Maximum Likelihood Estimation of Compound-Gaussian Clutter and ...

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

Maximum Likelihood Estimation of Compound-Gaussian Clutter and Target Parameters Jian Wang, Student Member, IEEE, Aleksandar Dogandˇzic´, Member, IEEE, and Arye Nehorai, Fellow, IEEE

Abstract—Compound-Gaussian models are used in radar signal processing to describe heavy-tailed clutter distributions. The important problems in compound-Gaussian clutter modeling are choosing the texture distribution, and estimating its parameters. Many texture distributions have been studied, and their parameters are typically estimated using statistically suboptimal approaches. We develop maximum likelihood (ML) methods for jointly estimating the target and clutter parameters in compound-Gaussian clutter using radar array measurements. In particular, we estimate i) the complex target amplitudes, ii) a spatial and temporal covariance matrix of the speckle component, and iii) texture distribution parameters. Parameter-expanded expectation–maximization (PX-EM) algorithms are developed to compute the ML estimates of the unknown parameters. We also derived the Cramér–Rao bounds (CRBs) and related bounds for these parameters. We first derive general CRB expressions under an arbitrary texture model then simplify them for specific texture distributions. We consider the widely used gamma texture model, and propose an inverse-gamma texture model, leading to a complex multivariate clutter distribution and closed-form expressions of the CRB. We study the performance of the proposed methods via numerical simulations. Index Terms—Compound-Gaussian model, Cramér–Rao bound (CRB), estimation, parameter-expanded expectation–maximization (PX-EM).

I. INTRODUCTION

W

HEN 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 Rayleigh distribution [1], i.e., the echo can be modeled as a complexGaussian 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 reflection from a large number of randomly phased independent scatterers. However, in high-resolution and low-grazing-angle radar, the real clutter data show significant deviations from the complex Gaussian model, see [2], because only a small sea surface area is illuminated by the narrow radar beam. The behavior of the small patch is nonstationary [1] and the number of scatterers is random, see [3]. Due to the different waveform

Manuscript received May 3, 2005; revised November 29, 2005. The work of J. 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 associate editor coordinating the review of this manuscript and approving it for publication was Prof. Yuri I. Abramovich. J. Wang and A. Nehorai are with the Electrical and Computer Engineering Department, University of Illinois, Chicago, IL 60607 USA (e-mail: jwang@ece. uic.edu; [email protected]). A. Dogandˇzic´ is with the ECE Department, Iowa State University, Ames, IA 50011 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 two scales [4], [5]. To take into account different scales of roughness, a two-scale sea surface scattering model was developed, see [6]–[8]. In this two-scale model—a compound-Gaussian model—the fast-changing component, which accounts for local scattering, is referred to as speckle . It is assumed to be a stationary complex Gaussian process with zero mean. The is used to describe the slow-changing component, texture variation of the local power due to the tilting of the illuminated area, and it is modeled as a nonnegative real random process. The complex clutter can be written as the product of these two components (1) The compound-Gaussian model is a model widely used to characterize the heavy-tailed clutter distributions in radar, especially sea clutter, see [2], [6], [9], and Section II. It belongs to the class of the spherically invariant random process (SIRP), see [10] and [17]. Note that the compound-Gaussian distribution is also often used to model speech waveforms and various radio propagation channel disturbance, see [10] and the references therein. Modeling of clutter using a compound-Gaussian distribution involves these important aspects: choosing the texture distribution, estimating its parameters, and evaluating the efficiency of the estimations. Many texture distributions have been studied, but their parameters were typically estimated using the method of moments, which is statistically suboptimal, see [2]. We present 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 parameters. We compute the Cramér–Rao bounds (CRBs) for the general compound-Gaussian model and simplify them for two texture distributions in Section IV. In Section V, we verify our results through Monte Carlo numerical simulations. II. MODELS We extend the radar array measurement model in [11] to account for compound-Gaussian clutter. Assume that an -element radar array receives pulse returns, where each pulse provides samples. We collect the spatiotemporal data from the th 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., scalar pound-Gaussian clutter was considered in [15].

1053-587X/$20.00 © 2006 IEEE

X ) in com-

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

where is an spatiotemporal steering matrix of the targets, is the temporal response mais an matrix of unknown complex amplitudes trix, of the targets. Here is the number of possible directions that the reflection signals will come from, and is the number of range gate that covers the target.2 The additive noise vectors , are independent, identically distributed (i.i.d.) and come from a compound-Gaussian probability distribution, see, e.g., [3], [10] and [14]–[17]. We now represent the above measurement scenario using the are conditionally indepenfollowing hierarchical model: dent random vectors with probability density functions (pdfs):

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in [12]. Since EM algorithms often converge slowly in some situations, we propose a PX-EM algorithm. Because of the introduction of new parameter, PX-EM algorithm can capture extra information from the complete data in the PX-E step. Also because its M step performs a more efficient analysis by fitting the expanded model, PX-EM has a rate of convergence at least as fast as the parent EM [23]. , The proposed PX-EM algorithm estimates by treating as the unobserved data. First we add an auxil(the mean of ) to the set of parameters . iary parameter in the original model. Hence the augmented Note that , where and are reparameter set is . Note that and are not lated as follows: unique whereas their product is. Under this expanded model, is (for ) the pdf of

(3) denotes the Hermitian (conjugate) where the superscript transpose, is the (unknown) covariance matrix of the speckle , are the unobserved component, and texture components (powers). We assume the texture to be fully correlated during the coherent processing interval (CPI) [18]. This assumption is reasonable since the radar processing time is not too long. We consider the following texture distributions follow a gamma distribution [2], [3], [14] • gamma: • inverse gamma: follow a gamma distribution [19]–[21]. III. MAXIMUM LIKELIHOOD ESTIMATION We develop the ML estimates of the complex ampli, speckle covariance matrix , and textude matrix ture distribution parameter from the measurements , see [22]. In the following, we present the PX-EM algorithms for ML estimation of these parameters under the above three texture models. The PX-EM algorithms share the same monotonic convergence properties as the “classical” expectation-maximization (EM) algorithms, see [23, Theorem 1]. They outperform the EM algorithms in the global rate of convergence, see [23, Theorem 2]. In our problem, the computations are confined to the PX-E step of the PX-EM algorithm. The PX-M step follows as a straightforward consequence of the PX-E step.

(5) denotes the gamma function. The conditional pdfs where are unchanged, see (3). The underlying statistical prinof ciple of PX-EM is to perform a “covariance adjustment” to correct the M step. In this problem, we adjust the covariance matrix to a product of and . More specifically, we use a expanded complete-data model that has a larger set of identifiable parameters, but leads to the original observed-data model with the original parameters identified from the expanded parameters via a many-to-one mapping [23]. We present the details of the derivation of the PX-EM algorithm in Appendix A. To summarize it, in the PX-E step, we calculate the conditional expectations of the complete-data are sufficient statistics assuming all unknown parameters known from the complete data log-likelihood. In the PX-M step, we estimate the unknown parameters from these expectations. The derivation of these estimates from the sufficient statistics are explained in [12] in details. The PX-EM algorithm for the above expanded model consists of iterating between the following PX-E and PX-M steps. PX-E Step: Compute the conditional expectations of the natural sufficient statistics

A. PX-EM Algorithm for Gamma Texture , as We model the texture components gamma random variables with unit mean (as, e.g., in [3]) and , i.e. unknown shape parameter

(6a)

(4)

(6b)

hence, the unknown parameters are . (The shape parameter is also known as the Nakagami- parameter in the communications literature, see, e.g., [24, Ch. 2.2.1.4].) This choice of texture distribution leads to the well-known clutter model, see [2] and [3] and references therein. The method for deriving EM- algorithm from complete-data sufficient statistics for a similar GMANOVA model is presented 2In high resolution radar, target can usually distribute in more than one range gates, see [13] and references therein.

(6c) (6d) (6e)

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

(7a)

B. PX-EM Algorithm for Inverse Gamma Texture We now propose a complex multivariate -distribution model for the clutter and apply it to the measurement scenario in Section II. A similar clutter model was briefly discussed in [17, Sec. IV.B.3], where it was also referred to as the generalized Cauchy , are distribution. Assume that gamma random variables with mean one and unknown shape pa. Consequently, follows an inverse gamma rameter distribution and the conditional distribution of given is , see also (3). Integrating out the un, we obtain a closed-form expression for the observed data marginal pdf of

(7b) (7c) (7d) (10)

where

(7e) (7f) and find

that maximizes

The above iteration is performed until , , and conrequires maximizing (7c), verge. The computation of which is accomplished using the Newton–Raphson method (embedded within the “outer” EM iteration, similar to [26]). The conditional-expectation expression (8), shown at the bottom of the page, is obtained by using the Bayes rule, (3) and (4), and .The integrals in change-of-variable transformation the numerator and denominator of (8) are efficiently and accurately evaluated using the generalized Gauss-Laguerre quadrature formula (see [27, Ch. 5.3])

which is the complex multivariate distribution with location vector , scale matrix , and shape parameter . Here, the unknown parameters are . We first estimate and assuming that the shape parameter is known and then discuss the estimation of . Known : For a fixed , we derive a PX-EM algorithm for and by treating , as the estimating unobserved 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 for gamma texture in Appendix A. Here, the resulting PX-EM algorithm consists of iterating between the following PX-E and PX-M steps: PX-E Step: Compute

(11a) for

and (11b)

(9) where is an arbitrary real function, is the quadrature and , are the aborder, and scissas and weights of the generalized Gauss-Laguerre quadra. ture with parameter

(11c) (11d)

(8)


PX-M Step: Compute

(12a)

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denotes the Frobenius norm. Also, and where denotes a Hermitian square root of a Hermitian matrix . , the concatenated Given an arbitrary radius is uniformly disvector of real and imaginary parts of -dimensional ball with radius , tributed on the surface of a centered at the origin. Denote by and the partial derivawith respect to its first and second entries, i.e. tives of and . For any , changing the order of differentiation and well-behaved integration leads to

(12b) (16a) where (16b) (12c) (12d) The above iteration is performed until and converge. Denote by and the estimates of and obtained upon convergence, where we emphasize their dependence on . Unknown : We compute the ML estimate of by maximizing the observed-data log-likelihood function concentrated with respect to and

(13) see also (10).

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 and operators create a single column vector by stacking elements below the main diagonal columnwise; vech includes the main omits it. The Fisher information matrix diagonal, whereas (FIM) for is computed by using [29, eqs. (3.21) and (3.23)]:

IV. CRAMÉR–RAO BOUND (CRB) AND RELATED BOUNDS (18a)

In this section, we first derive the CRB with general texture pdf assumption. Then we apply it for different texture distributions, see [28]. We also consider the hybrid CRB, which is not as tight as CRB.

(18b) where eters and

A. General CRB Results

denotes the FIM entry with respect to the param, and

Denote by the pdf of the texture . Then, according to the above measurement model, is a complex spherically invariant random vector (SIRV) with marginal pdf

(14) where

(18c) is the log-likelihood function. Then the CRB for

is

(18d)

(15a)

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

(15b)

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

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derivation and give the final FIM expressions (see Appendix B for details) (19a)

for the signal parameters are uncoupled from the clutter parameters and . Hence, the CRB matrix for remains the same whether or not and are known. Similarly, the CRBs for and remain the same whether or not is known. Also, (19a) and (19b) simplify to the FIM expressions for complex , and , Gaussian clutter when see also [29, eq. (15.52)]. B. CRB for Specific Texture Distributions

(21a)

Computing the texture-specific terms in (21) typically involves two-dimensional (2–D) integration that cannot be evaluated in closed form. This integration can be performed using Gauss quadratures, see, e.g., [27, Ch. 5.3]. Here we use the gamma texture as an example. Gamma Texture: Here we use the same model as in Section III-A. After applying a change-of-variable transformain (15a) and (16), we evaluate both integrals in tion (21) using the generalized Gauss–Laguerre quadrature formula is given (see (9).) For example, the formula used to compute in (22), shown at the bottom of the page, where, to simplify the notation, we omit the dependencies of the abscissas and . In Appendix C, we derive other coefficients weights on for gamma-distributed texture. Inverse-Gamma Texture: We use the model discussed in Section III-B. In this case, (15a) and (16) can be evaluated in closed form, leading to the following simple expressions for the texture-specific terms in (21a)–(21e) (see Appendix C)

(21b)

(23a)

(19b) (19c) (19d) (19e) (19f) where

(20) and

(23b) (21c)

(23c)

(21d)

(23d) (23e)

(21e) Here, (19a) and (19b) have been computed by using (18a) and the lemma, whereas (19c)–(19f) follow by using (18b). Interestingly, the FIMs of compound-Gaussian models with different texture distributions share the common structure in (19a)–(19f) where the texture-specific quantities are the scalar coefficients in (21). The above FIM and CRB matrices are block-diagonal [see (19e) and (19f)], implying that the CRBs

is the trigamma funcwhere tion. Interestingly, the CRB matrix for the signal parameters is proportional to the corresponding CRB matrix for complex Gaussian clutter, with the proportionality factor always greater than one. , the inverse gamma texture distribution degenerAs in (14) reduces to the ates to a constant, the marginal pdf of complex Gaussian distribution in (3) with , and (19a)

(22)


and (19b) simplify to the FIM expressions for complex Gaussian clutter. C. Hybrid (HCRB) The CRB is a lower bound of the covariance of all unbiased estimators of an unknown parameter vector. However, in some scenarios, we need to assess the estimation performance quickly but not so tightly. Thus, we also consider the computation of a less optimal bound, the HCRB. The HCRB is defined in [30]

(24a) (24b)

HCRB

Note that the HCRB takes the expectation over the unobserved for the whole product of two complete data score funcdata tions while the CRB takes the expectations, respectively. This usually reduces the calculation effort at the cost of degraded bound tightness. Similarly, we omit the dependencies of the information matrix and HCRB on the model parameters. With the derivation presented in Appendix D, the information entries of the HCRB for general texture are

(25a) (25b) (25c) (25d) Compared with FIM, the information matrix of HCRB is much simpler and easier to compute. It is interesting to observe the following. • , and are decoupled from each other. The HCRB is a block diagonal matrix with three blocks. Note that in the CRB, and are coupled. is constant. It does not change over the choice of • texture models. affects in a simple way—by multiplying a con• . stant with

V. NUMERICAL EXAMPLES The numerical examples presented here assess the estimation accuracy of the ML estimates of , , and the shape parameters of the texture components. We consider a measurement scenario pulses, implying that with a 3-element radar array and . We select a rank-one target scenario with , , complex target amplitude , and (26)

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with nor, and with spatial frequency . Here, denotes the Kronecker product. The speckle covariance matrix was generated using a model similar to that in th element of the [31, Sec. 2.6] with 1000 patches. The covariance matrix of the speckle component was chosen as

where malized Doppler frequency

(27) which is the correlated noise covariance model used in [33] (see also references therein). In the simulations presented here, we . The order of the Gauss-Hermite and generselect . alized Gauss-Laguerre quadratures was We compare the average mean-square errors (mse) of the ML estimates of , and over 2000 independent trials with the corresponding CRBs derived in Section IV. We also show the HCRBs in the results. Note: we just shown the average of elements of , , and in this paper. First we study the performance of the ML estimation for gamma texture in Section III-A. We have set the shape param. The Fig. 1 shows the mse for the ML estimates eter to and and the average mse for the ML estimates of the of speckle covariance parameters as functions of . In Fig. 2, we show the performance of the ML estimation for the inverse gamma texture in Section III-A. Here, the shape . Fig. 2 shows the mse for the ML parameter was set to estimates of and and the average mse for the ML estimates of the speckle covariance parameters as functions of . In Figs. 1 and 2, the mse matches the CRBs very well when the number of observations increases, which indicates that the PX-EM is the optimal asymptotically efficient estimation for target and the clutter parameters. HCRBs show their loose estimation to the lower bound of estimation variance as aforementioned. The average signal power and clutter power can be calculated by their definitions (28a) (28b) Thus, the signal-to-noise ratio (SNR) in these examples are 6.70 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 . In Fig. 3, we show the average mse of the estimates under the inverse-gamma texture model for four different values. The results are the averaged mse among 500 independent trials. When decreases, i.e., the clutter becomes spikier, the results show that there is no much difference for the performance of estimate of , while the estimate for becomes worse and the estimate for becomes more accurate. VI. CONCLUDING REMARKS In this paper, we developed maximum likelihood algorithms for estimating the parameters of a target with

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

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

compound-Gaussian distributed clutter. The algorithms are potentially useful to mitigate sea-clutter in high-resolution and low-grazing-angle radar. The proposed maximum likelihood

estimation is based on the parameter-expanded expectation-maximization algorithm. We also computed the CRBs and their hybrid versions for the unknown parameters. Our results


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texture. Numerical simulations confirmed the asymptotic efficiency of our estimates. APPENDIX A PX-EM ALGORITHM DERIVATION FOR GAMMA TEXTURE We derive the PX-EM algorithm to estimate the parameter set given the observations . With the auxiliary parameter , the augmented parameter , and the augmented model can be set is written as (A29a) (A.29b) Denote . Instead of maximizing , we the intractable likelihood function for the measurement maximize the complete data log-likelihood

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

(A.31a) are natwhere ural complete-data sufficient statistics [25, ch.1.6.2]: (A.32a) (A.32b) (A.32c) (A.32d) Fig. 3. Average mse for the ML estimates of , , under the inverse-gamma texture model as functions of N for different values.

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

(A.32e) We first assume that is a known constant. Take derivative respectively and let these of (A.31a) with respect to , , derivatives equal to zero, we get a set of equations. Solving these

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

(see ) (A.33a)

, where and are real and Proof: Let imaginary parts of vector . By applying Lemma 1, the proof is trivial. independently uniformly disLemma 3: For and any matrices and tributed on sphere

(A.33b) (A.33c)

(B.38)

(A.33d) where (B.39)

(A.33e) Proof: Note that

(A.33f) With these estimates, we can find the ML estimate of that maximizes the concentrated complete data log-likelihood

and

are independent

By symmetry In the PX-E step, we calculate conditional expectations of sufficient statistics , , , , (see (6a)–(6e)). Then in the PX-M step, we use these expectations to calculate the ML estimates of parameters in . The iteration goes between PX-E and PX-M steps until estimation results converges.

APPENDIX B DERIVATION OF THE SCORE FUNCTIONS AND FISHER INFORMATION MATRIX (FIM)

The first equation is proved. By applying the first equation in Lemma 2, the second equation is also easily proved. Now we start the derivation of FIM. First, recall the complete data log-likelihood (18c). We can get the contribution of each parameter to the score vector through straightforward calculations

Lange et al. derived the FIM of multivariate real -distribution in Appendix B of [19]. Here we follow the same procedure to derive the entries of the FIM of complex compound-Gaussian distribution. Before starting to derive the FIM entries, we list some preliminary results that will be used in the derivation here. uniformly distributed on real sphere Lemma 1: For and any real matrices and

(B.40a)

(B.40b) (B.34) (B.40c) These rules are used in the derivation (B.35) Proof: See [19] Appendix B. uniformly distributed on sphere Lemma 2: For and any matrices and

(B.36) and

(B.37)


The entry of FIM corresponding to

is

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Finally, we prove that (B.45) and (B.46) (B.41a)

Using Lemma 2 and Lemma 3, we can get the following results:

Proof: Since , for fixed , tion of (B.40a) while and of (B.40b).

and is an odd funcare even functions

APPENDIX C CALCULATION OF EXPECTATIONS In this section, we propose the calculation method of expectations derived in Appendix A. First recall that for any well-beand SIRV real vector with pdf in haved function the form of

(C.47)

(B.42a) where for details. The entry of the FIM matrix related to directly

. See [32]

where is the surface area of the unit sphere in . See [19], Appendix A. by letting Now build a 1-1 map , 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

can be derived

(C.48) Since

(B.43) Also, see (B.44a) at the bottom of the page.

, we can get following result

(C.49)

(B.44a)

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Define

Gamma Distribution: From the pdf of the gamma distribution (4), we can get the following results easily: (C.50) (C.51a)

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

(C.52a)

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

is the digamma function. For where notation simplification, we define . In the calculations, we change variables with and use the general Gauss–Laguerre quadrature for


both inner and outer integration. The results are shown in the first equation at the bottom of the page. Similarly, see the second equation at the bottom of the page and the equation at the bottom of the next page. Inverse-Gamma Distribution: Fortunately, we have a closed form for the functions of , , and in the inverse-gamma distribution with pdf

(C.53)

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(C.54a)

(C.54b)

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and

1) Expectations: Since (D.58) Similarly to Appendix C, we build a map Then

.

(C.54c) (D.59a) The calculations yield

(D.59b) (C.55a) (C.55b) (C.55c)

(C.55d) (C.55e)

(D.60) It is not hard to get (D.61a) (D.61b) is used. Here where the recurrence relation is the surface area of a unit ball in . Before deriving the entries of the FIM, we note that the firstorder partial derivatives are

APPENDIX D HCRB A. General Results In the compound-Gaussian model

(D.56a) (D.56b)

(D.62a)

The complete data log-likelihood function is

(D.57) Let . is SIRV.

,

(D.62b)


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2) Inverse Gamma Distribution ( -Distributed Clutter):

(D.62c) We follow the same procedure of Appendix B and get (see [32] for details)

(D.67) (D.68) and

(D.69a) (D.63a)

(D.63b) (D.69b) Thus (D.63c) where

, and (D.70a) (D.63d) (D.70b)

B. Application to Specific Texture Distributions (D.70c)

1) Gamma Distribution: From gamma pdf

We can derive

(D.64) . Also

(D.65a)

(D.65b) Substitute the above results into (D.63a) and (D.63c), we get

(D.66a) (D.66b) (D.66c)

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Jian Wang (S’05) received the B. Sc. and M. Sc. degrees in electrical engineering from the University of Science and Technology of China (USTC), Hefei, in 1998 and 2001, respectively. He is currently working toward the Ph.D. degree with the Electrical and Computer Engineering Department, University of Illinois at Chicago (UIC), under the guidance of Prof. A. Nehorai. His research interests are in statistical signal processing and its application to radar and array signal processing.

Aleksandar Dogandˇzic´ (S’96–M’01) received the Dipl. Ing. degree (summa cum laude) in electrical engineering from the University of Belgrade, Yugoslavia, in 1995, and the M.S. and Ph.D. degrees in electrical engineering and computer science from the University of Illinois at Chicago (UIC) in 1997 and 2001, respectively, under the guidance of Prof. A. Nehorai. In August 2001, he joined the Department of Electrical and Computer Engineering, Iowa State University, 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. Student Award by the Chicago Chapter of the IEEE Communications Society in 1996. He was awarded the Aileen S. Andrew Foundation Graduate Fellowship in 1997, the UIC University Fellowship in 2000, and the 2001 Outstanding Thesis Award in the Division of Engineering, Mathematics, and Physical Sciences, UIC. He is the recipient of the 2003 Young Author Best Paper Award and 2004 Signal Processing Magazine Award by the IEEE Signal Processing Society.

Arye Nehorai (S’80–M’83–SM’90–F’94) received the B.Sc. and M.Sc. degrees in electrical engineering from the Technion, Israel, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA. From 1985 to 1995, he was a faculty member with the Department of Electrical Engineering, Yale University, New Haven, CT. In 1995, he was a Full Professor with the Department of Electrical Engineering and Computer Science, University of Illinois 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 University Scholar of the University of Illinois. In 2006, he assumed the Chairman position of the Department of Electrical and Systems Engineering, Washington University, St. Louis, MO, where he is also the inaugural holder of the Eugene and 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 Publications Board, member of the Board of Governors, and member of the Executive Committee of this Society. He is the founding Editor of the Special Columns on Leadership Reflections in the IEEE Signal Processing Magazine. He was corecipient 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 the 2004 Magazine Paper Award with A. Dogandzic. He was elected Distinguished Lecturer of the IEEE SPS for the term 2004 to 2005. He is the Principal Investigator of the new multidisciplinary university research initiative (MURI) project entitled Adaptive Waveform Diversity for Full Spectral Dominance. He has been a Fellow of the Royal Statistical Society since 1996.