detector (8) versus K for L ¼ 6 and v ¼ 0.5.

4.3. Numerical results for the training data set With a sufficient number of secondary data vectors, the performance of two-step GLRT detector becomes very close to the performance of known-M GLRT detector [5], but it is impossible to collect enough training data for the realistic scenario. Here, we would like to give some numerical results for discussing the relation between detection probability and the parameter K. The detection probabilities of the adaptive detector (8) with covariance matrix estimator (11) are plotted in Figs. 6–8 versus the number of the secondary data vectors K. In particular, Fig. 6 refers to different L, Fig. 7 refers to different r and Fig. 8 refers to different v. The values of parameters are noted in each figure. The curves in Figs. 6–8 indicate that the performance of detector (8) is poor for K ¼ 2N. The detector (8) obtains an important performance gain when K changes from 2N to 3N, while the performance increases with K and reaches steady state after K45N. This feature holds true no matter

-20

-15

-10

-5 SCR (dB)

0

5

10

Fig. 9. Performance of detector (7) and (8) for K ¼ 2N.

how the parameters L, SCR, r and v change. It is also clear that, with numerous secondary data vectors, the recursive ^ requires heavy computation. These ML-estimator M results may be introductive for the issue that how to choose the secondary data.

4.4. The performance loss with different L The detection probabilities of detectors (7) and (8) are plotted in Figs. 9 and 10 for different L. Here, the values of the parameters are v ¼ 0.5, r ¼ 0.9 and N ¼ 8. For detector (8), we choose the recursive ML-estimator (11) to estimate the unknown covariance matrix. In Fig. 9, the loss of the detector (8) (dashed line) with respect to the detector (7) (solid line) is about 3 dB for L ¼ 1, about 5 dB for L ¼ 4 and about 7 dB for L ¼ 6, whereas in Fig. 10 it is about 0.6 dB for L ¼ 1, about 1 dB for L ¼ 4 and about 2 dB for L ¼ 6. Although this result suggests that the performance loss due to estimating M

ARTICLE IN PRESS 22

X. Shuai et al. / Signal Processing 90 (2010) 16–23

1 0.9

0.7



L=4

0.8 L=6

Pd

0.6 0.5 0.4

Acknowledgments

L=1

0.3 0.2 0.1 0 -20

known-M GLRT estimated-M GLRT

-15

-10

-5

0

5

10

SCR (dB) Fig. 10. Performance of detector (7) and (8) for K ¼ 5N.

increases with L since the loss is contributed by L scatterers, the curves in Figs. 9 and 10 still show that increasing the radar resolution capabilities can produce a significant detection gain. 5. Conclusion An adaptive detector has been developed for distributed targets in compound-Gaussian clutter. When the clutter distribution is perfectly known, we first derived the optimum NP detector. Unfortunately, the NP detector is difficult to implement since it involves on-line heavy computation to integrate with respect to the texture PDF and the values of the parameters of the texture PDF are always unknown in a realistic scenario. To overcome this difficulty, we considered that the texture of a determined range cell is an unknown deterministic parameter and is ML-estimated so that the two-step GLRT which is absolutely independent of the distribution of the texture and is fully adaptive for compound-Gaussian disturbance can be derived. For the unknown covariance matrix, we studied three estimators: SCM, NSCM and the recursive ML-estimator. Precisely, we analyzed the CFAR properties and computation complexity issues of the above three estimators and analytically showed their performances by Monte Carlo trials. The numerical results highlight the following conclusions:

 Modeling the texture as an unknown deterministic





vectors K. We also observed that the performance increases with K. The loss due to estimating covariance M increases with the number of scatterers L, while increasing the radar resolution capabilities can produce a significant detection gain.

parameter and estimating the texture component are viable since the loss due to adaptivity is negligible even for large number of scatterers L. Moreover, this strategy not only decreases the computation complexity but also makes the GLRT be independent of texture. ^ ensures CFAR property The recursive ML-estimator M and has the best detection performance for the realistic environment (0.9rrr0.99). Thus, the recursive ML^ is the most suitable estimator to impleestimator M ment the adaptive detection algorithm. The detection probability of adaptive detector (8) increases with the number of the secondary data

The authors are very grateful to the anonymous referees for their many helpful comments and constructive suggestions on improving the exposition of this paper, and to Dr. Guolong Cui for his several useful discussions. This work is supported in part by Program of Cultivating Academic Leaders of UESTC (Y02018023601055). References [1] E. Conte, A.D. Maio, G. Ricci, GLRT-based adaptive detection algorithms for range-spread targets, IEEE Transactions on Signal Processing 49 (7) (July 2001) 1336–1348. [2] S.M. Kay, Fundamentals of Statistical Signal Processing, Volume 1: Estimation Theory; Volume 2: Detection Theory, Publishing House of Electronics Industry, Beijing, China, 2006. [3] E.J. Kelly, An adaptive detection algorithm, IEEE Transactions on Aerospace and Electronic Systems AES–2 (1) (March 1986) 115–127. [4] F.C. Robey, D.R. Fuhrmann, R. Nitzberg, E.J. Kelly, A CFAR adaptive matched filter detector, IEEE Transactions on Aerospace and Electronic Systems 28 (1) (January 1992) 208–216. [5] N. Bon, A. Khenchaf, R. Garello, GLRT subspace detection for range and doppler distributed targets, IEEE Transactions on Aerospace and Electronic Systems 44 (2) (April 2008) 678–695. [6] F. Gini, M. Greco, Covariance matrix estimation for CFAR detection in correlated heavy tailed clutter, Signal Processing 82 (2002) 1847–1859 (special section on Signal Processing with heavy tailed distribution). [7] F. Gini, Performance analysis of two structured covariance matrix estimators in compound-Gaussian clutter, Signal Processing 80 (2000) 365–371. [8] F. Gini, M. Greco, Suboptimum approach for adaptive coherent radar detection in compound-Gaussian clutter, IEEE Transactions on Aerospace and Electronic Systems 35 (3) (July 1999) 1095–1104. [9] E. Conte, A.D. Maio, G. Ricci, Asymptotically optimum radar detection in compound Gaussian clutter, IEEE Transactions on Aerospace and Electronic Systems 31 (2) (April 1995) 617–625. [10] K. Gerlach, Spatially distributed target detection in non-Gaussian clutter, IEEE Transactions on Aerospace and Electronic Systems 35 (3) (July 1999) 926–934. [11] E. Conte, A.D. Maio, G. Ricci, Covariance matrix estimation for adaptive CFAR detection in compound-Gaussian Clutter, IEEE Transactions on Aerospace and Electronic Systems 38 (2) (April 2002) 415–426. [12] E. Conte, A.D. Maio, G. Ricci, Recursive estimation of the covariance matrix of a compound-Gaussian process and its application to adaptive CFAR detection, IEEE Transactions on Signal Processing 50 (8) (August 2002) 1908–1915. [13] E. Conte, A.D. Maio, G. Ricci, CFAR detection of distributed targets in non-Gaussian disturbance, IEEE Transactions on Aerospace and Electronic Systems 38 (2) (April 2002) 612–621. [14] F. Pascal, Y. Chitour, P. Forster, P. Larzabal, Covariance structure maximum-likehood estimates in compound Gaussian noise: existence and algorithm analysis, IEEE Transactions on Signal Processing 56 (1) (January 2008) 34–48. [15] F. Gini, M.V. Greco, A. Farina, Clairvoyant and adaptive signal detection in non-Gaussian clutter: a data-dependent threshold interpretation, IEEE Transactions on Signal Processing 47 (6) (June 1999) 1522–1531. [16] F. Gini, M.V. Greco, A. Farina, P. Lombardo, Optimum and mismatched detection against K-distributed plus Gaussian clutter, IEEE Transactions on Aerospace and Electronic Systems 34 (3) (July 1998) 860–876.

ARTICLE IN PRESS X. Shuai et al. / Signal Processing 90 (2010) 16–23

[17] A. Farina, F. Gini, M.V. Greco, K.J. Sangston, Optimum and suboptimum coherent radar detection in compound Gaussian clutter: a data-dependent threshold interpretation, in: Proceedings of IEEE National RADAR Conference, Ann Arbor, MI, May 1996, pp. 160–165. [18] E. Conte, M. Lops, G. Ricci, Radar detection in K-distributed clutter, IEE Proceedings of Radar, Sonar and Navigation 141 (2) (April 1994) 116–118. [19] E. Conte, A.D. Maio, C. Galdi, Signal detection in compoundGaussian noise: Neyman–Pearson and CFAR detectors, IEEE Transactions on Signal Processing 48 (2) (February 2000) 419–428.

23

[20] E. Conte, G. Ricci, Performance prediction in compound-Gaussian clutter, IEEE Transactions on Aerospace and Electronic Systems 30 (April 1994) 611–616. [21] F. Gini, A. Farina, Vector subspace detection in compound-Gaussian clutter, part I: survey and new results, IEEE Transactions on Aerospace and Electronic Systems 38 (October 2002) 1295–1311. [22] P. Lombardo, M. Greco, A. Farina, F. Gini, D. Pastina, Rada clutter modeling, in: Proceedings of RADAR 2005 in Washington D.C., Technical Report, May 2005, pp. 1–231.