The generalized likelihood ratio test (GLRT) is a very widely used technique for detecting signals of interest amongst noise, when some of the parameters describing the signal (and possibly the noise) are unknown. The threshold of such a test is set from a desired probability of false alarm $\Pfa$ and hence this threshold depends on the statistical assumptions made about noise. In practice however, the noise statistics are seldom known and it becomes crucial to characterize $\Pfa$ under a mismatched distribution. In this letter, we address this problem in the case of a simple binary composite hypothesis testing problem (matched filter) when the threshold is designed under a Gaussian assumption while the noise actually follows an elliptically contoured distribution. We also consider the inverse situation. Generic expressions for the assumed and actual $\Pfa$ are derived and illustrated on the particular case of Student distributions for which simple, closed-form expressions are obtained. The latter show that the GLRT based on Gaussian assumption is not robust while that based on Student assumption is

Besson, Olivier

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                               This is an author-deposited version published in: http://oatao.univ-toulouse.fr/  Eprints ID: 12170  To link to this article: DOI: 10.1109/LSP.2014.2352362 URL: http://dx.doi.org/10.1109/LSP.2014.2352362    To cite this version: Besson, Olivier On false alarm rate of matched filter under distribution mismatch. (2015) IEEE Signal Processing Letters, vol. 22 (n° 2). pp. 167-171. ISSN 1070-9908 Open Archive Toulouse Archive Ouverte (OATAO)  OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.   Any correspondence concerning this service should be sent to the repository administrator: staff-oatao@inp-toulouse.fr  On False Alarm Rate of Matched FilterUnder Distribution MismatchOlivier Besson, Senior Member, IEEEAbstract—The generalized likelihood ratio test (GLRT) is a verywidely used technique for detecting signals of interest amongstnoise, when some of the parameters describing the signal (andpossibly the noise) are unknown. The threshold of such a testis set from a desired probability of false alarm and hencethis threshold depends on the statistical assumptions made aboutnoise. In practice however, the noise statistics are seldom knownand it becomes crucial to characterize under a mismatcheddistribution. In this letter, we address this problem in the case ofa simple binary composite hypothesis testing problem (matchedÞlter) when the threshold is designed under a Gaussian assumptionwhile the noise actually follows an elliptically contoured distribu-tion. We also consider the inverse situation. Generic expressionsfor the assumed and actual are derived and illustrated onthe particular case of Student distributions for which simple,closed-form expressions are obtained. The latter show that theGLRT based on Gaussian assumption is not robust while thatbased on Student assumption is.Index Terms—Distribution mismatch, elliptically contoureddistributions, generalized likelihood ratio test, probability of falsealarm.I. INTRODUCTION AND PROBLEM STATEMENTC ONTROL of the false alarm rate is a crucial issue in mostradar systems, as a mismatch between the presumed (de-signed) probability of false alarm ( ) and the actual one hasvery detrimental effects in any post-detection step, e.g., targettracking. Usually, a desired is speciÞed and a threshold, as-sociated to a given detection scheme, is computed based on sta-tistical assumptions regarding the data under test. In practice,however, one cannot ensure that the received data will matchits assumed distribution, and therefore it is highly desirable thata detection scheme be robust to a possible mismatch betweenthe assumed and the actual data distribution. In this preliminarystudy, we investigate the relation between the designed andits actual value under distribution mismatch, for a conventionalhypothesis testing problem, namely(1)In (1), stands for the signature of the signal of interest anddenotes its unknown complex amplitude. is a random noisevector, whose assumed probability density function isherein considered to be known. The generalized likelihood ratiotest (GLRT) for the problem at hand is given by [1], [2](2)The threshold is computed from andthus depends on . Therefore, can be ensured only ifis distributed according to . In this letter, we investigatewhat becomes of when is drawn from .More precisely, we Þrst investigate the usual case wherecorresponds to a Gaussian distribution while correspondsto an elliptically contoured (EC) distribution [3], [4], [5]. ECdistributions have been used in a number of engineering appli-cations, including radar where they encompass the widely usedcompound-Gaussian model [6]. We refer the reader to [5] andreferences therein for a detailed discussion. In a second step,we consider the inverse situation, viz. the probability of falsealarm of the GLRT based on EC distributed data when appliedto Gaussian data. We provide closed-form expressions for theactual versus assumed in the case of Student distributions,and show that the GLRT based on a Gaussian assumption is notrobust at all, while the GLRT based on a Student assumption israther robust.II. ANALYSIS OF GAUSSIAN GLRT (MATCHED FILTER)When follows a complex Gaussian distribution, i.e.,, and the covariance matrix is known, theGLRT, also referred to as matched Þlter is given by [1], [2], [7](3)The probability of false alarm is related to throughwhere the superscript over in-dicates that the probability of false alarm is obtained for aGaussian distributed .Our aim is to study the robustness of the test (3) when appliedto elliptically distributed . Sincewe address probability of falsealarm in this paper, we consider only hypothesis for which, and we assume that , under , admits the followingstochastic representation [3], [4], [5](4)where means “as the same distribution as”. In (4), the vectoris uniformly distributed on the complex -sphere, whereis the size of vector . This means that can be written aswith . Note that andare independent [3], [8]. is a random positive scalar whoseprobability density function (p.d.f.) iswhere is the so-called density generator of the elliptic distri-bution and is a constant which ensures that integratesto one. Assuming that is positive deÞnite, the probability den-sity function of , under , is given by(5)where means “proportional to”. We Þrst examine how robustis the test in (3) when applied to in (4). Let ,denote the projection onto , andthe projection onto its orthogonal complement.Then, one can write that(6)where denotes the complex central chi-square distribu-tion with degrees of freedom (d.o.f.) whose p.d.f. isand is the complex beta distribu-tion, whose p.d.f. isConsequently, if the threshold is used as in (3), the actualprobability of false alarm is given by (the superscript meansthat the data follows an EC distribution with as the densitygenerator)(7)The previous formula allows one to recover the usual Gaussiancase for which , , which yields(8)Fig. 1. Actual probability of false alarm of the Gaussian GLRT when followsa Student distribution, as a function of . and .The two equations (7) and (8) enables one to relate the actual andassumed . Let us illustrate the following general formulaon the speciÞc case of a Student distribution with degreesof freedom, for which and. Then, one has (9), shown at the bottom of thepage, where, to obtain the last line, we made use of [9, 3.194.3].Therefore, the actual probability of false alarm is related to theone assuming is Gaussian distributed through the relation(10)The dependence of versus is illustrated in Fig. 1,for and . Accordingly, Fig. 2presents versus for various values of. Clearly, the Gaussian matched Þlter is not robust to deviationfrom the assumed data distribution as the actual can signif-icantly depart from the presumed one, making this detector notdesirable in practical situations.III. ANALYSIS OF EC GLRTSuppose now that, under , is distributed according to (4).In this case, assuming that is decreasing, the matched Þltertakes the form [10](11)(9)Fig. 2. Actual probability of false alarm of the Gaussian GLRT when followsa Student distribution, as a function of . Varying . .Then, using the fact that(12)it follows that(13)Consequently(14)Since , and is decreasing, we necessarily haveand thusTherefore, the integral in the previous equation is over the do-main . Finally, we get(15)Suppose now that is EC distributed with a density generator. Since the distribution of is independent of , the rep-resentation in (13) holds with nowand hence(16)For instance, when is Gaussian distributed, andit ensues that(17)Let us now apply these general formulas to the case of a Stu-dent distribution with degrees of freedom, for whichand . It is straightfor-ward to show that, in this case,where . Consequently, is obtainedas (18), shown at the bottom of the page. Hence, the thresholdcan be computed very easily from . Let us ex-amine now what happens if is used with the same thresholdbut is now Gaussian distributed. Then, we simply have(18)Fig. 3. Actual probability of false alarm of the Student GLRT when followsa Gaussian distribution. and .(19)This allows to obtain the following relation between the de-signed probability of false alarm and the actual one:(20)For illustration purposes, in Fig. 3, we displayversus for . Accordingly, Fig. 4 presentsversus for various values of . Clearly,the Student matched Þlter, when applied to Gaussian distributeddata, exhibits a very good robustness, since the actual isvery close to the designed one. In other words, even if thedata is truly Gaussian distributed, there is no signiÞcant loss inassuming that it is Student distributed as the resulting willbe quite close to the desired one.Finally, as a last example, consider a mismatch in the degreesof freedom of the Student distribution. In other words, the GLRTis constructed assuming a Student distribution with degreesof freedom (and subsequently the threshold is computed basedon this hypothesis) and it is applied to data following a Studentdistribution with degrees of freedom. Using (7) and (16), andafter some straightforward calculations it can be shown that(21)Fig. 4. Actual probability of false alarm of the Student GLRT when followsa Gaussian distribution, as a function of . Varying . .Fig. 5. Actual probability of false alarm of the Student GLRT (assumingd.o.f.) when follows a Student distribution with d.o.f., as a function of ., and .The relation between these two is illustrated in Fig. 5. Itcan be observed that the GLRT is not very sensitive to a wrongguess of the number of d.o.f. in the Student distribution, whichis a desirable feature.IV. CONCLUDING REMARKSIn this letter, we considered the (non-adaptive) detection ofa signal of interest embedded in noise. We studied the sensi-tivity of the probability of false alarm of the GLRT, when thelatter is designed based on some noise distribution, while thedata obeys another distribution. More precisely, we consideredthe case where data is assumed to be Gaussian [respectivelyelliptically] distributed while it is actually elliptically [respec-tively Gaussian] distributed. Conclusions are that the Gaussianmatched Þlter is not robust while the GLRT based on ellipticalassumption guarantees a rather close to the designed one.This preliminary work should be pursued and extended to themore relevant case of adaptive detectors.REFERENCES[1] L. L. Scharf, Statistical Signal Processing: Detection, Estimation andTime Series Analysis. Reading, MA, USA: Addison-Wesley, 1991.[2] S. Kay, Fundamentals of Statistical Signal Processing: DetectionTheory. Englewood Cliffs, NJ, USA: Prentice-Hall, 1998.[3] K. T. Fang and Y. T. Zhang, Generalized Multivariate Analysis.Berlin, Germany: Springer-Verlag, 1990.[4] T.W. Anderson and K.-T. Fang, Theory and applications of ellipticallycontoured and related distributions Dept. Statistics, Stanford Univ.,Stanford, CA, USA, Tech. Rep. 24, Sep. 1990.[5] E. Ollila, D. Tyler, V. Koivunen, and H. Poor, “Complex ellipticallysymmetric distributions: Survey, new results and applications,” IEEETrans. Signal Process., vol. 60, no. 11, pp. 5597–5625, Nov. 2012.[6] E. Conte and M. Longo, “Characterisation of radar clutter as a spheri-cally invariant process,” Proc. Inst. Elect. Eng., Radar, Sonar Navig.,vol. 134, no. 2, pp. 191–197, Apr. 1987.[7] L. L. Scharf and B. Friedlander, “Matched subspace detectors,” IEEETrans. Signal Process., vol. 42, no. 8, pp. 2146–2157, Aug. 1994.[8] Y. Chikuse, Statistics on Special Manifolds. New York, NY, USA:Springer-Verlag, 2003.[9] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Prod-ucts, ser. ser., A. Jeffrey, Ed., 5th ed. ed. New York, NY, USA: Aca-demic, 1994.[10] C. D. Richmond, “Adaptive array signal processing and performanceanalysis in non-Gaussian environments,” Ph.D. dissertation, Mass.Inst. Technol., Cambridge, MA, USA, 1996.

On false alarm rate of matched filter under distribution mismatch

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On false alarm rate of matched filter under distribution mismatch

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