Maximum Likelihood Estimation of Compound-Gaussian Clutter and Target Parameters

3884 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 10, OCTOBER 2006

Maximum Likelihood Estimation ofCompound-Gaussian Clutter and Target Parameters

Jian Wang, Student Member, IEEE, Aleksandar Dogandzic, Member, IEEE, and Arye Nehorai, Fellow, IEEE

Abstract—Compound-Gaussian models are used in radar signalprocessing to describe heavy-tailed clutter distributions. Theimportant problems in compound-Gaussian clutter modelingare choosing the texture distribution, and estimating its param-eters. Many texture distributions have been studied, and theirparameters are typically estimated using statistically suboptimalapproaches. We develop maximum likelihood (ML) methodsfor jointly estimating the target and clutter parameters in com-pound-Gaussian clutter using radar array measurements. Inparticular, we estimate i) the complex target amplitudes, ii) aspatial and temporal covariance matrix of the speckle component,and iii) texture distribution parameters. Parameter-expandedexpectation–maximization (PX-EM) algorithms are developedto compute the ML estimates of the unknown parameters. Wealso derived the Cramér–Rao bounds (CRBs) and related boundsfor these parameters. We first derive general CRB expressionsunder an arbitrary texture model then simplify them for specifictexture distributions. We consider the widely used gamma texturemodel, and propose an inverse-gamma texture model, leadingto a complex multivariate clutter distribution and closed-formexpressions of the CRB. We study the performance of the proposedmethods via numerical simulations.

Index Terms—Compound-Gaussian model, Cramér–Rao bound(CRB), estimation, parameter-expanded expectation–maximiza-tion (PX-EM).

I. INTRODUCTION

WHEN a radar system illuminates a large area of the sea,the probability density function (pdf) of the amplitude

of the returned signal is well approximated by the Rayleighdistribution [1], i.e., the echo can be modeled as a complex-Gaussian process. That distribution is a good approximation.This can be proved theoretically by the central limit theorem,since the returned signal can be viewed as the sum of the re-flection from a large number of randomly phased independentscatterers. However, in high-resolution and low-grazing-angleradar, the real clutter data show significant deviations from thecomplex Gaussian model, see [2], because only a small seasurface area is illuminated by the narrow radar beam. The be-havior of the small patch is nonstationary [1] and the numberof scatterers is random, see [3]. Due to the different waveform

Manuscript received May 3, 2005; revised November 29, 2005. The work ofJ. Wang and A. Nehorai was supported by AFOSR Grants F49620-02-1-0339,FA9550-05-1-0018 and DoD/AFOSR MURI Grant FA9550-05-1-0443. The as-sociate editor coordinating the review of this manuscript and approving it forpublication was Prof. Yuri I. Abramovich.

J. Wang and A. Nehorai are with the Electrical and Computer Engineering De-partment, University of Illinois, Chicago, IL 60607 USA (e-mail: [email protected]; [email protected]).

A. Dogandzic is with the ECE Department, Iowa State University, Ames, IA50011 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TSP.2006.880209

characteristics and generation mechanism, the sea surface wave,i.e., the roughness of the sea surface, is often modeled in twoscales [4], [5]. To take into account different scales of rough-ness, a two-scale sea surface scattering model was developed,see [6]–[8]. In this two-scale model—a compound-Gaussianmodel—the fast-changing component, which accounts for localscattering, is referred to as speckle . It is assumed to bea stationary complex Gaussian process with zero mean. Theslow-changing component, texture is used to describe thevariation of the local power due to the tilting of the illuminatedarea, and it is modeled as a nonnegative real random process.The complex clutter can be written as the product of these twocomponents

(1)

The compound-Gaussian model is a model widely used to char-acterize the heavy-tailed clutter distributions in radar, especiallysea clutter, see [2], [6], [9], and Section II. It belongs to the classof the spherically invariant random process (SIRP), see [10] and[17]. Note that the compound-Gaussian distribution is also oftenused to model speech waveforms and various radio propagationchannel disturbance, see [10] and the references therein.

Modeling of clutter using a compound-Gaussian distributioninvolves these important aspects: choosing the texture distribu-tion, estimating its parameters, and evaluating the efficiency ofthe estimations. Many texture distributions have been studied,but their parameters were typically estimated using the methodof moments, which is statistically suboptimal, see [2]. Wepresent our measurement model in Section II. In Section III,we develop the parameter-expanded expectation–maximization(PX-EM) algorithms to estimate the target and clutter param-eters. We compute the Cramér–Rao bounds (CRBs) for thegeneral compound-Gaussian model and simplify them for twotexture distributions in Section IV. In Section V, we verify ourresults through Monte Carlo numerical simulations.

II. MODELS

We extend the radar array measurement model in [11] to ac-count for compound-Gaussian clutter. Assume that an -ele-ment radar array receives pulse returns, where each pulse pro-vides samples. We collect the spatiotemporal data from theth range gate into a vector of size and model

as (see [11] and [12])1

(2)

1A special case of the model (2) for rank-one targets (i.e., scalarX) in com-pound-Gaussian clutter was considered in [15].

1053-587X/$20.00 © 2006 IEEE

WANG et al.: COMPOUND-GAUSSIAN CLUTTER AND TARGET PARAMETERS 3885

where is an spatiotemporal steering matrix of the tar-gets, is the temporal response ma-trix, is an matrix of unknown complex amplitudesof the targets. Here is the number of possible directions thatthe reflection signals will come from, and is the number ofrange gate that covers the target.2 The additive noise vectors

, are independent, identically distributed(i.i.d.) and come from a compound-Gaussian probability distri-bution, see, e.g., [3], [10] and [14]–[17].

We now represent the above measurement scenario using thefollowing hierarchical model: are conditionally indepen-dent random vectors with probability density functions (pdfs):

(3)

where the superscript denotes the Hermitian (conjugate)transpose, is the (unknown) covariance matrix of the specklecomponent, and , are the unobservedtexture components (powers). We assume the texture to be fullycorrelated during the coherent processing interval (CPI) [18].This assumption is reasonable since the radar processing timeis not too long. We consider the following texture distributions

• gamma: follow a gamma distribution [2], [3], [14]• inverse gamma: follow a gamma distribution

[19]–[21].

III. MAXIMUM LIKELIHOOD ESTIMATION

We develop the ML estimates of the complex ampli-tude matrix , speckle covariance matrix , and tex-ture distribution parameter from the measurements

, see [22]. In the following, wepresent the PX-EM algorithms for ML estimation of theseparameters under the above three texture models. The PX-EMalgorithms share the same monotonic convergence propertiesas the “classical” expectation-maximization (EM) algorithms,see [23, Theorem 1]. They outperform the EM algorithms inthe global rate of convergence, see [23, Theorem 2]. In ourproblem, the computations are confined to the PX-E step of thePX-EM algorithm. The PX-M step follows as a straightforwardconsequence of the PX-E step.

A. PX-EM Algorithm for Gamma Texture

We model the texture components , asgamma random variables with unit mean (as, e.g., in [3]) andunknown shape parameter , i.e.

(4)

hence, the unknown parameters are . (The shapeparameter is also known as the Nakagami- parameter inthe communications literature, see, e.g., [24, Ch. 2.2.1.4].) Thischoice of texture distribution leads to the well-known cluttermodel, see [2] and [3] and references therein.

The method for deriving EM- algorithm from complete-datasufficient statistics for a similar GMANOVA model is presented

2In high resolution radar, target can usually distribute in more than one rangegates, see [13] and references therein.

in [12]. Since EM algorithms often converge slowly in some sit-uations, we propose a PX-EM algorithm. Because of the intro-duction of new parameter, PX-EM algorithm can capture extrainformation from the complete data in the PX-E step. Also be-cause its M step performs a more efficient analysis by fitting theexpanded model, PX-EM has a rate of convergence at least asfast as the parent EM [23].

The proposed PX-EM algorithm estimates by treating ,as the unobserved data. First we add an auxil-

iary parameter (the mean of ) to the set of parameters .Note that in the original model. Hence the augmentedparameter set is , where and are re-lated as follows: . Note that and are notunique whereas their product is. Under this expanded model,the pdf of is (for )

(5)

where denotes the gamma function. The conditional pdfsof are unchanged, see (3). The underlying statistical prin-ciple of PX-EM is to perform a “covariance adjustment” to cor-rect the M step. In this problem, we adjust the covariance matrix

to a product of and . More specifically, we use a ex-panded complete-data model that has a larger set of identifiableparameters, but leads to the original observed-data model withthe original parameters identified from the expanded parametersvia a many-to-one mapping [23].

We present the details of the derivation of the PX-EM al-gorithm in Appendix A. To summarize it, in the PX-E step,we calculate the conditional expectations of the complete-datasufficient statistics assuming all unknown parameters areknown from the complete data log-likelihood. In the PX-Mstep, we estimate the unknown parameters from these expec-tations. The derivation of these estimates from the sufficientstatistics are explained in [12] in details. The PX-EM algorithmfor the above expanded model consists of iterating between thefollowing PX-E and PX-M steps.

PX-E Step: Compute the conditional expectations of thenatural sufficient statistics

(6a)

(6b)

(6c)

(6d)

(6e)

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where is the estimate ofin the th iteration and (6a)–(6e) are computed using (8)with , , and .PX-M Step: Compute

(7a)

(7b)

(7c)

(7d)

where

(7e)

(7f)

and find that maximizes

The above iteration is performed until , , and con-verge. The computation of requires maximizing (7c),which is accomplished using the Newton–Raphson method (em-bedded within the “outer” EM iteration, similar to [26]). Theconditional-expectation expression (8), shown at the bottom ofthe page, is obtained by using the Bayes rule, (3) and (4), andchange-of-variable transformation .The integrals inthe numerator and denominator of (8) are efficiently and accu-rately evaluated using the generalized Gauss-Laguerre quadra-ture formula (see [27, Ch. 5.3])

(9)

where is an arbitrary real function, is the quadratureorder, and and , are the ab-scissas and weights of the generalized Gauss-Laguerre quadra-ture with parameter .

B. PX-EM Algorithm for Inverse Gamma Texture

We now propose a complex multivariate -distribution modelfor the clutter and apply it to the measurement scenario in Sec-tion II. A similar clutter model was briefly discussed in [17, Sec.IV.B.3], where it was also referred to as the generalized Cauchydistribution. Assume that , aregamma random variables with mean one and unknown shape pa-rameter . Consequently, follows an inverse gammadistribution and the conditional distribution of givenis , see also (3). Integrating out the un-observed data , we obtain a closed-form expression for themarginal pdf of

(10)

which is the complex multivariate distribution with locationvector , scale matrix , and shape parameter . Here,the unknown parameters are . We first estimate

and assuming that the shape parameter is known and thendiscuss the estimation of .

Known : For a fixed , we derive a PX-EM algorithm forestimating and by treating , as theunobserved data and adding an auxiliary mean parameter for

, similar to the gamma case discussed in Section III-A.The derivation of PX-EM algorithm is analogous to the one forgamma texture in Appendix A. Here, the resulting PX-EM al-gorithm consists of iterating between the following PX-E andPX-M steps:

PX-E Step: Compute

(11a)

for and

(11b)

(11c)

(11d)

(8)

WANG et al.: COMPOUND-GAUSSIAN CLUTTER AND TARGET PARAMETERS 3887

PX-M Step: Compute

(12a)

(12b)

where

(12c)

(12d)

The above iteration is performed until and converge.Denote by and the estimates of andobtained upon convergence, where we emphasize their depen-dence on .

Unknown : We compute the ML estimate of by maxi-mizing the observed-data log-likelihood function concentratedwith respect to and

(13)

see also (10).

IV. CRAMÉR–RAO BOUND (CRB) AND RELATED BOUNDS

In this section, we first derive the CRB with general texturepdf assumption. Then we apply it for different texture distribu-tions, see [28]. We also consider the hybrid CRB, which is notas tight as CRB.

A. General CRB Results

Denote by the pdf of the texture . Then, ac-cording to the above measurement model, is a complexspherically invariant random vector (SIRV) with marginal pdf

(14)

where

(15a)

(15b)

and denotes the Frobenius norm. Also,where denotes a Hermitian square root of a Hermitianmatrix .

Given an arbitrary radius , the concatenatedvector of real and imaginary parts of is uniformly dis-tributed on the surface of a -dimensional ball with radius ,centered at the origin. Denote by and the partial deriva-tives of with respect to its first and second entries, i.e.

and . For anywell-behaved , changing the order of differentiation andintegration leads to

(16a)

(16b)

Define the vector of signal and clutter parameters

(17a)

where the subscript denotes a transpose

(17b)

(17c)

and is the texture parameter.3 Here, the vech andoperators create a single column vector by stacking elementsbelow the main diagonal columnwise; vech includes the maindiagonal, whereas omits it. The Fisher information matrix(FIM) for is computed by using [29, eqs. (3.21) and (3.23)]:

(18a)

(18b)

where denotes the FIM entry with respect to the param-eters and , and

(18c)

is the log-likelihood function. Then the CRB for is

(18d)

To simplify the notation, we omit the dependencies of the FIMand CRB on the model parameters. We also omit details of the

3We parameterize the texture pdf using only one parameter. The extension tomultiple parameters is straightforward.

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derivation and give the final FIM expressions (see Appendix Bfor details)

(19a)

(19b)

(19c)

(19d)

(19e)

(19f)

where

(20)

and

(21a)

(21b)

(21c)

(21d)

(21e)

Here, (19a) and (19b) have been computed by using (18a) andthe lemma, whereas (19c)–(19f) follow by using (18b).

Interestingly, the FIMs of compound-Gaussian models withdifferent texture distributions share the common structure in(19a)–(19f) where the texture-specific quantities are the scalarcoefficients in (21). The above FIM and CRB matrices areblock-diagonal [see (19e) and (19f)], implying that the CRBs

for the signal parameters are uncoupled from the clutterparameters and . Hence, the CRB matrix for remains thesame whether or not and are known. Similarly, the CRBsfor and remain the same whether or not is known. Also,(19a) and (19b) simplify to the FIM expressions for complexGaussian clutter when , and ,see also [29, eq. (15.52)].

B. CRB for Specific Texture Distributions

Computing the texture-specific terms in (21) typicallyinvolves two-dimensional (2–D) integration that cannot beevaluated in closed form. This integration can be performedusing Gauss quadratures, see, e.g., [27, Ch. 5.3]. Here we usethe gamma texture as an example.

Gamma Texture: Here we use the same model as inSection III-A. After applying a change-of-variable transforma-tion in (15a) and (16), we evaluate both integrals in(21) using the generalized Gauss–Laguerre quadrature formula(see (9).) For example, the formula used to compute is givenin (22), shown at the bottom of the page, where, to simplifythe notation, we omit the dependencies of the abscissas andweights on . In Appendix C, we derive other coefficientsfor gamma-distributed texture.

Inverse-Gamma Texture: We use the model discussed inSection III-B. In this case, (15a) and (16) can be evaluated inclosed form, leading to the following simple expressions forthe texture-specific terms in (21a)–(21e) (see Appendix C)

(23a)

(23b)

(23c)

(23d)

(23e)

where is the trigamma func-tion. Interestingly, the CRB matrix for the signal parame-ters is proportional to the corresponding CRB matrix forcomplex Gaussian clutter, with the proportionality factor

always greater than one.As , the inverse gamma texture distribution degener-

ates to a constant, the marginal pdf of in (14) reduces to thecomplex Gaussian distribution in (3) with , and (19a)

(22)

WANG et al.: COMPOUND-GAUSSIAN CLUTTER AND TARGET PARAMETERS 3889

and (19b) simplify to the FIM expressions for complex Gaussianclutter.

C. Hybrid (HCRB)

The CRB is a lower bound of the covariance of all unbiasedestimators of an unknown parameter vector. However, in somescenarios, we need to assess the estimation performance quicklybut not so tightly. Thus, we also consider the computation of aless optimal bound, the HCRB.

The HCRB is defined in [30]

(24a)

HCRB (24b)

Note that the HCRB takes the expectation over the unobserveddata for the whole product of two complete data score func-tions while the CRB takes the expectations, respectively. Thisusually reduces the calculation effort at the cost of degradedbound tightness. Similarly, we omit the dependencies of the in-formation matrix and HCRB on the model parameters. With thederivation presented in Appendix D, the information entries ofthe HCRB for general texture are

(25a)

(25b)

(25c)

(25d)

Compared with FIM, the information matrix of HCRB ismuch simpler and easier to compute. It is interesting to observethe following.

• , and are decoupled from each other. The HCRB is ablock diagonal matrix with three blocks. Note that in theCRB, and are coupled.

• is constant. It does not change over the choice oftexture models.

• affects in a simple way—by multiplying a con-stant with .

V. NUMERICAL EXAMPLES

The numerical examples presented here assess the estimationaccuracy of the ML estimates of , , and the shape parametersof the texture components. We consider a measurement scenariowith a 3-element radar array and pulses, implying that

. We select a rank-one target scenario with ,, complex target amplitude ,

and

(26)

where with nor-malized Doppler frequency , and

with spatial frequency .Here, denotes the Kronecker product. The speckle covari-ance matrix was generated using a model similar to that in[31, Sec. 2.6] with 1000 patches. The th element of thecovariance matrix of the speckle component was chosen as

(27)

which is the correlated noise covariance model used in [33] (seealso references therein). In the simulations presented here, weselect . The order of the Gauss-Hermite and gener-alized Gauss-Laguerre quadratures was .

We compare the average mean-square errors (mse) of the MLestimates of , and over 2000 independent trials with thecorresponding CRBs derived in Section IV. We also show theHCRBs in the results. Note: we just shown the average of ele-ments of , , and in this paper.

First we study the performance of the ML estimation forgamma texture in Section III-A. We have set the shape param-eter to . The Fig. 1 shows the mse for the ML estimatesof and and the average mse for the ML estimates of thespeckle covariance parameters as functions of .

In Fig. 2, we show the performance of the ML estimationfor the inverse gamma texture in Section III-A. Here, the shapeparameter was set to . Fig. 2 shows the mse for the MLestimates of and and the average mse for the ML estimatesof the speckle covariance parameters as functions of .

In Figs. 1 and 2, the mse matches the CRBs very well whenthe number of observations increases, which indicates that thePX-EM is the optimal asymptotically efficient estimation fortarget and the clutter parameters. HCRBs show their loose esti-mation to the lower bound of estimation variance as aforemen-tioned. The average signal power and clutter power can be cal-culated by their definitions

(28a)

(28b)

Thus, the signal-to-noise ratio (SNR) in these examples are6.70 and 7.95 dB respectively. Note that in these examples,

the SNRs do not change with number of snapshots .We also investigate the performance of the clutter spikiness,

which can be indicated by the clutter texture parameter . InFig. 3, we show the average mse of the estimates under the in-verse-gamma texture model for four different values. The re-sults are the averaged mse among 500 independent trials. When

decreases, i.e., the clutter becomes spikier, the results showthat there is no much difference for the performance of estimateof , while the estimate for becomes worse and the estimatefor becomes more accurate.

VI. CONCLUDING REMARKS

In this paper, we developed maximum likelihood al-gorithms for estimating the parameters of a target with

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Fig. 1. Average mse for the ML estimates of ���, ���, � and corresponding CRBsand HCRBs under the gamma texture model, as functions of N .

compound-Gaussian distributed clutter. The algorithms arepotentially useful to mitigate sea-clutter in high-resolution andlow-grazing-angle radar. The proposed maximum likelihood

Fig. 2. Average mse for the ML estimates of ���, ���, � and corresponding CRBsand HCRBs under the inverse-gamma texture model, as functions of N .

estimation is based on the parameter-expanded expecta-tion-maximization algorithm. We also computed the CRBs andtheir hybrid versions for the unknown parameters. Our results

WANG et al.: COMPOUND-GAUSSIAN CLUTTER AND TARGET PARAMETERS 3891

Fig. 3. Average mse for the ML estimates of ���, ���, � under the inverse-gammatexture model as functions of N for different � values.

are based on the general compound-Gaussian model and canbe applied to various texture distributions. We obtained com-pact closed-form results of the bounds for the inverse-gamma

texture. Numerical simulations confirmed the asymptotic effi-ciency of our estimates.

APPENDIX APX-EM ALGORITHM DERIVATION FOR GAMMA TEXTURE

We derive the PX-EM algorithm to estimate the parameter setgiven the observations .

With the auxiliary parameter , the augmented parameterset is , and the augmented model can bewritten as

(A29a)

(A.29b)

Denote . Instead of maximizingthe intractable likelihood function for the measurement , wemaximize the complete data log-likelihood

(A.30)

Substitute (3) and (5) into (A.30), we can write the completedata log-likelihood as

(A.31a)

where are nat-ural complete-data sufficient statistics [25, ch.1.6.2]:

(A.32a)

(A.32b)

(A.32c)

(A.32d)

(A.32e)

We first assume that is a known constant. Take derivativeof (A.31a) with respect to , , respectively and let thesederivatives equal to zero, we get a set of equations. Solving these

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equations, we can obtain the ML estimates of , , (see[12] for the derivations of the ML estimates for and )

(A.33a)

(A.33b)

(A.33c)

(A.33d)where

(A.33e)

(A.33f)With these estimates, we can find the ML estimate of that

maximizes the concentrated complete data log-likelihood

In the PX-E step, we calculate conditional expectationsof sufficient statistics , , ,, (see (6a)–(6e)). Then in the PX-M step,

we use these expectations to calculate the ML estimates ofparameters in . The iteration goes between PX-E and PX-Msteps until estimation results converges.

APPENDIX BDERIVATION OF THE SCORE FUNCTIONS AND FISHER

INFORMATION MATRIX (FIM)

Lange et al. derived the FIM of multivariate real -distributionin Appendix B of [19]. Here we follow the same procedure toderive the entries of the FIM of complex compound-Gaussiandistribution.

Before starting to derive the FIM entries, we list some pre-liminary results that will be used in the derivation here.

Lemma 1: For uniformly distributed on real sphereand any real matrices and

(B.34)

(B.35)

Proof: See [19] Appendix B.Lemma 2: For uniformly distributed on sphereand any matrices and

(B.36)

(B.37)

Proof: Let , where and are real andimaginary parts of vector . By applying Lemma 1, the proof istrivial.

Lemma 3: For independently uniformly dis-tributed on sphere and any matrices and

(B.38)

(B.39)

Proof: Note that and are independent

By symmetry

The first equation is proved. By applying the first equation inLemma 2, the second equation is also easily proved.

Now we start the derivation of FIM. First, recall the com-plete data log-likelihood (18c). We can get the contributionof each parameter to the score vector through straightforwardcalculations

(B.40a)

(B.40b)

(B.40c)

These rules are used in the derivation

and

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The entry of FIM corresponding to is

(B.41a)

Using Lemma 2 and Lemma 3, we can get the followingresults:

(B.42a)

where . See [32]for details.

The entry of the FIM matrix related to can be deriveddirectly

(B.43)

Also, see (B.44a) at the bottom of the page.

Finally, we prove that

(B.45)

and

(B.46)

Proof: Since and, for fixed , is an odd func-

tion of (B.40a) while and are even functionsof (B.40b).

APPENDIX CCALCULATION OF EXPECTATIONS

In this section, we propose the calculation method of expec-tations derived in Appendix A. First recall that for any well-be-haved function and SIRV real vector with pdf inthe form of

(C.47)

where is the surface area of the unit sphere in . See [19],Appendix A.

Now build a 1-1 map by letting, where and are the real and imaginary parts of

, respectively. Clearly, if is SIRV in , will be SIRV in. Also note that , where denotes the

norm. Applying (C.47), we get

(C.48)

Since , we can get following result

(C.49)

(B.44a)

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Define

(C.50)

By applying (C.48) and (C.49),

Similarly, see the equation at the bottom of the page.

Gamma Distribution: From the pdf of the gamma distribu-tion (4), we can get the following results easily:

(C.51a)

(C.52a)

where is the digamma function. Fornotation simplification, we define

. In the calculations, we change variables withand use the general Gauss–Laguerre quadrature for

WANG et al.: COMPOUND-GAUSSIAN CLUTTER AND TARGET PARAMETERS 3895

both inner and outer integration. The results are shown in thefirst equation at the bottom of the page. Similarly, see the secondequation at the bottom of the page and the equation at the bottomof the next page.

Inverse-Gamma Distribution: Fortunately, we have a closedform for the functions of , , and in the inverse-gammadistribution with pdf

(C.53)

(C.54a)

(C.54b)

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and

(C.54c)

The calculations yield

(C.55a)

(C.55b)

(C.55c)

(C.55d)

(C.55e)

APPENDIX DHCRB

A. General Results

In the compound-Gaussian model

(D.56a)

(D.56b)

The complete data log-likelihood function is

(D.57)

Let ,. is SIRV.

1) Expectations: Since

(D.58)

Similarly to Appendix C, we build a map .Then

(D.59a)

(D.59b)

(D.60)

It is not hard to get

(D.61a)

(D.61b)

where the recurrence relation is used. Hereis the surface area of a unit ball in .

Before deriving the entries of the FIM, we note that the first-order partial derivatives are

(D.62a)

(D.62b)

WANG et al.: COMPOUND-GAUSSIAN CLUTTER AND TARGET PARAMETERS 3897

(D.62c)

We follow the same procedure of Appendix B and get (see [32]for details)

(D.63a)

(D.63b)

(D.63c)

where , and

(D.63d)

B. Application to Specific Texture Distributions

1) Gamma Distribution: From gamma pdf

(D.64)We can derive . Also

(D.65a)

(D.65b)

Substitute the above results into (D.63a) and (D.63c), we get

(D.66a)

(D.66b)

(D.66c)

2) Inverse Gamma Distribution ( -Distributed Clutter):

(D.67)

(D.68)

and

(D.69a)

(D.69b)

Thus

(D.70a)

(D.70b)

(D.70c)

Interestingly, the inverse-gamma texture and the gamma tex-ture share the same block in the FIM of and .

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[2] F. Gini, M. V. Greco, M. Diani, and L. Verrazzani, “Performanceanalysis of two adaptive radar detectors against non-Gaussian realsea clutter data,” IEEE Trans. Aerosp. Electron. Syst., vol. 36, pp.1429–1439, Oct. 2000.

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Jian Wang (S’05) received the B. Sc. and M. Sc. de-grees in electrical engineering from the University ofScience and Technology of China (USTC), Hefei, in1998 and 2001, respectively.

He is currently working toward the Ph.D. degreewith the Electrical and Computer EngineeringDepartment, University of Illinois at Chicago (UIC),under the guidance of Prof. A. Nehorai. His researchinterests are in statistical signal processing and itsapplication to radar and array signal processing.

Aleksandar Dogandzic (S’96–M’01) received theDipl. Ing. degree (summa cum laude) in electricalengineering from the University of Belgrade, Yu-goslavia, in 1995, and the M.S. and Ph.D. degreesin electrical engineering and computer science fromthe University of Illinois at Chicago (UIC) in 1997and 2001, respectively, under the guidance of Prof.A. Nehorai.

In August 2001, he joined the Department of Elec-trical and Computer Engineering, Iowa State Univer-sity, Ames, as an Assistant Professor. His research

interests are in statistical signal processing theory and applications.Dr. Dogandzic received the Distinguished Electrical Engineering M.S. Stu-

dent Award by the Chicago Chapter of the IEEE Communications Society in1996. He was awarded the Aileen S. Andrew Foundation Graduate Fellowship in1997, the UIC University Fellowship in 2000, and the 2001 Outstanding ThesisAward in the Division of Engineering, Mathematics, and Physical Sciences, UIC.He is the recipient of the 2003 Young Author Best Paper Award and 2004 SignalProcessing Magazine Award by the IEEE Signal Processing Society.

Arye Nehorai (S’80–M’83–SM’90–F’94) receivedthe B.Sc. and M.Sc. degrees in electrical engineeringfrom the Technion, Israel, and the Ph.D. degreein electrical engineering from Stanford University,Stanford, CA.

From 1985 to 1995, he was a faculty memberwith the Department of Electrical Engineering,Yale University, New Haven, CT. In 1995, he wasa Full Professor with the Department of ElectricalEngineering and Computer Science, University ofIllinois at Chicago (UIC). From 2000 to 2001, he

was Chair of the Department of Electrical and Computer Engineering (ECE)Division, which then became a new department. In 2001, he was named Uni-versity Scholar of the University of Illinois. In 2006, he assumed the Chairmanposition of the Department of Electrical and Systems Engineering, WashingtonUniversity, St. Louis, MO, where he is also the inaugural holder of the Eugeneand Martha Lohman Professorship of Electrical Engineering.

Dr. Nehorai was Editor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL

PROCESSING during 2000–2002. During 2003–2005, he was Vice President(Publications) of the IEEE Signal Processing Society, Chair of the PublicationsBoard, member of the Board of Governors, and member of the ExecutiveCommittee of this Society. He is the founding Editor of the Special Columnson Leadership Reflections in the IEEE Signal Processing Magazine. He wascorecipient of the IEEE SPS 1989 Senior Award for Best Paper with P. Stoica,as well as coauthor of the 2003 Young Author Best Paper Award and of the2004 Magazine Paper Award with A. Dogandzic. He was elected DistinguishedLecturer of the IEEE SPS for the term 2004 to 2005. He is the PrincipalInvestigator of the new multidisciplinary university research initiative (MURI)project entitled Adaptive Waveform Diversity for Full Spectral Dominance. Hehas been a Fellow of the Royal Statistical Society since 1996.