This letter considers adaptive detection of a signal in a

nonhomogeneous environment, more precisely under a covariance mismatch between the test vector and the training samples, due to an interference that is not accounted for by the training samples, e.g., a sidelobe target or an undernulled interference. We assume that the covariance matrices of the test vector and the training samples verify the so-called generalized eigenrelation. Under this assumption, we derive the generalized likelihood ratio test and show that it coincides with Kelly’s detector

Besson, Olivier

Orlando, Danilo

English

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IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 10, OCTOBER 2007 731Adaptive Detection in NonhomogeneousEnvironments Using the Generalized EigenrelationOlivier Besson, Senior Member, IEEE, and Danilo OrlandoAbstract—This letter considers adaptive detection of a signal in anonhomogeneous environment, more precisely under a covariancemismatch between the test vector and the training samples, due toan interference that is not accounted for by the training samples,e.g., a sidelobe target or an undernulled interference. We assumethat the covariance matrices of the test vector and the training sam-ples verify the so-called generalized eigenrelation. Under this as-sumption, we derive the generalized likelihood ratio test and showthat it coincides with Kelly’s detector.Index Terms—Adaptive detection, covariance mismatch, gener-alized eigenrelation, generalized likelihood ratio test, undernulledinterference.I. PROBLEM STATEMENTI N a conventional adaptive detection problem, it is desired todetect the presence of a target signal, whose presumed arrayresponse is , in a observation vector , contaminated byGaussian noise whose covariance matrix is unknown. Ad-ditionally, it is assumed that training samples are available,which share the same covariance matrix . When the environ-ment is homogeneous, i.e., when , the reference detec-tors are Kelly’s generalized likelihood ratio test (GLRT) [1], theadaptive matched filter (AMF) [2], and the adaptive coherenceestimator (ACE) [3]–[5]. These detectors have been thoroughlystudied, and their performances are now well understood in thiscanonical case, as well as in the practically important case ofsteering vector mismatch (i.e., the actual array response differsfrom ) or covariance matrix mismatch (i.e., ); see,e.g., [6]–[9]. All detectors have to face a conventional tradeoffbetween robustness to slightly mismatched signals and rejectionof unwanted signals, such as sidelobe targets or undernulled in-terference. Indeed, assume that a signal is present with signature. Then there are mainly two possibilities. Either is a perturbedversion of —which can occur, e.g., with non perfectly cali-brated arrays—and then a detection should be declared, even if, since there is a target in the look direction. Another pos-sibility is that corresponds to a sidelobe target, in which caseno detection should be declared. The performance of Kelly’sGLRT, the AMF, and the ACE have been assessed in the caseManuscript received February 9, 2007; revised March 16, 2007. The work ofO. Besson was supported by the Mission pour la Recherche et l’Innovation Sci-entifique (MRIS) of the Délégation Générale pour l’Armement (DGA) underGrant no. 06-60-034. The associate editor coordinating the review of this man-uscript and approving it for publication was Dr. Stefano Buzzi.O. Besson is with the Department of Avionics and Systems, ENSICA, 31056Toulouse, France (e-mail: besson@ensica.fr).D. Orlando is with the Dipartimento di Ingegneria dell’Innovazione, Univer-sità del Salento, 73100 Lecce, Italy (e-mail: danilo.orlando@unile.it).Digital Object Identifier 10.1109/LSP.2007.898355of mismatched signals, and they behave quite differently. TheAMF, which is an energy detector, has excellent mainlobe sensi-tivity but is highly sensitive to sidelobe signals. On the contrary,the ACE that measures the angle between the received signaland has excellent sidelobe rejection capabilities at the priceof a lower detection for slightly mismatched signals. Kelly’sGLRT appears to offer an excellent tradeoff in terms of robust-ness versus sidelobe rejection. Ideally, it would be highly de-sirable to control the rate at which the probability of detectionfalls when departs from . To this end, some strategies havebeen proposed. Reference [10] proposes a blend of the GLRTand the AMF; a single scalar sensitivity parameter is used tocontrol the degree to which unwanted signals are rejected. In[11] and [12], the actual target’s signature is assumed to be-long to a cone, whose axis is the presumed signature. The coneangle is a user parameter that can be selected so as to ensurethe desired tradeoff between mainlobe sensitivity and sidelobesenergy rejection. Another solution is to use a two-stage detec-tion scheme: the first stage uses a detection scheme that shouldlet pass most signals—so as not to miss a slightly mismatchedtarget signal—while the second stage is more selective and re-jects signals that are deemed not to impinge from the look direc-tion; see, e.g., [13] and [14]. A third solution consists in modi-fying the hypothesis testing problem, so as to include a fictitioussignal under the null hypothesis, which is sufficiently far from; see, e.g., [15] and [16]. Hence, if is too far from , no de-tection will be declared.However, the previous references consider the case whereand a single signal is impinging. In some situations,such as undernulled interference, there might be an additionalsignal in the test vector, which is not accounted for by secondarydata, resulting in , where is unknown. In [17],we considered the following hypothesis testing problem:(1)where and are independent, zero-mean complex-valuedGaussian distributed random vectors with respective covariancematrices(2)where is unknown and arbitrary. In this letter, we proposea modification of the detection problem (1), where andare now constrained to satisfy the so-called generalized eigen-relation (GER) [7], [18], which states that .1070-9908/$25.00 © 2007 IEEE732 IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 10, OCTOBER 2007In the case where , this amounts to enforcethat , i.e., that falls in a null of the asymp-totic optimal beampattern. The GER was primarily introducedby Richmond as a means to simplify the analysis of conven-tional adaptive detectors in nonhomogeneous environments andto derive closed-form expressions for the distribution of the cor-responding test statistics. However, as convincingly argued byRichmond, the GER, even though it is not likely to be perfectlysatisfied in practice, is a viable approach to modelling under-nulled interference [7], [18].Another motivation for studying the type of detectionproblem described in (1) is that it can also be helpful to dis-criminate between a slightly mismatched target signal and asidelobe target. Consider first the case where is arbitrary. Ifa single signal with response is present, and if is not closeto , then it is likely that will be assimilated to , and thus,will be in force. Therefore, we can expect this test to haveexcellent sidelobe rejection capabilities. Indeed, we showedthat the GLRT for the detection problem (1) with arbitraryis the ACE, which provided a new explanation of the behaviorof ACE. However, the price to be paid is a lower detectioncapability of slightly mismatched signals. In order to remedythis problem, introducing the constraint maybe a solution. Indeed, only those signals that are sufficiently“far” from can be assimilated to , and therefore, one canexpect a better detection of slightly mismatched target signals.In other words, enforcing the GER may offer a good tradeoffbetween robustness to steering vector uncertainties and sideloberejection capabilities.In the next section, we derive the GLRT for the hypothesistesting problem (1), assuming that the GER holds, i.e., that.II. GENERALIZED LIKELIHOOD RATIO TESTLet denote the secondary data matrix.Under the stated assumptions, the joint likelihood function ofand , under hypothesis , is given by [17](3)where(4)with , , and(5)(6)In (3), stands for the exponential of the trace of the ma-trix between braces. We maximize the likelihood function (LF)successively with respect to (w.r.t.) —under the constraint— and w.r.t. under . In order to maximizethe LF w.r.t. , we need to maximize(7)under the constraint that . Let ,, and let be anorthonormal basis for . The GER constraint implies thatfor some unknown vector . can thusbe rewritten as(8)Using arguments similar to those presented in [17], it can beshown that(9)Let us now observe that(10)where and are the orthogonal projectors on and, respectively. It ensues that(11)In order to obtain , we need now to maximize(12)with respect to . Using the readily verified fact that(13)BESSON AND ORLANDO: ADAPTIVE DETECTION IN NONHOMOGENEOUS ENVIRONMENTS 733with(14)it follows that(15)from which we infer that satisfies the following implicit equa-tion:(16)with(17)We now show that if satisfies (16), then and can be ex-pressed as functions of and only. In order to determine ,note from (16) that(18)Since from (14) , post-multiplying (18)by leads to(19)Therefore, if satisfies (16), in (14) is necessarily given by(20)Accordingly, (16) yields(21)Pre-multiplying the previous equation by and post-multiplying it by , and after simple calculations, it canbe shown that(22)where(23)Therefore, the matrix that maximizes is given by(16), with and given in (20) and (22), respectively. Re-porting this value in the LF, and after some tedious but straight-forward derivations, we obtain(24)We need now to derive a simpler expression for (24) that in-volves and only. For that, we notice thatUsing the previous equation, it is not difficult to show that(25)Consequently(26)Under , the LF maximization is terminated. Under , itremains to minimize . However, it isknown that [1](27)and therefore the GLR isGLR (28)Finally, the GLRT is equivalent toGLRT (29)which is nothing but Kelly’s detector [1]. It is amazing to seethat Kelly’s detector, which is one of the most celebrated adap-tive detectors, whose good properties and performances have734 IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 10, OCTOBER 2007been thoroughly studied, emerges as the solution of this new de-tection problem. The result presented here partly explains whyKelly’s detector performs well in the case of undernulled inter-ference [7]. Moreover, as we intuitively discussed in the intro-duction, it explains why it is less selective than ACE (in termsof sidelobe rejection capabilities) but more performant for de-tection of slightly mismatched signals, hence providing a goodsolution for trading robustness against sidelobe rejection.III. CONCLUSIONWe considered a variation on our previous work [17] on adap-tive detection in the case of a covariance mismatch due to aninterfering signal that contaminates the test vector but not thetraining samples. In contrast to [17], where the interference’ssignature could be arbitrary, we assumed here it is such that thegeneralized eigenrelation between and is satisfied. As wediscussed, this constraint may also help to offer a good tradeoffbetween robustness to slightly mismatched signals and rejectionof unwanted signals. We showed that the GLRT for this problemis Kelly’s detector, providing one more argument to use it.REFERENCES[1] E. J. Kelly, “An adaptive detection algorithm,” IEEE Trans. Aerosp.Electron. Syst., vol. 22, no. 1, pp. 115–127, Mar. 1986.[2] F. C. Robey, D. R. Fuhrmann, E. J. Kelly, and R. Nitzberg, “A CFARadaptive matched filter detector,” IEEE Trans. Aerosp. Electron. Syst.,vol. 28, no. 1, pp. 208–216, Jan. 1992.[3] 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.[4] L. L. Scharf and T. McWhorter, “Adaptive matched subspace detec-tors and adaptive coherence estimators,” in Proc. 30th Asilomar Conf.Signals Systems Computers, Pacific Grove, CA, Nov. 3–6, 1996, pp.1114–1117.[5] S. Kraut and L. L. Scharf, “The CFAR adaptive subspace detector is ascale-invariant GLRT,” IEEE Trans. Signal Process., vol. 47, no. 9, pp.2538–2541, Sep. 1999.[6] E. J. Kelly, “Performance of an adaptive detection algorithm; rejectionof unwanted signals,” IEEE Trans. Aerosp. Electron. Syst., vol. 25, no.2, pp. 122–133, Apr. 1989.[7] C. D. Richmond, “Performance of a class of adaptive detectionalgorithms in nonhomogeneous environments,” IEEE Trans. SignalProcess., 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] G. A. Fabrizio, A. Farina, and M. D. 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Pulsone and C. M. Rader, “Adaptive beamformer orthogonal re-jection test,” IEEE Trans. Signal Process., vol. 49, no. 3, pp. 521–529,Mar. 2001.[16] F. Bandiera, A. De Maio, and G. Ricci, “Adaptive CFAR radar de-tection with conic rejection,” IEEE Trans. Signal Process., to be pub-lished.[17] O. Besson, “Detection in the presence of surprise or undernulled inter-ference,” IEEE Signal Process. Lett., vol. 14, no. 5, pp. 352–354, May2007.[18] C. D. Richmond, “Statistics of adaptive nulling and use of thegeneralized eigenrelation (GER) for modeling inhomogeneities inadaptive processing,” IEEE Trans. Signal Process., vol. 48, no. 5, pp.1263–1273, May 2000.

Adaptive beamformer orthogonal rejection test,”

Adaptive CFAR radar detection with conic rejection,”

Adaptive matched subspace detectors and adaptive coherence estimators,” in

An adaptive detection algorithm,”

Asymptotically optimum radar detection in compound-Gaussian clutter,”

Detection in the presence of surprise or undernulled interference,”

Detection of a signal in linear subspace with bounded mismatch,”

Exact performance of STAP algorithms with mismatched steering and clutter statistics,”

G.A.Fabrizio,A.Farina,andM.D.Turley,“Spatialadaptivesubspace detection in OTH radar,”

Performance analysis of the adaptive sidelobe blanker detection algorithm in homogeneous environments,”

Performance of a class of adaptive detection algorithms in nonhomogeneous environments,”

Performance of an adaptive detection algorithm; rejection of unwanted signals,”

Robust adaptive radar detection in the presence of steering vector mismatches,”

Statistics of adaptive nulling and use of the generalized eigenrelation (GER) for modeling inhomogeneities in adaptive processing,”

The CFAR adaptive subspace detector is a scale-invariant GLRT,”

Adaptive detection in nonhomogeneous environments using the generalized eigenrelation

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Adaptive detection in nonhomogeneous environments using the generalized eigenrelation

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