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IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008 281

The Adaptive Coherence Estimator is the

Generalized Likelihood Ratio Test for a Class

of Heterogeneous Environments

Stéphanie Bidon, Student Member, IEEE, Olivier Besson, Senior Member, IEEE, and

Jean-Yves Tourneret, Member, IEEE

Abstract—The adaptive coherence estimator (ACE) is known to

be the generalized likelihood ratio test (GLRT) in partially homo-geneous environments, i.e., when the covariance matrix of the secondary data is proportional to the covariance matrix of the vector under test (or = ). In this letter, we show that ACE is indeed the GLRT for a broader class of nonhomoge-neous environments, more precisely when is a random matrix, with inverse complex Wishart prior distribution whose mean only is proportional to . Furthermore, we prove that, for this class of heterogeneous environments, the ACE detector satisfies the con-stant false alarm rate (CFAR) property with respect to and .

Index Terms—Adaptive coherence estimator, adaptive detection,

generalized likelihood ratio test, nonhomogeneous environments.

I. INTRODUCTION

I

N many applications such as sonar and radar, it is desired to detect the presence of a signal of interest embedded in colored noise. The standard detection problem is formulated as a binary hypothesis test

(1a) (1b) where and stand for the amplitude and signature of the target in the cell under test, respectively. In (1),

denotes the complex normal distribution of a -dimensional vector with mean and covariance matrix , and is the pri-mary data vector with covariance matrix . The training sam-ples are assumed to be independent with covari-ance matrix . The primary data vector is also independent from the training samples.

When the signature and the structure of the covariance ma-trix are both known (i.e., is known up to a scaling factor), the generalized likelihood ratio test (GLRT) was shown to be the constant false alarm rate (CFAR) matched subspace

Manuscript received october 18, 2007; revised December 10, 2007. This work was supported in part by the Délégation Générale pour l’Armement (DGA) and in part by Thales Systèmes Aéroportés. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Xiaoli Ma. S. Bidon and O. Besson are with the University of Toulouse, ISAE, Depart-ment of Electronics, Optronics and Signal, 31055 Toulouse, France (e-mail: sbidon@isae.fr; besson@isae.fr).

J.-Y. Tourneret is with IRIT/ENSEEIHT, 31071 Toulouse, France (e-mail: jean-yves.tourneret@enseeiht.fr).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/LSP.2007.916044

detector [1], [2]. The detector makes no use of the secondary data but requires prior knowledge on the noise statistics. To circumvent this drawback, an adaptive version of the detector was introduced in [3] and referred to as the adaptive coherence estimator. It consists in replacing the primary covariance ma-trix of the CFAR matched subspace detector by the estimate , which is the sample covariance matrix (SCM), and

. The detection test then reduces to

(2) The test (2) appeared to be very attractive for radar, sonar, and communication applications as it is invariant to data scaling. Later the adaptive coherence estimator (ACE) was shown to be the GLRT in a partially homogeneous environment [4], i.e., when the secondary covariance matrix is proportional to the pri-mary covariance matrix

(3) with and unknown. Additionally, in [5], the authors proved the ACE to be the uniformly most powerful in-variant (UMPI) detection test. References [6]–[8] provide a performance analysis of the ACE under homogeneous and nonhomogeneous environment. The ACE test was also presented independently for detecting targets embedded in compound-Gaussian clutter [9]. It emerges naturally as the adaptive counterpart of an asymptotic approximation of the GLRT when the structure of the clutter covariance matrix is known.

However, real-world environments lead to a large amount of heterogeneities which cannot be all embraced by the model (3). In the context of space time adaptive processing, Melvin proposed in [10] models of amplitude and spectral clutter het-erogeneities (e.g., due to clutter edges or intrinsic clutter mo-tion) that cause profound structural mismatches between pri-mary and secondary covariance matrices. In a previous work [11], we proposed a new model of heterogeneous environments in a Bayesian framework. Our aim was to have a model that allows one to keep around while having mathemat-ical tractability to derive new detectors. More precisely, the sec-ondary covariance matrix was assumed to be a random ma-trix distributed according to a complex inverse Wishart distribu-tion with mean , i.e.,

(4) This model ensures that with probability one. Fur-thermore, the parameter scales the distance between and : the larger , the more homogeneous the environment [11].

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282 IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008

We now extend this Bayesian model by introducing a power scaling factor

(5) In this way, the heterogeneity level is increased compared to (4). However, on average, we recognize the partially homogenous deterministic environment

(6)

In this letter, we show that the GLRT for the detection problem of (1) with the environment described by (5) is also the ACE.

The test is shown to be independent of the primary covariance matrix and the power ratio and hence possesses the CFAR property.

II. GENERALIZEDLIKELIHOODRATIOTEST

In this section, we show that the GLRT for the partially ho-mogeneous Bayesian environment described by (5) is the ACE. The GLRT is classically defined as follows:

(7)

where stands for the joint distribution of and under hypothesis , .

A. Distributions

This section derives the distributions required in (7). Let us denote the centered primary snapshot. Since and are independent, the joint distribution of is

(8) The density of conditional to and was set to

and thus can be written as

(9) where and stand for the determinant and the exponen-tial of the trace of a matrix, respectively. In order to derive the density of conditioned on and , note that

(10)

Using the independence of the ’s, the conditional density of is given by

(11) The conditional distribution of given is defined by the heterogeneity model (5), i.e.,

(12) with

(13) Using (11) and (12), the distribution of can thus be expressed as (see [11] for similar derivations)

(14) Finally, the joint distribution of conditioned on , is given by

(15)

B. Maximum Likelihood Estimate (MLE) of

Differentiating the logarithm of (15) and setting the derivative to zero implies that the MLE of verifies

(16) By multiplying this equation by on the left-hand side and by on the right-hand side, one recognizes a quadratic matrix equation

(17) with

(18) (19) We proceed as in [11] to solve (17). Note that the matrix is Hermitian positive definite. Consider one of its eigenvectors associated with the eigenvalue . Multiplying (17) by , we obtain

(20) Hence, is necessarily an eigenvector of , and

is an eigenvalue of associated with . As is a rank one Hermitian positive matrix, there exists a unitary matrix such that

(21)

Thus, has two different eigenvalues, and verifies one of the following equations:

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BIDON et al.: ADAPTIVE COHERENCE ESTIMATOR IS THE GENERALIZED LIKELIHOOD RATIO TEST 283

The first equation yields a value of proportional to : (23) The second equation is a second-order polynomial equation

(24) where and . This equa-tion has a unique real positive soluequa-tion whose explicit form is not required for our analysis. The remaining derivations will use (24) only.

The solution of the quadratic matrix equation (17) is thus given by

(25) with

(26) Finally, using (19) and (25), the MLE of is shown to depend on only through the matrix

(27)

C. MLE of

Let us denote

(28) Using the previous expressions (27) and (15), one obtains

(29) Then noticing that does not depend on [use (23) and (26)], the above expression can be simplified to

(30) Differentiating the logarithm of and equating the derivative to zero implies that the MLE of verifies

(31) Then, gathering the terms which depend on and multi-plying by , one obtains

(32) Using (24), we observe that the coefficient of is equal to zero. So the MLEs of and are proportional

(33)

Plugging (33) in (24), we obtain the following expression for the MLE of :

(34)

D. MLE of

Noticing that is proportional to , see (33), and that the product is constant [cf (34)], we have

(35) The MLE of under amounts to minimizing the quantity

(36) The minimum is well known to be achieved for [12]

(37) and is equal to

(38)

E. GLRT Statistic

The th root of the GLR can be expressed as

(39) and hence the GLRT is the ACE defined in (2).

III. DETECTORPERFORMANCES

A. CFAR Behavior

We show in this section that under , the distribution of the test statistic (2) is independent of and , and hence that ACE has the enjoyable CFAR property. We proceed as in [12] and consider the unitary matrix such that

(40) (41) Let us define (42) (43) (44) Then the test statistic can be rewritten as

(45) Under , has a complex normal distribution

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284 IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008

Fig. 1. Probability of detection versus SNR—Influence of.

Conditioned on , , and , the matrix has a complex Wishart distribution with degrees of freedom

(47) Therefore, the distribution of given is

(48) which is recognized as a multivariate -distribution.

We first observe that in (41) is a fixed vector. Moreover, and are independent —since and are independent— and the distributions of in (46) and in (48) are parameter free. Since the test statistic in (45) depends only on , , and , it follows that its distribution under does not depend on or , which proves that ACE is CFAR.

B. Numerical Simulations

In this section, we study the influence of on the detector performances. The dimension of the observation space is set to , and training samples are available. The probability of false alarm is set to . The detector threshold is computed from Monte Carlo runs, with a different value of at each run, drawn from the conditional distribution in (12). The signal-to-noise ratio (SNR) is defined

as SNR . Fig. 1 displays the probability of de-tection , obtained from runs, versus SNR for different values of .

From inspection of this figure, we observe that, as expected, the more homogenous the environment (i.e., the larger ), the better the detector performances. There is almost 3 dB differ-ence between the case and the case

which corresponds to the partially homogeneous model (3). This shows that the Bayesian environment (5) models a larger de-gree of heterogeneity than the deterministic environment. Fur-thermore, the figure shows that the heterogeneity level does not depend linearly on . Indeed when is small, a slight variation in results in a large variation of the homogeneity level. The trend is inverted for larger values of .

IV. CONCLUSIONS

The adaptive coherence estimator is a well-known detection scheme which has proved to be effective in a number of non-homogeneous environments. It is the UMPI test in partially ho-mogeneous environments and is known to perform well also in compound-Gaussian clutter with fully correlated texture. In this letter, we showed that it is also the GLRT in nonhomogeneous environments such that the conditional distribution of

is a complex inverse Wishart distribution with . This seems to indicate that ACE is (close to) optimum for a large class of nonhomogeneous environments, at least it is rather ro-bust to covariance mismatches characterized by scaling ambi-guities. This is to be contrasted with the marked selectivity of ACE with respect to steering vector mismatches. Indeed, ACE is known to possess strong rejection capabilities for signals whose signatures differ from the presumed ones.

REFERENCES

[1] L. L. Scharf, Statistical Signal Processing: Detection, Estimation and

Time Series Analysis. Reading, MA: Addison Wesley, 1991. [2] L. L. Scharf and B. Friedlander, “Matched subspace detectors,” IEEE

Trans. Signal Process., vol. 42, no. 8, pp. 2146–2157, Aug. 1994.

[3] L. L. Scharf and T. McWhorter, “Adaptive matched subspace detec-tors and adaptive coherence estimadetec-tors,” in Proc. 30th Asilomar Conf.

Signals Systems Computers, Pacific Grove, CA, Nov. 3–6, 1996, pp.

1114–1117.

[4] S. Kraut and L. L. Scharf, “The CFAR adaptive subspace detector is a scale-invariant GLRT,” IEEE Trans. Signal Process., vol. 47, no. 9, pp. 2538–2541, Sep. 1999.

[5] S. Kraut, L. L. Scharf, and R. W. Butler, “The adaptive coherence estimator: A uniformly most powerful invariant adaptive detection statistic,” IEEE Trans. Signal Process., vol. 53, no. 2, pp. 427–438, Feb. 2005.

[6] S. Kraut, L. L. Scharf, and T. McWhorter, “Adaptive subspace detec-tors,” IEEE Trans. Signal Process., vol. 49, no. 1, pp. 1–16, Jan. 2001. [7] C. D. Richmond, “Performance of a class of adaptive detection algorithms in nonhomogeneous environments,” IEEE Trans. Signal

Process., vol. 48, no. 5, pp. 1248–1262, May 2000.

[8] K. F. McDonald and R. S. Blum, “Exact performance of STAP algo-rithms with mismatched steering and clutter statistics,” IEEE Trans.

Signal Process., vol. 48, no. 10, pp. 2750–2763, Oct. 2000.

[9] E. Conte, M. Lops, and G. Ricci, “Asymptotically optimum radar de-tection in compound-Gaussian clutter,” IEEE Trans. Aerosp. Electron.

Syst., vol. 31, no. 2, pp. 617–625, Apr. 1995.

[10] W. L. Melvin, “Space-time adaptive radar performance in heteroge-neous clutter,” IEEE Trans. Aerosp. Electron. Syst., vol. 36, no. 2, pp. 621–633, Apr. 2000.

[11] S. Bidon, O. Besson, and J.-Y. Tourneret, “A Bayesian approach to adaptive detection in non-homogeneous environments,” IEEE Trans.

Signal Process., vol. 56, no. 1, pp. 205–217, Jan. 2008.

[12] E. J. Kelly, “An adaptive detection algorithm,” IEEE Trans. Aerosp.

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