We consider the problem of detecting a signal of interest

in the presence of colored noise, in the case of a covariance mismatch between the test cell and the training samples. More precisely, we consider a situation where an interfering signal (e.g., a sidelobe target or an undernulled interference) is present in the test cell and not in the secondary data. We show that the adaptive coherence estimator (ACE) is the generalized likelihood ratio test for such a problem, which may explain the previously observed fact that theACE has excellent sidelobe rejection capability, at the price of low mainlobe target sensitivity

Besson, Olivier

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352 IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 5, MAY 2007Detection in the Presence of Surpriseor Undernulled InterferenceOlivier Besson, Senior Member, IEEEAbstract—We consider the problem of detecting a signal of in-terest in the presence of colored noise, in the case of a covariancemismatch between the test cell and the training samples. More pre-cisely, we consider a situation where an interfering signal (e.g., asidelobe target or an undernulled interference) is present in thetest cell and not in the secondary data. We show that the adaptivecoherence estimator (ACE) is the generalized likelihood ratio testfor such a problem, which may explain the previously observed factthat the ACE has excellent sidelobe rejection capability, at the priceof low mainlobe target sensitivity.Index Terms—Adaptive coherence estimator (ACE), adaptivedetection, covariance mismatch, generalized likelihood ratio test(GLRT), sidelobe target.I. INTRODUCTIONTHE PROBLEM of detecting the presence of a signal ofinterest embedded in noise is a fundamental one in manyapplications and has been studied extensively in the literature[1]. Given the 1 output of an array of sensors—referredto as the primary data vector or the test cell—it amountsto deciding whether or not a component with array responsevector is present, in addition to noise, whose covariancematrix will be denoted as . In an adaptive detection sce-nario, training samples , which share the same covariancematrix , are assumed to be available, and used to infer thenoise covariance matrix in the primary data. Assuming thatthe environment is homogeneous, i.e., , the gener-alized likelihood ratio test was derived by Kelly [2]. Later,the adaptive matched filter (AMF) was presented in [3]. First,the GLRT for known is derived; next, the sample covari-ance matrix obtained from the ’s is substituted for toyield the AMF. Kelly’s GLRT and the AMF are consideredas “reference” detectors for this homogeneous environment.Finally the adaptive coherence estimator [4] was introducedin the case of a partially homogeneous environment for whichis only proportional to . It turned out that the ACE isthe GLRT for this problem [5], as well as the uniformly mostpowerful invariant test [6].Both detectors perform well in the homogeneous case butincur a loss of performance in the presence of steering vectormismatch (i.e., the actual array response differs from ) or co-variance matrix mismatch (i.e., ), see e.g., [7]–[11] forManuscript received August 23, 2006; revised October 5, 2006. The associateeditor coordinating the review of this paper and approving it for publication wasDr. Hongbin Li.The author is with the Department of Avionics and Systems, ENSICA, 31056Toulouse, France (e-mail: besson@ensica.fr).Digital Object Identifier 10.1109/LSP.2006.888295thorough analyses of these detectors performance in the pres-ence of both types of mismatch. A particular scenario of co-variance mismatch which is of interest to us occurs when aninterference is present, and not accounted for by the secondarydata: this is the case when a sidelobe target is present or with un-dernulled interference. This unaccounted interference may thentrigger a detection, which is an undesirable result as it corre-sponds to a false alarm. Ideally, one would like a high sensitivityto a target in the mainlobe, and capability to reject mismatchedsignals. In order to meet these conflicting goals several strate-gies have been investigated. For instance, a solution is proposedin [12] which consists of a blend of the GLRT and the AMF;a single scalar-sensitivity parameter is used to control the de-gree to which unwanted signals are rejected. Another solutionis to use a 2-stage detection scheme, see, e.g., [13], [14], wheretwo successive tests should be passed before the presence of atarget is declared. For example, the adaptive sidelobe blanker(ASB) of [13] consists of an AMF test followed by an ACE test.The principle behind this choice is that the AMF possesses anexcellent target sensitivity while the ACE has excellent side-lobe energy rejection capability. Hence, the 2-D ASB allows totrade off target sensitivity versus sidelobe rejection capability.A third alternative is to modify the hypotheses testing problemso as to account for targets with response orthogonal to underthe null hypothesis. This is the principle of the adaptive beam-former orthogonal rejection test (ABORT) of [15]. When thereis no target in the pointing direction but one in another (e.g.,sidelobe) direction, the test will more likely incline towards thenull hypothesis, which is the desired result. In [16], a similar butmore general approach is taken, in which the steering vector issubject to uncertainties.In this letter, we focus on a situation where the test vectormay contain an interference which is not accounted for bythe training samples, resulting in , whereis unknown. The classical hypotheses testing problem is thusmodified to account for this situation. The main result of thisletter is to show that the ACE is the GLRT for this detectionproblem. Therefore, this result provides a theoretical explana-tion of the observed fact that the ACE has excellent sideloberejection capabilities.II. GENERALIZED LIKELIHOOD RATIO TESTAs explained previously, we consider the problem of de-tecting a signal of interest in the presence of: 1) colorednoise with covariance matrix and 2) an interference as-sociated with an unknown steering vector , using training1070-9908/$25.00 © 2007 IEEEBESSON: DETECTION IN THE PRESENCE OF SURPRISE OR UNDERNULLED INTERFERENCE 353samples whose covariance matrix is . Formulated mathe-matically, this amounts to deciding between the following twohypotheses:(1)with the following statistical assumptions:• are independent, zero-mean complex-valued Gaussiandistributed random vectors with covariance matrix(2)• is a zero-mean complex-valued Gaussian distributedrandom vector with covariance matrix(3)Furthermore, we assume that is independent of the ’s.Note that we choose to absorb the interference power inand do not assume that has a given norm.• is the array response for the pointing direction anddenotes the deterministic and unknown amplitude.We let denote the secondary data matrix.Under the stated assumptions, the joint likelihood function ofand , under hypothesis , is given by [1](4)where under , , 1, stands for the traceof the matrix between braces and(5)is the sample covariance matrix. In order to derive the GLRT,one must maximize the likelihood function with respect to(w.r.t.) all unknown parameters, namely , under , and ,, under . Let us first observe that(6)(7)so that can be rewritten as(8)where(9)does only depend on (and possibly ). In order to maximizethe likelihood function w.r.t. , we need to maximize(10)where . For the sake of convenience, let usintroduce the quantities , , and, so that the logarithm of can be rewritten as(11)Differentiating w.r.t. yields(12)Setting this derivative to zero implies that is proportional to, i.e., . In order to obtain , let us first observe that,with , and . Hence,in (11) can be rewritten as(13)where we emphasize the dependence towards . Differentiatingthe previous equation w.r.t. yields(14)Therefore, the derivative in (14) is null when such that(15)With given by (15), it can be readily verified that(16)and therefore in (15) corresponds to a maximum of . Itfollows that the maximizer of is . Reporting thisvalue in (10), and using (15), it ensues that(17)which implies that(18)In order to obtain , we need now to maximize(19)354 IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 5, MAY 2007with respect to . It is straightforward to show that(20)Setting the previous derivative to zero, it follows that the maxi-mizer of must satisfy(21)Therefore, the solution is necessarily of the form(22)where should be chosen so that (21) is fulfilled. However, withof the form (22), it can readily be shown that(23)Using the previous equation along with (21)–(22), it ensues thatthe value of for which (21) holds true is(24)and the corresponding covariance matrix is(25)It can readily be verified that for any ,and, hence, in (25) is the maximizer of . Additionally,with the matrix of (25), one has ,and and are constants which do not depend on. Gathering these results, we finally obtain(26)Under , the likelihood function has been maximized underall unknown parameters. Under , it remains to minimize. However, it is well known [2] that(27)Consequently, the GLR is(28)and the GLRT is equivalent to(29)which is exactly the ACE!Theoretical and numerical performance analysis of the ACEdetector for arbitrary covariance mismatch can be found in [10].When and satisfy the so-called generalized eigenrelation,i.e., when -which, in the caseamounts to -, closed-form expressions for the dis-tribution of the ACE test statistic are derived in [9]. These tworeferences also provide analyses of and comparisons with theAMF and the GLRT.III. CONCLUSIONSIn this letter, we considered the adaptive detection of a signalwhen there exists a covariance mismatch between the test celland the training samples, due to a surprise or undernulledinterference. We showed that the ACE is the GLRT for sucha problem. This result thus shades a new light on the alreadyobserved fact that the ACE has excellent sidelobe rejectioncapability.REFERENCES[1] L. L. Scharf, Statistical Signal Processing: Detection, Estimation andTime Series Analysis. Reading, MA: Addison-Wesley, 1991.[2] E. J. Kelly, “An adaptive detection algorithm,” IEEE Trans. Aerosp.Electron. Syst., vol. AES-22, no. 1, pp. 115–127, Mar. 1986.[3] 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.[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] S. Kraut, L. L. Scharf, and R. W. Butler, “The adaptive coherenceestimator: A uniformly most powerful invariant adaptive detectionstatistic,” IEEE Trans. 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Adaptive beamformer orthogonal rejection test,”

Adaptive CFAR radar detection with conic rejection,”

Adaptive matched subspace detectors and adaptive coherence estimators,” in

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

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

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Performance of a class of adaptive detection algorithms

Performanceofanadaptivedetectionalgorithm;Rejection of unwanted signals,”

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Statistical Signal Processing: Detection, Estimation and Time Series Analysis.

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Detection in the presence of surprise or undernulled interference

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Detection in the presence of surprise or undernulled interference

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