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1 RD-AI6S 213 ON THE OPTINALITY OF DATA PROCESSORS FOR SIGNAL I/1 DETECTION-OVER A CLASS OF CONTAMINATED NOISES(U) TEXAS UNIV AT AUSTIN D R HALVERSON ET AL. 39 MAY 85 UNCLASSIFIED RFOSR-TR-85-U775 RFOSR F/G 9/2 NL MENOMONEEh

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3 G& MAMIE OF PUNDINGISPONSORING 9b. OFFICE SYMBOL S. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER ORGANIZATION 0Itappmembki) AFOSR NM AFOSR k. A Mg.;S Titf Sese and ZIP Code) 10. SOURCE OF FUNDING NOS1.AKWOKUI Bolling AFB, D.C ELE ME NTNO. NO.C NTO. N. 11. TITLE (inlude Securely ChmwillesaionI ON THE OPTIMALITY OF-~ ~ 2304 A5 I DATA PROCESSORS FOR SIGNAL continued on revers 12. PERSONAL AUTIIORISI111 D. R. Halverson (Texas A&M University) and Gar L. Wise 13&. TYPE OF REPORT 1S. TIME COVERED (14. DATE OF REPORT fyg'.* M.. DpJ) 1S. PAGE COUNT Inei.iOMIJMTL3QLB 19S5, May SUPPLEMENTARY NOTATION PROCEEDINGS OF THE TWENTY-SECOND ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, Monticello, Illinois, October 3-5, 1984, pp II 117. COSATI CODIES IS. SUDJCT TERMS O'Conlinut on, orm. If aeammy end lden~fy by blot* numbeti FIEL GROP *111.on.signal detection, likelihood ratio, optimal detection, lt. ABSTRACT (Ciue on news. I rw#iugy and identify fy Stee numsepri ---- PWe consider the effect induced on the data processor of a signal detection system when the underlying noise distribution functions are varied about their nominal values. We first consider the detection of a time varying deterministic signal in additive noise, and then extend our results to a more general situation in which the signal possesses a random amplitude. Our results characterize a class of contaminants of an arbitrary nominal distribution over which the data processor can be designed using the nominal distribution, and it is seen, for example, that such a class can contain distribution functions which can-differ greatly from the nominal distribution.-:, OTIC-I[ILL COPY EET 2&. DesuReSTomiAV~ILAS LOTV OF ABSTRACT 21. ABSTRACT SECURITY CLASSIFICATIO" UNCLASSIFIED/UNLIMITID 9)SAME AS RIPT. 0 OTIC USERS 0 UNCLASSIFIED % : 22& NAMIE OF RESPONSIBLE INDI1VIDUAL 22b. TELEPHONE NUMBER OPP OFICE SYMBOL Brian W. Woodruff, Major, USAF flalud., Ame Cedet (202) AFOSR/NM

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5 . AFOSR-TR ON THE OPTINALITY OF DATA PROCESSORS FOR SIGNAL DETECTION OVER A CLASS OF CONTAMINATED NOISES D. R. HALVERSON Department of Electrical Engineering Texas AIM University College Station, Texas and G. L. WISE Departments of Electrical and Computer Engineering and Mathematics University of Texas at Austin Austin, Texas ABSTRACT We consider the effect induced on the data processor of a signal detection system when the underlying noise distribution functions are varied about their nominal values. We first consider the detection of a time varying deterministic signal in additive noise, and then extend our results to a more general situation in which the signal possesses a random amplitude. Our results characterize a class of contaminants of an arbitrary nominal distribution over which the data processor can be designed using the nominal distribution, and it is seen, for example, that such a class can contain distribution functions which can differ greatly from the nominal distribution. I. INTRODUCTION Consider the discrete time detection of a signal in corrupting noise. It is well known that a typical corresponding detector consists of a data processor followed by a threshold comparator. In fact, it is also well known that under a variety of fidelity criteria the data processor is characterized in terms of the relevant likelihood ratio. We note that it is often convenient in practice to employ an appropriate function of the likelihood ratio for the data processor. However, in order to present specific results about the general situation we will focus attention on the likelihood ratio itself. An important factor which often limits the employment of the likelihood ratio within the data processor is inexact knowledge of the underlying statistical distributions. One way of resolving this difficulty is to employ a robust detector, such as the kind obtained from Huber's results (e.g. see [1]). Numerous authors have investigated various aspects of robust detection [2-6]. While robustness approaches can achieve the. desired goal of desensitizing the data processors to inexact statistical knowledge, the would-be user of such detectors quickly becomes aware of their inherent limitations (for example, the extension of Huber's results to account for realistic forms of dependency and effects of nonstationarity, the question of the loss in performance occasioned by the robustness, etc.)o For this reason, it is beneficial to consider situations where a simpler approach can be employed. In this paper we consider two commnonly encountered situations; the first consists of the detection of a time varying deterministic signal in additive (not necessarily Gaussian) noise, whereas the second generalizes this situation to include a time varying signal with random amplitude. Our results allow determining when the likelihood ratio is invariant to perturbations in the underlying distributions from their nominal values, thus admitting the employment of the Presented at the 7int-Second Annual Alerton Conference on Conmm Control, and Computing, October 3-5, 1984; to be published in the Proceedings of the Conference. 30 cation,

6 * -2- nominal distributions to design the data processor. HO: jo * H 1 : 11 II. DEVELOPMENT The situation under consideration can be modeled as a choice between where 10 and U, are finite measures defined on the Borel sets of tn. X be a a-finite dominating measure for p0 and u,. e.g. we could take X = 1 1 (+o1y). Consider first the detection of a time varying deterministic signal; that is, for a fixed element 8 of IRn we set pi(b) = po(b-{e}) for all Borel sets B of Mn. Note that the time varying signal 8 is represented as a point in IRn. We define the real, nonnegative, Borel measurable functions fo = du and f dl- and note that fl(.) = fo(--e) a.e. with respect to X. We also let S = {xe]rn :fo(x)fl(x)>o}, and note that the likelihood ratio A(-) given by A(x) = f 1 (x) exists and is positive for all xes. Note that S contains those elements of IRn for which we obtain a nondegenerate test; the cases where A(x) =0 or A(x) =- are easily checked. Let ( denote the class of all real, nonnegative, Borel measurable functions defined on IRn; we will call an element of "8 a density. We then note that a wide class W of variations in the nominal density fo( - ) is enerated by considering all densities f: ]Rn JIR such that f(.) =.- c)fo 0 Hcg(, where gew and c is a real number satisfying 0<c <1 (cf. the c-contamination class of [1]). In this case the actual density under H 0 is not necessarily the nominal density f (- 0 ) but rather an arbitrary element of V, with an associated ge' and ce[o,1). In the following we consider only those g belonging to '8. We thus are actually testing between H 0 : (1-) jo + cr O, HI: (1-) Il +cr 1, where.for each Borel set B in IRn ro(b) - f gdx and r(b) - ro(b-{e}). Equivalently, we test between the densities H 0 : (-c fo(.)+cg() HI: (I-c) fo(.-e) +cg(.-e). Let 4 -,?:, ;,,.:;., r,. :,,.,.:.. ;,.; : , _-,....-,, ;..,

7 -3- 'We are interested in the relevant likelihood ratio Ag(.) given by, for xes, x f (x-e) +c[g(x-e)-f (x-)] Ag, fo(x) +E[g(x)-fo(x) Note that A g,(x) exists for all xes via our restrictions on c. Note also that, for xes, Ag E(x) (),(glx)-folx))][g(x-e)-fo(x-b)j 3C[f(x) +c(g(x)-fo(x))] 2 [Efo(x-e) * (g(x-e)-fo(x-o))][g(x)-fo(x)] We therefore have, for xes, [fo(x) +e(g(x)-f 0(x))] 2 h )= 0 if and only if g(x-e) = g(x) f o(x-e) fo(x) This is true if and only if there exists a Borel measurable function p: IRn I XR such that for all xes we have g(x) =p(x)fo(x) and p(x) =p(x-e). Since the above equivalence is true for all ee[0,l), we have established the following result: Theorem 1: Let A be a nonempty open subset of [0,1) under the induced topology, and let gee. We then have A gc(x) = A(x) for all xes and for all eea if and only if there exists a Borel measurable function p: IRn -'IR such that for all xes, g(x) = p(x)fo(x) and p(x) = p(x-e). Note that the above result characterizes the subclass of contaminants g(-) for which the likelihood ratio A g,(x) is invariant for each xes as c ranges over nonempty subsets which are open in [0,1). This is true regardless of the nonempty open set A and in particular we can simply let c be any number in [0,1). The data processor of the detector could then be designed based on the nominal density with no degradation in performance for such contaminants g. More generally, let us consider the extended situation where H 1 is nominally governed by the (Borel measurable) density fl (- ) given by fl(x) f nfo(x-t)di(t), where for a fixed subset C of the nonzero integers the finite measure p is defined on the Borel sets of IRn and is purely atomic with atoms ({cej:cec}. Let S = {xelrn: fo(x)f 1 (x) >01. This case includes the previous one and extends it to the situation where the signal is allowed to possess a certain form of random amplitude. As before, we consider variations in the nominal density fo( - ) which lie in W. In particular, let ge be such that there exists a Borel measurable function p: In-R. with g(x) - p(x)fo(x) and

8 -4- p(x) - p(x-e) for xeg. We then note that the likelihood ratio which corresponds to the nominal density f 0 (.) is given on S by?l(x) whereas the one which corresponds to the variation?(-) (l-c)f 0 (.) +cg(-) is given, for all xes and for all ce[091), by YOlx +C Jn glx-t)-fo(x-t)]d1-lt) -9C fo(x) +cg(x)-f 0 (x)j cec fo(x-ce) +C[g(x-ce)-fo(x-ce)] fo(x) +Clg(x)-f 0 (x)c for some countable collection of.real constants k c which depend only on c,, and j. Note thdt it then follows from Theorem 1 that for each xes, Ag(x) = A(x) for all ce[0,1). We have therefore established the following result: Theorem 2: Suppose gew and that there exists a Borel measurable function p: R n -P.M with g(x) = p(x)fo(x) and p(x) = p(x-0) for all xeg. It then follows that A (x) = A(x) for all xes and for all ce[0,1). The above result is thus an extension of Theorem I to a more general situation; however, we retain only the sufficiency condition. This, of course, is the condition we often might want to employ in practice. Note that a key element in both Theorem 1 and Theorem 2 is that a contaminant which is a periodic multiple (with period equal to the signal) of the nominal density leads to an unperturbed likelihood ratio. The results of Theorem 1 and Theorem 2 provide us with an easily used tool to construct remarkedly unintuitive examples of noise contaminants, all of which employ the same optimal data processor for a wide variety of fidelity criteria. For example, let n=l and 6=0.1. If I' if xe[-1,1 ] Sfo(x) =. I[-. 1, 1 J(x) =( fx[ll '~ 0 otherwise i.e. the associated random variable is uniform on [-1,1], and if the perturbation of f 0 is given by 4.5x107 +5xl0-2 f xe 10- +' 10l ifxci - -T,T-+ T for I =-10,-9,...,l0 l,) f(x) - 0 if xe(--l) U(1 5x10 " 2 otherwise then the resultant likelihood ratios are the same over S =[-0.9,1], even though the two densities are very dissimilar. Notice in this example that f(x) is nowhere equal to fo(x) on the interval (-1,1]. For another example, let n =1, 6-2, and q ". e, e.,,%,'%' V..' '.., '-,-'.-.. ' ".. '...'..-.. ".*...-,

9 If xe[-10,10] fo (x) [1.10o(X) 0 otherwise ~1 Let D be a Cantor set (see, for example, [7]) in [0,1] with Lebesgue measure Xe(0,1). If the perturbation of f is given by Sif xed+{21) for l=-5,-4,...,4 f(x) - 0 if xe(--,-10)u(lo,-) A, otherwise then the resultant likelihood ratios are the same over S =[-8,10], even though the Cantor set results in a perturbation of f which is obviously very dissimilar to f 0 Notice that the maximum of the perturbation f can be made to exceed any preassigned real number. III. CONCLUSION We have considered the effect induced on the data processor of a signal detection system when the underlying distribution functions are varied about their nominal values. Having noted the crucial role played by the likelihood ratio in the detection system, we have therefore focused attention on the variation in the likelihood ratio itself. When testing first for the presence of a time varying deteministic signal, and then in a more general situation, we showed that the likelihood ratio is invariant over a class of perturbations of the nominal noise distribution. ACKNOWLEDGEMENT This research was supported in part by the Air Force Office of Scientific Research under Grants AFOSR and AFOSR REFERENCES 1. P. J. Huber, Robust Statistics, Chap. 10. New York: Wiley, R. D. Martin and S. C. Schwartz, "Robust detection of a known signal in nearly Gaussian noise," IE Trans. Inform. Theory, vol. IT-17, pp , Jan S. A. Kassam and J. B. Thomas, "Asymptotically robust detection of a known signal in contaminated non-gaussian noise," ME Trans. Inform. Theory, vol. IT-22, pp , Jan A. H. E1-Sawy and V. D. VandeLinde, "Robust detection of known signals," IE Trans. Inform. Theory, vol. IT-23, pp , Nov H. V. Poor, "Robust decision design using a distance criterion," ME Trans. Inform. Theory, Vol. IT-26, pp , September H.. V. Poor, "Signal detection In the presence of weakly dependent noise, part 11: robust detection," ie Trane. Inform. Theory, vol. IT-28, pp , September B. R. Gelbaum and J. M. H. Olmsted, Counterexample in AnaZysis, pp San Francisco: Holden-Day, ",..t,..."".*. -. p- ".-,d = *." - ".%S" r.' "" ""T',-*' "..-. ''""' ' - - ',. - " ".*. "...',

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