ENTGINDEIING RESEARCH INSTITUJTE U1TIVERST2MY OF MICHIGAN AITJT ARBOR TEE TECORY OF SIG1AJL DTECTABILTTY PART I. TIEE GENERAL TIECRY ISSUED SEPARAPTELY: PAIR II. APPLICATIONS WITH GAUSSIAI NOISE Technical Report No. 13 Electronic Defense Group Department of Electrical Engineering By: W. W. Peterson Approved by: -/. &L)A 9 T. G. Birdsall H. W. Welch, Jr. Project M970 TASK OR1ER INO. EDG-3 CONTRACT NO. DA-36-039 sc-15358 SIGNAL CORPS, DEPATIEWNT OF TEE ARMY DEPARTMENT OF ARMY PROJECT NO. 3-99- 04-042 SIGNAL CORPS PROJECT NO. 29-194B-0 June, 1953

ERRATA FOR EDG TECHNICAL REPORT NO. 13 Part I pg. 10 The notation A2(k) means an optimum criterion such that PN(A2(k)) = k. This notation is defined on page 16. pg. 11 Line 3, The sentence should read "If at any point (PN(A), P (A)) on curve (1) a line is drawn with slope Ok given by the operating level graph, it will be tangent to the curve and will intersect the axis at the value PSN(A) - kPSN(A)." pg. 24 The seoond line from the bottom of the page should start "PSN(A2 - A1) = 0" pg. 35 Omit the xo between lines 4 and 5. pg. 40 Line 6, This should read "measurable set Bo contained in A such that P(Bo) =y Part II pg. 5 Footnote 2 should read "If exp...... etc." 2:xN pg. 357 Line 5, should read "times the amplitude squared of its envelope, etc." pg. 64 Line 1, replace "when" by "for which". Note: An introduction to the theory of signal detectability, using as little mathematics as possible and including discussions of the applications of sequential analysis as well as the types of optimum criteria discussed in Part I, has been prepared as EDG Technical Report No. 19. Enough theoretical material will be included so that this report could be used in place of Part I as an introduction to Part II,

TABLE OF COIf:[EI[TS PART I. TIFE GEIRAL THEORY Page ABSTRACT V A Ci-IOTJ,.ED.IC,[I S vi 1. COIICEPTS AI) TIEOREETICAL rESULTS 1 1.1 Introduction 1 1.2 Detectability Criteria 3 1.3 A Posteriori Probability and Signal Detectability 5 1.4 Optimum Criteria 6 1.5 Theoretical Results 7 1.6 Receiver Evaluation 8 2. I4ATHZEnLATICAL THEORY 12 2.1 Introduction 12 2.2 Mathematical Description of Signals and NToise 12 2.3 A Posteriori Probability 14 2.4 Criteria and Optimum Criteria 15 2.4.1 Definitions 15 2.4.2 Theorems on Optimum Criteria 16 2.5 Evaluation of Optirum Receivers 26 2.5.1 Introdcuction 26 2.5.2 Evaluation of Criterion Type Receivers 26 2.5.3 55 valuation of A Posteriori Probability Woodward and Davies Type Receivers 29 2.6 Conclusions 30 APPEIDIX A 31 APPENIDIX B 33 APPENDIX C 42 BIBLIOGRAPiY 428 LIST OF SYIfIBOLS 51 DISTRIB3UTIOIT LIST 55 iii

TABLE O COiiTL7ITS (cont.) PAIU II. _APPLICLTIOiTS WITH CII USSLIi NOISE (Issued Separately) ABSTRACT ACItTOWLEDGE1 E I-ITS 3. IN2RODUCTION AND GAUSSIAN NOISE 3.1 Introduction 3.2 Gaussian TNoise 3.3 Likelihood Ratio writh Gaussian INoise 4. LILIIIOOD PATIO AI) IT S DISTRIDBUTION FOR SPECIAL CASES 4.1 InYtroduction 4.2 Sigal Irlovm Ex-actly 4.3 Signal Knovmn Except for Carrier Phase 4.4 Si-maal Consisting of a Sample of Whitoe Gaussian lioise 14.5 Video Design of a B3road Band Receiver 4.6 A Radar Case 4.7 Approximate Evaluation of a Receiver 4.8 Signal ihich is One of M Orthogonal Signals 4.9 Signal hiich is Ono of M Orthogonal Sigrmals with Unknowr Carrier Phase 5. DISCUSSION OF THE SPECIAL CASES 5.1 Receiver Evaluation 5.2 Receiver Design 5.3 Conclusions APPENDICES iv

AZBSTPrAC' PARIT I The several statistical approaches to the problem of signal detectability which have appeared in the literature are showrn to be essentially equivalent. A general theory based on likelihood ratio embraces the criterion approach, for either restricted false alarn probability or miniumr weighted error type optinwum, and the a posteriori probability approach. Receiver reliability is shownm to be a funct;ion of the distribution functions of likelihood ratio. The existence and uniqueness of solutions for the various approaches is proved under general hypothesis. PAP&T II The xfull power of the theory of signal detectability can be applied to detection in Gaussian noise, and several general results are given. Six special cases are considered, and the expressions for likelihood ratio are derived. The resulting optimum receivers are evaluated by the distribution functions of the likelihood ratio. In two of the special cases studied, the uncertainty of the signal ensemble can be varied, throwing some light on the effect of uncertainty on probability of detection.

ACiOWLE ED t\IM'S In the work reported here, the authors have been influenced greatly by their association with the other members of the Electronic Defense Group. In particular, MIr. H. W. Batten contributed much to the early phases of the work on signal detectability. The authors are indebted to M1r. W. C. Fox and Mrr. Paul Roth for the proofs of Lenmma 1 and Lemma 2 in Appendix B and also to Mr. Fox for the proof of Lemma 4 and for the many helpful suggestions and corrections resulting from his careful reading of the text. The authors also wish to aclmowledge their indebtedness to Dr. A. B. Macnee, Dr. H. W. Welch, and Mr. C. B. Sharpe for the many suggestions resulting from their reading the report, and to Geraldine L. Preston for her assistance in the preparation of the text. vi

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN THE TIEORY OF SIGNAL DETECTA.BILITY PASRT I. TIM GEMJflPAL THiEORY ISSUED SEPARATELY: PART II. APPLICPTIOITS WITH GAUSS I.AT NOISE,. Cone e tS ancd Theoretical Rosults 1.1 Introduction RandCon interference plays the key role in the, theory of sigral dctectability. It not only places a limit on the ener, which a sigal must have to be detected reliably, but it also limits the bandwidth of a receiver for strong siglas, or generally the variety of sigals which can be detected consistently in a given receiver. Part I of this report presents the basic theory of detecting signals in random i;terfcrence and Part II applies it to some simple problems in design and evaluation of receivers. The signal detectability problem is represented schematically in Fig. 1.1. The operator has available a voltage varying with time, which will be referred to as the receiver input. This voltage is in some way different when a signal is present from when there is noise alone.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN TRANSMITTER NOISY CHANNEL RECEIVER OPERATOR I RECEIVER INPUT FIG. 1.1. BLOCK DIAGRAM OF SIGNAL DETECTION PROBLEM. The receiver is the operator' s tool or analyzing system; it enables him to study the input to the receiver by observing the receiver output. He can use the receiver input to his advantage only if (1) the receiver input is differcnt when there is a si`ual than when there is no sigmal, and (2) he knows enough about the signals and the noise to Eanalyze thle input so as to recognize the difforence. The operator can do better than random guessing in deci(Vlri whether or not there is a signal present only when he has information about the sigal-s, the noise, anld his receiver; this must be recognized before treating this problem. The information about the sirlial a-Jc. aebout; the noise is usually of a statistical nature because of thle random nature of noise, and the ucertiainlty as to the exact signal that will be transmitted. Signal detectability has been recognized as a statisti cal p-roblem by a number of authors. There have been two distinct approaches to the problem. The first, the criterion approach, is first presented in Threshold Signals by J. L. Lawson and G. I. Ullenbeck. The second, using a posteriori probability, Lawson and Uhlenbeck;, Ref. 1; Woodward and Davies, Refs. 2, 3, 4, and 5; Reich and Swrerling, Ref. 6; Middleton, Ref. 7; Slattery,r Ref. 8; Hanse, Ref. 9; Schwartz, Ref. 10; INorthl, Ref. 11; Kaplan and Fall, Ref. 12. 2Lawson and Uhlonbeck, Ref. 1.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN was studied by P. M. Woodward and I. L. Davies. Trhe difference between the two nlethods lies mainly in the approach. Both are presented in this report, and the very close connection between the results of the two will be demonstrated in Section 2; namrely, the basic receiver required can be the sanle for either case, only the final manner of analysis and presentation of thle output is different. The criterion approach requires less of this analysis, and has been given more attention in this report because it is somewhat simpler. 1.2 Detectability Criteria Suppose the operator is required to guess whether or not there is a signal present. He wTill, for certain receiver inputs, say that a signal is 2 present. Such receiver inputs will be said to satisfy the criterion, or to be in the criterion. Those receiver inputs which lead him to guess that there is no signal present are not in the criterion. T.here are two distinct kinds of errors which the operator mkay makxe. He may say there is a signal present if there is only noise; this is a false alarm. He my say there is only noise when signal plus noise is present; he misses the signal. One of these errors may be more serious than the other, so that they must be considered separately. It will be convenient to use the ordinary notation of probability theory. Events will be represented by letters, and in particular, the following symbols will be used for the following events: iDavies, Ref. 2., and Woodward and Davies, Ref. 3.,lWe shall assume the operator is sciontifically logical, i.e., for the same receiver input he will always give the same response. An alternative approach is described in Appendix A. ____________________3_........

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN SN There is signal plus noise N There is noise alone A The operator says there is a sigal, i.e., the receiver input is in the criterion CA The operator says there is only noise, i.e., the receiver input is not in the criterion. If B and C are events, P(B) is the probabili.ty of occurrence of event Be P(B-C) is the probability of occurrence of events B and C together, and P,(C) is the (conditional) probability of occurrence of event C if event La is known to occur. Prom the statistical inforiation given about the sigma1l and the interference it turns out to be convenient to calculate I'1,(A) and PS1(A), because these quantl;ities do not depend upon the a priori probability that a signal is present. This rill be done in Part II of this report for some interesting cases. If these probabil.ities, PI-(A) and PSIT(A) are given as well as P(SNI), the a priori probability that a signal is present, then the probability of any combination of the events in this discussion can be calculated. In fact, any three (algcbraically) independent probabilities can be used to calculate all the others. That; there are just three (algebraically) independent probabilities can be seen by noting that all of the events discussed are combinations of the four events ST-A, WNA, SIT-CA, and N.CA, and any probabilities can be calculated from the probabilities of these four. But the sunl of the probabilities of these four is unity, so only three of these are independent. Thus, for example, P(SNIA) = P(SN) PsN(A), P(WITA) = [1 - P(Si)] NP:(A), P(SI'TCA)= P(SH) PSI:T(CA) = P(SI) [1 -'S!T(A)], (1.1) P(A) = P(SI-.A) + P(N'A), PA(SIT) = A),ectC. - __ ___ ___ __ ___ ___ ___ __ ___ ___ _ 4

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 1.3 iA Postoriori Probability and Signgal Detectability As an alternative to requiring the operator to say whether a signal is present or not, the operator might be asked what, to the best of his tmowledge, is the probability that a signal is present. This approach has the advantage of getting more information from the receiving cquipment. In fact Woodward anzd Davies point out that if the operator nmakes th-e best possible estimate of this probability for each possible transmitted message, ho is supplying all the information which his equipment can give him.l The method of making the best estimate of the a -posteriori probability tthat a signal is present will be discussed in this report. A good discussion of this approach is also fournd in the original 2 papers by Woodward and Davies. It is shamcm in Section 2 that the a posteriori probability is given by the following equation: P ())&x). P(S) (1.2) Px (sr ) = () P(SN) 1 ( ( P(SN.2) where Px(SN) is the a posteriori probability for the receiver input denoted by x and Q(x) is the likelihood ratio for the same receiver input. Likelihood ratio for a particular receiver input is usually defined as the ratio of probability density for that receiver input if there is signal plus noise to the probability density if there is noise alone. It is a measure of how likely that receiver input is when there is sigial plus noise as compared with when there is noise alone. It is a random variable; its value depends upon what the receiver input happens to be. If a receiver which has likelihood ratio as its output 1Ref. 3. |2Re. 2, 3, 4, and 5.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN can be built, and if the a priori probability P(SN) is known, a posteriori probability can be calculated easily. The calculation could be built into the receiver calibration, making the receiver an optimum receiver for obtaining a posteriori probability. 1.4 Optimum Criteria An iimportant question is whether or not it is possible to find the optimum criterion for a given situation. A first step toward the answer is to define what is meant by optimum, and this definition depends upon the situation. It may be possible to put a numerical value upon the correct responses and a numerical cost on the errors. Suppose VS.,A = Value of the correct response SN.A VNI.CA = Value of the correct response N-CA (1.3) KSN.CA = Cost of the error SN-CA KN.A = Cost of the error N.A Then V = VSN.AP(SWNA) + VN.CAP(N.CA) K- Sc P(SN.CA) - IT.A ) (1.4) is the expected value of the response of the equipment for a given criterion. An optimum criterion then would be one which would maximize this expression. Since the later sections will calculate PN(A) and PSI(A), it will be an advantage to express the expected value V of the response in terms of these quantities. V=Vs AP(SN) P. (A) V.CA[l - P(SI,)] [1 - P(A)] K [- S (CAP(SN) [1-(A) - P~ P(S) (] PN(A) V |Ps() S (sT) (VSN A~ SII CA) - P(A) [1 - P(SIN)] (VN CA + KN A) + VNCA [1 - P(SN)] - KSN CA P(SQ). (1.5) L ~~~~~~~~~~~6 ______________________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Thus maximizing V is equivalent to requiring that PS (A) - PIT(A) is a maximum, where (1.6) = - P(sN) (VN.CA + y.A) P(Si) (V +K SNI.A SIT. CA Note that P(SN) is the a priori probability that there is a signal present. In another case it may be required to limit the probability of a false alarm and to minimize the probability or a missed signal with this restriction. In symbols, it is required that, P(I.A) < Po (1.7) P(SN-CA) is a minimum. This also can be expressed in terms of Pw(A), PSI(A), and the a priori probability P (SW): P(I.-A) = - P(SN)J P1i,(A) -< Po, or PN(A)< k = 1 P(SN)' and. P(SN*CA) = P(SN) [1 - PSI(A)] is a mininum, i.e., PSN(A) is a maximum. 1.5 Theoretical Results Both of the above problems of finding an optimum criterion will be discussed in later sections, and it will be shown that under very general conditions both problems have essentially the same solution. The optimum criterion consists of all receiver inputs with likelihood greater than some number D. For the first type of optimum criterion, P is the parameter in Eq. (1.6), and for the second type of criterion, P can be determined from the value of the parameter k in Eq. (1.8). It has already been mentioned that a posteriori probability is the simple function of likelihood ratio given in Eq. (1.2). Thus a receiver which could calculate the likelihood ratio for each receiver input can be used as an a posteriori probability type receiver or as either of the criterion type 7

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN receivers. Part II of this report, which treats some specific cases, deals only with the likelihood ratio. 1.6 Receiver Evaluation1 Usually a receiver is judged on the basis of probability of false alarm if no signal is sent, i.e., PN(A), and the probability of detection if a signal is sent, PSN(A). The reliability of any receiver in any given situation can be summarized in one graph, called the receiver operating characteristic, on which P S(A) is plotted against PN (A). For any criterion and any fixed set of signals, there is fixed value for PsrI(A) and a fixed value for PN(A). Thus the criterion can be represented as a point on the receiver operating characteristic graph. A criterion-type receiver may operate at any level (i.e., any value of 3 or any value of K), and hence is represented by a curve. Two types of optimum criteria have been discussed, and the graph points up the relation between the two. In Fig. 1.2 curve (1) is based on optimum operation for which PSQN(A) is maximized for PI(A) fixed. Thus, no receiver can operate above the first curve. The third curve is a lower limit in operation found by rotating the optimum curve about the center point of the graph; it would result if an optimum receiver operator minimized PSN(A), i.e., said no whenever he should say yes, and vice versa. No receiver, no matter how poor, can be made to operate bet.w the third curve. The diagonal could be achieved by turning the receiver off and guessing, in which case PSN(A) = PI(A). In the next section it will be shown that the derivative of curve (1) sketched in the lower plot, is the operating level p of the optimum receiver; that is, if the slope at some point is A, then the corresponding optimum criterion |Only evaluation of criterion type receivers is discussed here. Evaluation of an a posteriori probability type receiver is considered in Section 2.5. ____ ___ ____ ___ ____ ___ ____ ___ ___8 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

~S-I 1-9 V 3 r L9-~9-V OL6 -V 0.6 0.5 0.4 0.3 0.2 1.0 PN(A ) 10 1 l I 2 W 0.. z I 0.8 Id 0.6 0.4 0.3 FIG. 1.2 TYPICAL RECEIVER OPERATING CHARACTERISTIC, 9

S~g-i-9 IN3r 8 99-9-vl OL6-J 1.0 0.9,- 00~~~~0.8 PsN (A2 (K) — t — 0.7 0.6 PsN(A2 (K)) -K K / I rol 0.5 A SMALLER VALUE,. PSN(A)19K PN(A 0.4 0.3 (3,o 0.2 1.0 0.1.2.3.4.5,6.7.8.9 1.0 PN(A) iO 8 6 I~~~~~~~~~~~ I~~~~~~~~~~~~~ -I 4 0. -I 0.4 0.3 FIG. 1 3I I2 I~~~~~~~~~~~ I~~~~~~~~~ \-I 0.8 13K — ~' ~- 0.6 -0.45 FIG. 1.35 TYPICAL RECEIVER OPERATING CHARACTERISTIC. 10

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN is made up of all inputs which have likelihood ratio greater than or equal to i. The relationship between the first and second types of optimum criteria is graphically illustrated in Fig. 1.3. If at any point (PN(A), PSN(A)) on curve (1) a line is drawmrn with slope P, it will be tangent to the curve and will intersect the axis at the value PSt (A) - pD (A). This is the quantity to be maximized for the first type of optimum criterion, and if a line with the same slope is drawm through any other point on or between curves (1) and (3), it will cut the axis below the point where the tangent cuts the axis. Thus, curve (1) is not only the curve for the optimcum of'the type when P;(A) is bounded and P S(A) SIT max imized, but also the curve for the optimum criterion trhen values are placed on the operator's responses. A non-optimun receiver can be evaluated in a given situation if its receiver operating characteristic is drawn together with that of the optimum. One receiver is better than another over a range if it is closer to the optimum than the other. In some instances the optimlur curve for a given situation will nearly match another receiver's operation in the same situation except that the optimum will require less signal energy. In this case, the non-optimum receiver can be given a db rating for that situation. Each application of the theory treated in Part II of this report is accompanied by the receiver operating characteristic of the optimum receiver. _11

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 2. MATIE MATICAL TIEORY 2.1 Introduction The method for handlin the signal detectability problem mathematically is described in this section. The first step is the presentation of the appropriate mathematical description of the signals and noise. In these terms the signal detectability problem is restated in several forms discussed in Section 1 of this report. It is then shoam that in each case, if the likelihood ratio canu be determined for each receiver input, the problem is essentially solved. Thus the conclusion is that the receiver design problem should be treated in terms of likelihood ratio; this is the approach used in Part II. 2.2 Mathematical Description of Si ls and Noise Any receiver input, noise or signal plus noise, is a voltage which is a function of time. Thus we shall be considering a set of functions. In this report it will be assumed that the receiver input is limited to bandwidth W, and that the observation is of finite duration T. By the sampling theorem, any such function is completely determined when its values at "sampling" points spaced 1/2W seconds apart through the observation interval are known. There are 2WT sampling points in all. Thus a receiver input can be considered as a point in a 2WT dimensional space, the values at the sample points being taken as coordinates. Let us call the space R. If there is noise at the receiver input, the receiver input voltage may usually be any of an infinite number of functions, i.e., any of an infinite number of points in the 2WT dimensional space R. With Gaussian noise any point is 1Shannon, C. E., "Cormunication in the Presence of Noise," Proc. IRE, Vol. 37, p. 10, January 1949; also Appendix D of Part II. l _ _ _ __ ~12......_

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN theoretically possible. It is a matter of chance which one occurs. Thus it appears that the appropriate way to describe the noise is to give the probability density for points in the space of receiver inputs. The same is true when there is signal plus noise, so that we shall deal with the space R and two probability density functions, fN(x) for the case of noise alone, and fSN(x) for the case of signal plus noise. Here x denotes a point of the space R. In a practical application, inforlmation will be given about the signals as they would appear without noise at the receiver input rather than about the signal plus noise probability density. Then fsN(x) must be calculated from this information and the probability density function f,,(x) for the noise. The noise and the signals will be assumed independent. If the signals can be described by a probability density function fs(x), fs (x) = j fN(x-s) f(s), (2.1) R where the integration is over the whole space R. The receiver input x(t) could be caused by any signal s(t), and noise x(t) - s(t). The probability density for x is the probability that both s(t) and x(t) - s(t) will occur at the same time, sunned over all possible s(t). If the signals cannot be described by a probability density function, a more general form must be used, in which the signals are described by a probability measure, PS; the formula for this case is fSN(X) = f fN(x-s) d Ps(s). (2.2) R This is what is called a Lebesgue integral, and it means essentially to average 1We shall assume that the probability density function exists. See Appendix A. 13

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN f (x-s) over all values of s in the whole space weighting according to the probability PS of the points s appearing as signals.l 2.3 A Posteriori Probability The approach of Woodward and Davies2 to the signal detectability problem is to ask the operator, "wMhat is the probability that a signal is present?" ie is to give the probability, using knowledge of the receiver input, i.e., he gives the a posteriori probability. If the probability density functions are continuous, the a posteriori probability P (SN) can be found for any particular receiver input x. Bayes' theorem3 is used, but not directly, since PsN(x) and PN (x) are both zero. Consider a small sphere U with radius r and center x. Then PU(SN) can be obtained by Bayes' theorem, and P (SN) can be defined as the x Px(SN) = im PU (SIT). (2.3) r -o Denote by P(SN-U) the probability that signal plus noise will be present and the receiver output will be in U. Then P(SN'U) = P(SN) ~ PSN(U) = PU(SN) ~ P(U) (2.4) and P(U) = PSN(U) P(SN) + PN(U) (1- P(SN)) (2.5) Solving for PU(SN), P(SN) PSI (U) P(sN) +P-N (2.6) P(SN) PSN(U) + (1 - P(SN)) Cramer, Ref.14 pp. 62, 188. 2Woodward and Davies, Ref. 3. Cramer, Ref.14t, p. 57. - ~~~~~~~~~~14

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY-.OF MICHIGAN By the definition of probability density function, PS,1 (U) = f SNI() dX U P((U) = f f(x) d, (2.7) U where the integal is really a multiple integral over the volume of the sphere U in the n-dimensional space. Then psT(U) SN(f)u dx -P U, (2.8) P (U) = f f (dx.... U and if fSN (x) and f. (x) are continuous, lim Ps( f()= (x). (2.9) The ratio of probability densities fSIN(x)/fN(x) = C(x) is called the likelihood ratio. It follows that Px(SN) r- 0 PU(S) = P(SN) J(x) + [1 P(SN)] This is the existence probability as defined by Woodward and Daviesl Notice that the likelihood ratio ~(x) is the all-important quantity. Px(SN) is a simple monotone increasing function of the likelihood ratio. Therefore if P(SN) is knowm and if the receiver produces 2(x), a calibration will convert this to PX(SN). 2.4 Criteria and the Optimum Criteria 2.4.1 Definitions. Suppose the operator is only required to guess whether or not there is a signal present. For certain receiver inputs he will guess there is a signal present. These receiver irnputs form a subset of 1Ref. 3. 1

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN the space R of all possible receiver inputs. Let us call this subset the criterion and denote it by A. That is, a point x is in the criterion A if the operator will say there is a signal present when x occurs as receiver input. It will be convenient to have a symbol for each of the two types of optimum criteria described in Section 1.4. The first type will be denoted by A1(f); that is, A1(P) is any subset of R such that for fixed 2 0, PsI [A1(P)] - PN [A1(3)] is maximum. (2.11) The second type will be denoted by A2(k); that is, A2(k) is any subset of R such that P11 (A2(k)) < k and (2.12) PSN (A2(k)) is maximu. The likelihood ratio ~ (x), which is defined as ratio of the probability density functions, fsN(x)/fj(x) plays an important role in the following discussion. It is a measure of hor much more likely the receiver input is to be if there is signal plus noise than if there is noise alone. 2.4.2 Theorems on Optimum Criteria. The optimum criterion is closely related to the likelihood ratio. For the first type of criterion the connection is given by the following theorems. Theorem 1: Denote by A the set of points for which the likelihood ratio.L(x)_> P. Then A is an optimum criterion A1(f). Proof: The condition that A be an optimum criterion A1(f) is that PsN(A) - p PN(A) is maximum; i.e., for any other set B of receiver inputs PS (A) - P PN(A) 2 PSN(B) - 3 PN(B). S__ 16...

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN PSN (A) - ( P 1(A) = J fSN(x) dx (3 fl (x) dx A A (2.13) = I [ fSN3(X) ( dx where the integration is over the set A, and so is really a multiple integral over a part of the space R which has 2.WT dimensions. Let B be any set different from A. Denote by A-B the set of points which are in A and not in B, by B-A the set of points which are in B but not in A, and by BnA the set of points which belong to both A and B. Then since A is the union of A-B and AflB, and A-B and AnfB have no points in connon, PSN(A) P- P P(A) = f [ fS X) - ( dx A J f r[iS'(X) - fN(X) ] dx (2.14) ACnB + f[fST(x) -P f'i(X)] dx A-B Likewise PsIT(B)- PITc-(B) = f [ SSI(x) -P (f(X)] dx AnB ~~AflBnD~~~ ~(2.15) + [sIr(x) - f' (X)] dx B-A Thus PsI(A) - 3 PN(A) - [P I5(B) - PFT(B)] (2.16) f [r(4x) - 3 fN (x)] dx: - [fSN(x) - (x f(X)] dx A-B B-A.....-~ 17

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The points in A-B are in A, and so for them fSlr(x)/fT(x) = I (x) _> 3, so that fsN(x) - 3 f (x) > 0, and the first integral in Eq. (2.16) is not loss than zero. The points in the set B-A are not in A, so f s(x)/fN(x) < i, and the second integral in Eq. (2.16) is no greater than zero. Thus PsN(A)- PI(A) PSIJ(B)- PI(B), (2.17) PSI(A) - D I.(A) is a raximurm, and A is an optimnum criterion Al(3) There is not a unique optimum criterion A (D). In the first place "optimumi' was defined in terrlis of probability. Thus a change in A1(3) which wotuld not change P'T [A 1()] or PN [AL(,3)] would result ir an equally good criterion. Such a change ight consist of adding or taking out a single point, a finite number of points, or generally any set of probability zero.1 ciore insight into the uniqueness is given by the followinlg theorem. Theorem 2: If A is an optimum criterion A1(1), then the set of points in A for which 2(x) < has probability zero, and the set of points not in A for which _ (x) > e has probability zero. Proof: We will show that cany criterion which does not have these two properties is not an optimun criterion. Consider any criterion B with a subset C, of non-zero probability, such that the likelihood ratio of each point in C is less than 5. There is a positive number E and a subset CE of C, having non-zero probability, such that e(x) < 3 - E for the points in CE. If this were not true, then for any positive srnkll number E, the subset CE would have probability zero. These subsets CE are monotone, that is, A set E will be said to have probability zero if both PSN(EB) and P(E3) are zero. ____________________________________ lb~~~~JIEI

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN if E2 < E, then CE contains CE1, and, since C contains no points with likelihood ratio equal to., the union of all CE is C itself, and would have probability zero. As in Eq. (2.14), PSI(CE)- Pi(CE) = J [fSNxPfiX1X= (X)[(X) - ]dx CIE CE and since -L(x) < D - E or Z(x) - D < - E, PSN(CE) - p P(Ce) < - e f(x) dx = - PN(Ce). (2.19) CE Therefore, if PN(C ) > 0O PS(CE) - p PI,(CE) < O. (2.20) But CE is a subset of A, and therefore PSN(B - CE) - p PN(B - CE) > PsN(B) - P PI(B) (2.21) and B is not an A1( ). It can be shown in an analogous manner that if there is a set D of non-zero measure outside of criterion B such that L(x) > P in D, then there is a subset DE of D such that PSN(DE) - p PIN(DE) > 0 (2.22) and therefore PSN(BUDe ) - p PIN(BUDe ) > PSN(B) - PN(B), (2.23) and B is not an A1(B). 1Cramer, Ref. 14, p. 50, Eq. 6.2.3; and p. 77, paragraph 8.2. 19..

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN This theorem says nothing about the points for which ~ (x) = I. It is not hard to shor that PSN(A) - P P?(A) is not affected by including or excluding points where P (x) = t. Thus a criterion A (P) must include all points for which f(x) > P (except perhaps a set of probability zero), none of the points where (x) < 3 (except perhaps a set of probability zero), and it ray or may not include a point for which ~(x) = 3. In the most general case, when the noise is Gaussian, the following two theorems shor the uniqueness of A1(f). Theorem 3: If the probability density function for noise alone, fIT(x), is an analytic function, then the set of points for which S(x)= 6 has probability zero.l1 A function is said to be analytic if it is analytic in the ordinary sense when considered as a function of each single coordinate. The proof of the theorem is quite involved, and so it is given in Appendix B. Theorem 4 follows immediately from Theorem 2 and Theorem 3. Theorem 4: If the probability density function for noise alone fN(x) is analytic, any two optimum criteria A1(p) can differ only by a set of probability zero. Now let us turn to the second type of optimum criterion. Theorem 5: Let A be a set such that if x is in A, the likelihood ratio ~(x) > }, while if x is not in A, a(x). Then if?1(A) = k, A is an optimum criterion 2 (k). Proof: An optimum criterion A2(k) must satisfy the conditions PN(A) < k, and PsN(A) is maximum. The first is satisfied by hypothesis. Suppose B is any other set such that PN(B) < k. Denote by A-B the set of points in A which are not in B3, by B-A 1A little more is needed in the hypothesis for Theorem 3 than that fN(x) is analytic. See Appendix B. 20

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN the set of points in B which are not in A, and by BnA the set of points common to B and A. Since A is the union of A-B and AfB, and since A-B and AlB have no points in common, PN(A) = f f(x) dx= fN(x) dx + f N(x) dx A A-B AOB (2.24) P(A-BN(A-B)+ P(AnB) = k Likewise P (B) - P(B-A) + P (ArOB) < k, (2.25) N N and thus P (A-B) > PN(B3-A). (2.26) Also, (B-A) (2.27) PSN(BA) = fsN(x) dx (.27) BfSN(x) and since any point x in B-A is not in A, (x) = (x) < 3 and fN(x) PMsI(B-A) -f x f (x),x 43 B-A B-A or PsN (B-A) S 3 PN(B-A). (2.28) Likewise PSN(A-B) > 3 PN(A-B). (2.29) Collecting Eqs. (2.26), (2.28), and (2.29), PSNZ(B-A) < 3 P N(B-A) < 4 PN(A-B) < PsN(A-B) (2.30). 21.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN As in Eq. (2.24), PS,1(A) = j Sf(x) dX = fJf(X) ax + fSf(x) x A A-B AnlB (2.31) Ps (A-B) + PSI (AnB) and PSN(B) = PS(B-A) + PSN (AnB). (2.32) Therefore PSN (A) - PSN (B) = PSN (A-B) - PSN(BA). (2-33) From nIqs. (2.30) and (2.33) it follows that PSN (A) 2 PSN (B) (2.34) and PSN(A) is a maximum. It follows from Theorem 5 that every optimum of the first type, A1(3), is an optimum of the second type. More precisely, if set A is an optimum of the first type it is associated with the fixed e for which it is an A1(3). By Theorem 2, the likelihood ratio in A is not less than P, and outside A the likelihood ratio is not greater than 3, except on a set of probability zero. But the introduction or omission of such a set has no effect on PsN(A) or PN(A). Since P1N(A) has some value, call it a; A will be an A2(a) by Theorem 5. Theorem 6: For every k between 0 and 1 there is an optinum criterion of the first type Ak, such that P (Ak) = k. Proof: For each value P we consider the maximal A1(3); by Theorem 2 this is the set consisting of all points of likelihood ratio not less than 3;, = {x (x)) }. (2.35) 22

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Now if for k there is a 1 such that PN(I4) = k, then because M4 is an A1(p) the proof is complete. Next we point out that 140 is the whole space R and MClO is the empty set, and therefore P (M ) =1 and PI(MD4 ) = O. For any value of k, if there is no Mi. such that PN(14 = k, let * = rain { J IP( (M k} = gZb { IPNJ (M) < k} that is, PN(I-IX) > k and if P > D*, P(1T(M) < k. Thus the jump in PI is due to those points in I4 * for which < (x) = 1*. Because the probability density functions exist, every point has probability zero and therefore there is a subset S of these points with ~(x) = 1* for which Pi = PN(Mi3*) - k. This is shown in Appendix B (Lerma 4). Removing this subset from nI*, P(4i2* - S) = k. (2.36) Because 14 - S satisfies Theorem 1, it is an A1(D*). Of course, by Theorem 5, it is an A2(k) also. The foollaring theorem completes this circle of proof. Theorem 7: For any k there is a Pk such that every A2(k) is an Al(13k). Proof: Let A be any A2(a). By Theorem 6 there exists a D1 and an Al(kw), which we will denote by A*, such that PIT(A*) = k. Then by Theorem 5, A* is also an A2((k), and hence for both A and A*, PS is maximum and P < k. Therefore PSI(A*) = I'Si(A) (2.37) P (A*) = - PIr(A) (2.) Itkultiplying'Eq. (2.38) by - P3k and adding gives PST(A*) - P, P.~(A-) < Sl(~)- k P'(A)' (2.39).. 23

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Since A* raximizes this expression, the equality must hold, and A is also an Al(k). In summnary, these theorems show that P can be written as a mulLtivalued function of k and that k can. be written as a multivalued:function of t. These relations can be sharpened somewhat. Theorem 8: Let a < b be two values taken on by ~(x). If no set of the form I |%1X < I (X) < }X 2 for a 5 X1 < e < b has probability zero, then Pk is a single valued function of k on some interval I, with a < Ok_ b, and d PSN(Al(1Bk))/dk exists and equals Pk for every k in I. Proof: 1) In general, if a function is monotone on an interval and its range of values is also an interval, then it is continuous. If it were not, then at some point the left and right hand limits would be unequal, which would introduce a gap in the range of values, contradicting the hypotheses. 2) If Pkl > 1k and if the interval from ik1 to P1k contains a subinterval of [ a b] of length greater than zero, then k2 > k1. There are, by Theorem 6, criteria of the first type Ai (for i = 1, 2), which, by Theorem 2, may be chosen so that A. contains all points for which C (x) > Ak and no points for which i 2 (x) < Ski. Also PN(Ai) = ki by Theorem 5. By applying PN to the equation A2 = A U (A2 -A1), one obtains r2 = 1 + PN(A2 - A1). If PN(A2 - Al) = 0, then from Eqs. 2.7 and the fact that i(x) is bounded on A2 - A1, it follows that PsH(A2 - Al) - 0 also. But, by hypotheses, Al - A2 cannot have probability zero. Hence k2> k1. l _ _ __ _ __ _ __ _ _ __ _ __ _.... _24l

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 3) Let I be the set of points k for which at least one Pk is in the open interval from a to b, and let Ok denote the possibly multivalued function defined on I. Then 2) says that Pk is both single valued and monotone, and Theorems 1 and 6 imply that the range of values of 1; is the interval from a to b. Hence I is an interval, for if it were not, there would exist three values kl < k2 < k3 with only the middle one not in I. Then Okl< a2 < k3 and Pk2 would not be in the interval from a to b, yet the other two would be —a contradiction. Thus 1) can be applied to Bk and Pk is therefore continuous on I. 4) To form the derivative, let D = A1(k) Al(Pk ) if k k o0 (2.42) =Al(k ) - Al(k) ifk ko Then PSN(Al(pk)) - PSN(Al(ko )) PsN(D) lim = li - (2.43) k-*ko+ k - ko k ko+ k- ko Since k> k < o and in D, k < (x) <, kf (x) 0 0 < fSN(x) E Ok fN(X). But SN (D)= fN(x) dx = e(x) fN(x) dx (2.44) D D and n)= = f N( dx (2.45) D 25

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN and therefore 3 P.(D) < PSN(D) < ko P (D). Similarly if k < k ON (D) PSN () kPI (D)* T-hus Ps,? ((D) 1 im k 2~46 rn — k - k (2.46) by virtue of the result that h is a continuous function of k. 2.5 Evaluation of Optimum Receivers 2.5.1 Introduction. This section treats the problem of deterrining how well a given receiver will perform its task of detecting signals. For the criterion type receiver, the probability of false alarm if no signal is sent, PN(A), and the probability of detection if a signal is sent, PSrT(Al), gsive a good measure of receiver performance. For the a posteriori probability type receivers, the average or mean a posteriori probability with signal plus noise and with noise alone describe the receiver's ability to discriminate between signal plus noise and noise alone. 2.5.2 Evaluation of Criterion Type Receivers. For simplicity, let us restrict this discussion to the case in which the probability density function for noise alone, fN(x) is analytic. Denote by FSN(P) the probability that the likelihood ratio C(x) is equal to or greater than P if there is signal plus noise, and similarly, let F1(D) be the probability that e (x) is equal to or greater than 5 if there is noise alone. These are the complimentary distribution functions for 2(x). Then for any A1(1), PSN (A1(P)) = FSN(5), and (2.47) PN (A1i()) = FN(), (2.48) 26

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN because the set of points for which i (x) >_ -, and differs front any A1(3) only by a set of probability zoro (Theorem 4). By Theorem 7, every A2(k) is an A1(). The Bk corresponding to k can be found from Eq. (2.48) PN(A1(Pk)) = FN(P3k) = k (2.49) Then PsN (A2(k)) = FSN (k) (2.50) Thus, if the distribution functions FsN(p) and FN(p) are known, any criterion type receiver can be evaluated. It turns out that not both FSN(B) and FN(p) are necessary. Theorem 8 states that d FN() =, (2.51) since PsN(Ai(tk)) = FSN(pk), and k = FN(k). Thus, if FN(1) is known, FSN(3) can be found by integrating Eq. (2.51).1 FSN() =- y d FN( Y) (2.52) As an alternative, FSN(3) might be given as a function of FN(j); this is the receiver operating characteristic graph. Then ~ can be found from Eq. (2.51); i.e., P is the slope of the graph.'The change in sign is because the functions FSN(B) and FN(1) are complimentary distribution functions. If the density function associated with FN(t) is g(3), then d. F - g(P) and FSN(,) = g() d. 27

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN A corollary of Theorem 8 is the following: The nth moment of the distribution for noise alone is the (n-l)st moment of the signal plus noise distribution. n-1. n-1. N y N Y S() N.5JS -CO _~CO) _CO As an example of the application of this corollary, note that the mean value of likelihood ratio with noise alone is always unity. If the variance with noise 2 2 alone is crJ, the second moment of F (P) is 1 + -; then the mean of the N N signal plus noise distribution is 1 + 2N, and the difference of the means is O'N N For detection corresponding roughly to Fig. 2.1, the difference of the means of the two distributions must be of the order of the standard deviation of the distributions, so that 2 0, (2.54) N N F FIG. 2.1 RECEIVER OPERATING CHARACTERISTIC -F28: Foro'~~~~~~~~~~~~N:I (/

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN or the variance of the distribution with noise alone must be of the order of unity. For better detection, oN must be greater. 2.5.3 Evaluation of A Posteriori Probability Woodward and Davies Type Receivers. Davies proposes the mean a posteriori probability as a measure of the efficiency of a receiver. The mean a posteriori probability is defined as: /sN (Px(SN)) = f Px(SN) fs(x) dx (2.55) R N (Px (SN)) Px (SN) f (x) dx (2.56) R These can be evaluated if the distribution functions FSN(P) and F N() for likelihood ratio are klmovm. Since P (SN X Px (SN) =P(SN).P (x) (2.57) P(sN) jz (x) + z- P(s)'I the mean a posteriori probabilities are ALSN(P (SN)) = f _+ - d F (y), and (2.58) x j P(SN) + 1 - P(S) (y) anSN kLN (Px(SN)) = Ty P(S + )T P(SN)S) d F (y) * (2.59) Davies presents the formula FSM [Px(SN)] +1 -P(SN) /N [Px(SN)] =1 (2.60) which enables one to calculate easily either one of the mean a posteriori probabilities once the other has been calculated. 29

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 2.6 Conclusions It is possible to combine the most common statistical approaches to the theory of signal detectability into one general theory. In this theory likelihood ratio plays the central role: the result of the theory is that a receiver built so that its output is likelihood ratio can be adapted easily to accomplish the task specified in any of the well-nownm approaches to signal detectability. If the probability distribution of likelihood ratio is known, then the receiver reliability can be evaluated. In Part II of this report, likelihood ratio and its distribution functions are calculated for a number of specific cases, and the problems of receiver design are discussed. 30

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN APPENSDLX A It was assumed throughout the discussion of the criterion approach to signal detectability that for any given receiver input, the operator would always give the same response. This is certainly not the case with threshold signals and a huarn operator. A more realistic approach might be to assume that for any receiver input x, the operator would say with probability p(x) that there is signal plus noise. Finding the optimaum receiver would then consist of finding the optimumu p(x). This approach does not lead to aly interesting new results; if p(x) = 1 on an optimum criterion and zero on its compliment, then p(x) is opt imwnu. Tlhe theorems on signal detectability are proved in Section II in more general form thman has yet been found necessary in an application. However, they can be generalized somewhat, and this appendix discusses some of the possibilities. It is certainly possible to consider more general spaces of signals. Ay space on which a probability measure can be defined might be used. In order to prove the theorems on optimumn criteria, harever, some sort of likelihood ratio seems necessary. One possibility is to assume the measure P (A) and the random variable E (x) are given and to define PSN(A) through the integral PSIT(A) J- e (x) d P(A). (A.1) A The, mean value of A(x) must be unity, of course. If the space is a Euclidean space of finite dimension, then it is possible to define an arbitLrary measure through distribution fnctions. These 31

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN functions, being monotone, have: a derivative almost everywhere, and thus afford a means of defining likelihood ratio. For any point which has measure zero, the likelihood is the ratio of the derivatives of the distribution function for signa plus noise and for noise alone. Points which do not have measure zero can always be treated separately. There can be only a countable number of these and likelihood ratio for such a point x can be defined as PsN(x) e(x) =P (x) (A.2) P1(x) Any point with infinite likelihood ratio belongs in the criterion, of course, and such a point has a posteriori probability unity. Then likelihood ratio is defined except for a set of points of measure zero. In any case where likelihood ratio is defined and satisfies Eq. (A.1), Theorems 1 and 2 can be proved. The lemma (Appendix BB, Lemma 1) which is needed for the proof of Theorem 5 can be proved for any space and measure for which sets of arbitrarily small measure can be found containing each point. If this holds and likelihood ratio is defined, then Theorems 5, 6, 7, and 8 can be proved........._ _ _ _ _ _ _ _ _32.. 2

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX B This appendix contains the proof of Theorem 3 and the lemmna required to complete the proof of Theorem 6. It is convenient to prove three lemmas from which Theorem 3 will follow directly. Lenna 1: Let S be a sphere (i.e., the set of all points whose distance to a fixed point is less than or equal to a fixed positive number) in n-dimensional Euclidean space En. Let f(x) be a continuous real function defined on S. Then the graph G [x x] of fx) of f(x) in E llas (n+l)-measure zero. Proof: Let the volume (the n-measure) of S be V. Since f(x) is uniformly continuous on S, for every E > O there is a 8 > O such that whenever the distance between xl and x2 is less than 8 it follows that i f(xl) - f(x2)J< E/4V. Moreover, for each 8 > 0 there is a decomposition of En into pairwise disjoint congruent n-dinensional cubes each with its greatest diagonal of length less than 8/2. This decomposition may be chosen so that, if {Ci i = 1, 2.,k are the cubes that touch S, then (volume Ci) < 2V. (B.1) Thus I. = f(Ci) is an interval of length less than 2(E/4V) = E/2V. Now, let Ci be the (n+l)-cube formed by the Cartesian product C. x I; by construction, the graph G is covered by the (n+l)-cubes Ci*. Also Z [(n+l)-volume Ci*< [(n)-volume Ci] /2V<2V. /2V (B.2) i Thus for each E > 0 there is a covering of G by (n+l)-cubes whose total (n+l)-volune is less than E. This means (n+l)-measure of G is zero. 33

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Lemma 2: Let D be an open set in Euclidean n-diensional space En and f(-) a real function defined for all points x in D which has continuous partial derivatives of all orders such that at each goint x in D at least one partial derivative (of any order) does not vanish. Then, if b is some value taken on by f, the set f-l(b) of all points x such that f(x) = b has n-measure zero. Proof: A point x in D is said to have "order zero" if some first order derivative of f does not vanish at x; x has "order r" (r a positive integer) if all partial derivatives of f of order < r vanish at x, but at least one partial derivative of f of order r+l does not vanish at x. By the hypotheses, every point of D has finite order. For each integer r > 0 let C be the set of points in f-l (b) of order 0O r r; then f-l(b) = U Cr. The theorem is proved if it is shoan that the n-measure rrO r=0 of Cr is zero for each r. This will be done in two steps. I. At each point xQ in Cr. there is a sphere S(x~) centered at x~ such that S (x~) 0 Cr has n-measure zero. II. There is a countable collection S(x )}, i = 1, 2,..., of such spheres such that Cr is contained in the union U S(x i=l Step II is an application of the Lindel]8ff theorem which asserts that every collection of spheres contains a countable subcollection whose union is equal to the union of all the original spheres.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN The proof of I follows: Since x~ is of order r, one of the derivatives of order r of f(x), say w (x), has a first order derivative which does not vanish at x~. By a change in notation, this can be written as: w does not vanish at xo n X~ xo = (xo10... x0n). The implicit function theorem can then be applied to w, yielding these results: 1) there is a sphere S(x~) centered at x~ and contained in D. 2) writing;C for the projection of S(x~) onto the xl,..., Xn_ "coordinate plane,"' n is an (n-l) sphere. There is a real valued continuous function X(xl,... Xnl) defined on - whose graph G = 1[ X... Xn-1l X(X1 *.. Xnl)]} is the set of all points x in S(x') such that wc(x~) = (x); that is G = s(xo)nl-1 [W(xO)]. Note: 2) says that, iL particular, w [xl}.... xlll X(xl... Xnl)] = w(x~). This is tile usual way of stating the theorem. By Lemma 1, the n-measure of G is zero. Thus step I is proved if S (x)nflrC G. Case I: r = 0. If x is in S(x0~)n Co, then x is of order r = 0 and f(x) = f (x). But in this case w must have been chosen to be f, so w(x) = w(x~), which implies that x is in G. Case 2: r > 0. If x is in S(x~)nfCr, then x is of order r, which means that in particular all r-order partials of f vanish at x. Hence w(x) = 0. Also, by the same argument w(x~) = O, and cw(x) = w(x~) implies that x is in G. This completes the proof of Lemmna 2. Lemma 3: If fN(XlX 2,..., Xn) is an analytic function defined on n-dimensional Euclidean space En, and if P(S1,S2,..., Sn) is a probability measure on En such that there exists a bounded set in En whose probability is unity, then 35

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN fsN(X1,..., Xn) = /f N(Xl-Sl,..., xn-sn) dP(S,..., Sn) (B.S4) En exists and is analytic. Proof': Let B be a bounded set such that P(B) = 1. Then B, the closure of B, is such a set also; it is certainly bounded, and it can be assigned the measure unity, since BcBcEn and 1 = P(B) < P(B) < (En) = 1. (B.5) The probability of the complement of B is zero, and hence the integration can be restricted to the set B rather than to the whole of En. For a fixed (x1,... Xn) and for (sil...~ sn) in B, f1N(xl-8l...} Xn-s ) is bounded, since fi is continuous and B is closed and bounded. The function fN is also measurable, since it is continuous. (This assumes open sets are measurable.) Then the integral exists.l The ftulction fN(x l..., xn) being analytic means that f (... x ) is an analytic function in the ordinary sense when considered as a function of any single coordinate xi. Let us forget about the other coordinates for the present. Then fN(xi) has a power series expansion at each point x i, which converges in a neighborhood of the point (x~i 0) in the complex plane. Thus fN(xi) can be extended for complex values of x. in a region containing the real axis. Formally, Cramer fs4,n(x.+h)-fs (x ii fsN(xi) =lirm h (..6) X. h -- p. Cramr, Ref. l4, Section 5.2, p. 37. 56

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN h= h f[(xl-Sl,..., xi+h-si,..., X Sn dP(Sl,, (o SI) hB.O1 1 1(B.7) h-*O Xl- 1 i i n n B f B(xflsl,... x -ssn) dP(sl' x -. ).8) h-.oLN i n n - ira ~ L1(X1-S..., x+ih- Xn-Sn) -B a~'.. (B.l) Bh 1 The only question now is wkhether or not it is permissible to interchange the order of integration and taking the limit of the difference quotient at step (B.9). This is permissible if the difference quotient converges unifornly, which turns out to be the case. The function fN(xi) is analytic in a domain which extends to complex values of xi near the real axis. The function f' (xi ~ h - si) can be considered as a function of h - sip and is analytic for complex values of h - si in a domain containing the real axis. Since the values of s = (sl..., sn) in B are a closed bounded set, and th"e values of h can certainly be bounded, the set V of 37

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN values h - si is bounded. V can also be taken as closed, and it can be chosen so that no point si is on its boundary. Then there will be a minimum distance ho > 0 from points s. to the boundary of V. Consider the function (s1,..., s, h) = [iijjji i i n5n n n i I -f(X -sly..., - X., Xn-s )] if h / 0, and afN - fN -, if h = 0 defined for I h J _ ho, and s in B. 4 is continuous at every point, and it is defined for all points (h, s) with h = u+iv and s = (s,..., s ) of a compact subset of En+2. is therefore uniformly continuous, and its convergence to x, as h approaches zero along any complex valued path is uniform in s. Thus the difference quotient converges unifornmLy. Lemma 3': Let fN(x1,..., Xn) be a function of n complex variables, and suppose that for each i, there is a domain Di in the complex plane and a number h such that the domain Di contains all points Within a distance of hlo of the real axis, and fN(xl,... Xi,..., xn) is an analytic function of xi in Di for all real values of the other coordinates. Then, if P(sl,... sn) is a probability measure on the n-dimensional Euclidean space En, fSN (X e..} Xn) = fS f(xr-sl,.r. n-n) dP(sl...', Sn) (B.11) En is analytic if it exists.1 1If f is bounded, the interal must exist, as in the previous case. 38

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The proof will be omittl;'ed. The idea of the proof is as follows: one must form the difference quotient for fsN(xl,...,~ xn) for each coordinate xi 1 [f- S(X-l..., i +h..., n) - fSN(Xl -... Xi... xn) and show that the limit as h - O exists, and is equal to what is obtained by differentiating under the integral sign. The space can be divided into two parts such that one will have arbitrarily small measure and contribute an arbitrarily small anmoust to the integrals, while the other will be closed and bounded miand hence on it the order of integration and taking the limit as h-.O can be interchanged, as in Lemma 3. The domain Di. is required so that differentiation in the complex plane will be possible. Now let us discuss Theorem 3. Suppose f (x) is analytic, and suppose either Lelma 3 or Lemma 3' holds. Then fs (x) is analytic, and their ratio s(x) (x) is analytic except where f.(x) = 0. This is a set of measure zero, by Lemma 2. Since ~(x) is analytic, the points where Q(x) = 5 form a set of measure zero, by Lemlma 21 This proves Theorem 3. Theorem 3: If the probability density function for noise alone, fIT(x), is an analytic function, (and if either Leiunaa 3 or Lemma 31 holds,) then the set of points for which 2(x) = p has measure zero. The restriction that Lemma 3 or Lemma 3' holds is not at all serious. If the signals have bounded energy, Lemma 3 holds. Lemia 3' would be expected to hold for most analytic probability density infctions, and in particular it does hold if the noise is Gaussian. 1Imote that Lebesgue measure zero implies probability zero, since the probability is defined throurth density functions....._~~~~1 39

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The following lemma is needed to complete the proof of Theorem 6. Lemma 4: Let f(x) be a probability density function defined on the n-dimensional Euclidean space En. Denote by P(A) the value of the integral f f(x) dx for all A subsets A of En for which the integral exists. If A is any P-measurable set 0 whose measure P(Ao) is finite, and if 0 < < p (Ao), then there is a Pmeasurable set Bo such that P(Bo) = Y. The follaoing proof maktes the theorem valid for any measure on any space M with the property "C" defined below. Proof: Under the hypotheses above, the measure P has a special property relative to the space En. Property "C": There is a countable class [Ci], i = 1, 2,..., of P-measurable sets such that if x is a point and E > 0 then there is a Ci containing x such that P(Ci) < E. One can obtain such a class by choosing all (n-dimensional) spheres of rational radcius centered at points whose coordinates are rational. This class is countable because the rational numbers are countable. Its members are P-measurable because f f(x) dx exists for any sphere A. That it has property A "C" is a way of stating a fuandamental property of integrals. The desired set B will be constructed as the union of a special 0 sequence [Di]of P-measurable sets. Define D1 to be ClnAo if P(CAnAO) < A; sequence i 1 1 if(eflobe y ); otherwise define D1 to be empty. If D has been defined, define Dn+ Dn U [Clln+ Ao] if P {Dn U[Cn+l A0] - Y; oterwise define Dn+l =Dn Since D CDn P(D) - (D ) < Y. Hence the sequence [P(D )] of real n n+l' n n+ Ln nunbers converges. A general property of measures yields the result that cco r n = lir P(D). Write B = U 1 Dn; then P(B) = ni P(D).

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN It renains to be shown that P(Bo) =. Suppose P(Bo) < Y; then writing E = y- P(Bo) > 0, one has P(Bo) = - E. Since P(Bo) < P(Ao) there is a point x in A0 bt not in Bo. By property "C", there is some Ck containing x such that P(Ck) < E. Return to the definition of Dk. If P {Dkl U [CknAo]}< Y then Dk was defined to be D_1 U [CknAo]. Here P{ lU[ kcnAo]}<P(D _l) + P(Ck) P(o) + P(Ck)< (Y- ) +E = Thus it was the case that Ck A CD CBo. But CkN A contains a point x not in Bo. k ok 0 k o This contradiction shows that P(BO) is actually equal to y and not less than as was supposed. 4.,

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN APPEndIX C The following theory was developed as the preparation of the text of this report neared completion. The subhject matter is appropriate to this report, and so it is included. The purpose of this material is to characterize uniformly best tests, or criteria. If there are a family of signal distributions (or hypotheses, in statistical terms), and if a criterion A is an A (k) for each of them, then A is a uniformly best test. Theorem C1 states that if all distributions in a family of signal distributions are k-equivalent, all optimum criteria are uniform best tests, and Theorem C2 states the converse. In the first three cases considered in Part II of The Theory of SigL Detecab the signal known exactly, the signal known except for carrier phase, and the signal a sample of white Gaussian noise, two signal distributions differing only in signal energy are k-equivalent. Thus, by Theorem C4., a signal distribution with fixed signal energy and one with the signal energy having an arbitrary distribution are k-equivalent in these three cases. These three cases have for the boundaries of their optinmu criteria, planes, cylinders, and spheres, respectively. For the other cases, with more complicated criterion boundaries, k-equivalence cannot be expected when energy is changed. Definition: If fSN((x) and fSN() (x) and f (x) are defined on En, and if there exists a set X of probability zero such that for any two points x and y in inl E, but not in X, il(x)-. l(y) if and only if ~2(x)> ~2(Y), then fSN(l)(x) and fSN(2)(x) are said to yield k-equivalent distributions. 1Neyman and Pearson, Ref. 13.,

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Theorem C1: If ~fs(1) (x) and (2) (x) give k-equivalent distributions, then a criterion is an A2(k) for the first if and only if it is an A2(k) for the second. Proof: Suppose A is an A2(k) for the first distribution. Then by Theorem 7, there is a 3 such that A is a A1(3). By Theorem 2, A contains all points for which ~(x) > D and none for which ~(x) < P, except for a set of probability zero. Except for a set of probability zero, if x and y are any two points such that x is in A and y is not in A, then Xl(x) _~1(y). By definition of kequivalence, there is a set X of probability zero, such that if x and y are also not in X}, ~(x) ~2(Y). Then there must exist a number P2 such that for any x except a set of probability zero, 2 (x)- 2 if x is in A and X2(x)L <2 if x is not in 32. If follows that A is an A1(P2) with respect to the second distribution. Furthermore, PN(A) = k, for either distribution since the probability dcensity with noise alone is the same for both distributions. It follows by Theorem 5 that A is an A2(k) for the second distribution. Thf (1) a (2) Theorem C2: If fSN(l1 (x) and fSN (x) lead to two distributions such that for every k, any criterion A is an A2(k) for one if and only if it is for the other 2 also, then fSi(l)(x) and fSN )(x) lead to k-equivalent distributions. Proof: Consid~er the family of sets A. where A x { 1(x)~ o}, and a takes on all rational number values greater than zero. Each Ax is an A2(k) for some k with respect to the first distribution, by Theorem 5. Then it is for the second. also, by hypothesis. Each AC is an A1 [p(a~)] for some P(ca) by Theorem 7. For each AC, the set of points C. such that x is in A and ~(x) < 3(ca) or x is not in A. and (x) > t (c~) has probability zero, by Theorem 2. Let X1 be the union of all the sets Co, and since each Cc- has probability zero, and the rational |numbers and hence the family C is countable, it follows the the set X1 has probability zero. 4l

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Now consider the family of sets A = n AI = (x) r} (C 1.) ally < r defined for every positive real numaber r. Also define g(r) = a. u. b. (ca) (c.2) all C <r Then for any point x not in X1, if x is in A r 2 (x) > g(r). Also consider the family of sets A*r = all A {x 1(x) > r} (C.3) all 1>r defined for every positive real number r. If x is a point not in X1} and if x is not in A*ri 2,(x)<= g. Y. b. g(r;-) (c.4) all r*> r For any value of r at which g(r) is continuous, g(r) = Xg.. b. g(r*). (C.5) all r*>r Any point x which is not in X1 and for which l (x) = r is in A but not in r A*-r and therefore g(r) 2(x) <- g(r), i.e., 22(x) = g(r). (C.6) Clearly g(r) is a monotone increasing function of r. It can therefore have at most a countable number of discontinuities. Let r denote a discontin0 uity in g(r) and suppose that the set of points B = {xI l(x) = r} has probability greater than zero. Define h(ro) = Se u. b. { P {x x B and 2(x) < 3} = 0} (c.7) h*(ro) =. b. { I P(x x B and 2(x)> } ) = o} The claim is made that h(ro) = h* (ro). Suppose h(ro) h*(ro). Then there (to

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN exists a number y such that h(ro) < y < h*(ro) Define C1 - {x Ih(r)- 2 2(x) r } (c.8) c2 = Jr < <2(x) ~ h* (r) } both C1 and C2 have probability greater than zero, by Eq. (C.7). Now consider the set Ar C2. It is an A2(k) for the first distribution, by Theorem 5. Clearly, by Theorems 7 and 2, it caznot be an A2(k) for the second distribution. The contradiction leads us to conclude that h(ro) = h*(ro). Then for each discontinuity r0 there exists a set of probability zero, say S(ro), such that if 4l(x) = r and x is not in S(ro), -2(x) = h(ro). Let X2 = U S(ro). Then 2all r 0 h2 has probability zero, since there are only a countable number of points of discontinuity ro. How define X = X1U X2, X also has probability zero. Let the function h(r) be defined as follows: h(r) = g(r) if g(r) is continuous at r (c.9) h(r) = h(ro) at r = ro, a discontinuity of g(r). The function h(r) has the following properties: (1) h(r) is a monotone increasing function of r, and (2) if xl(x) = r, and x is not in X, then 42(x) = h(r). The first assertion is an obvious consequence of the way in which h(r) is defined, and the fact that g(r) is monotone. The second assertion has been shoawn separately first for points where g, and hence h, is continuous, Eq. (C.6), secondly for the points of discontinuity of h, in the preceding paragraph. Now suppose x and y are not elements of X, and l(X) >- 1(Y) If L(x) = r and 21(Y) = r. then r = r It follows from the fact that h(r) is monotone increasing that h(rx) - h(ry), and since i2(x) = h(rx) and y4~

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN p2(Y) = h(ry), 42(x) _~2(Y)' Since X has probability zero, this completes the proof. Theorem C3. If fSN(1) (x) is k-equivalent to fSN (i)(x) for each value of i between 2 and n, (or between 2 and co), and ai are positive real numbers such that n n a. = 1 (or a = 1) then SN(1)(x) and ai fSN (x) 1 1 i i (or E ai fSNi) (x)) yield k-equivalent distributions. The set X (in the definition of k-equivalence) for the distribution given by the sum is taken as the union of the sets X for the individual distributions. Then the proof is obvious. Theorem C4: If fSN (x) is a continuous function of a in an interval [a, b], if for any two numbers al and a2, fSN l)(x) is k-equivalent to fSN(2) (x) and if F(ca) is a monotone function which is zero at the left end of the interval and 1 at the right end of the interval, then b a SN( )(x) dF(a) is k-equivalent to any fSN( (x). Proof: Choose any a in the interval [a, b]. Then for each rational value of in the interval a, b] fSN (x) and fSN() (x) are k-equivalent. There is a set XE, which has probability zero, such that if x, y are not in X..,'P(x)I - (y) if and only if ~o (x)_ Lao(y). The union X of all X~ with rational a also has probability zero, since the rational numbers are countable. Furthermore, if x and y are not in X, then >4(x)-La(y) for any rational value of a implies..(x)> ic (y), and o(x)~ (y) implies =(x)2z(y) for 46

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN all rational values of ca. Since fSN((x) is continuous in, (x) must be continuous in c also, and it must follow that for any real ca in [a} bJ and for any x, y not in X, ~a(X ) I ~a(y) if and only if 2a%(x) >2Rao(y). Then it is easy to show that if x and y are not in X, b'e [a(X) - e(y)] dF(c) 0 if and only if ao (x)~ a (y), and hence J fSN (x) dF(ca) is equivalent to fSN (o) (X). STT~~~~~~~~~~~~~~~~~~1.

BIBLIOGRAPHY On Statistical Approaches to the Signal Detectability Problem: 1. Lawson, J. L., and Uhlenbeck, G. E., Threshold Signals, McGraw-Hill, New York, 1950. This book is certainly the outstanding reference on threshold signals. It presents a great variety of both theoretical and experimental work. Chapter 7 presents a statistical approach of the criterion type for the signal detection problem, and the idea of a criterion which minimizes the probability of an error is introduced. (This is a special case of an optimum criterion of the first type.) 2. Davies, I. L., "On Determining the Presence of Signals in Noise," Proc. I.E.E. (London), Vol. 99, Part III, pp. 45-51, March, 1952. 3. Woodward, P. M., and Davies, I. L., "Information Theory and Inverse Probability in Telecommunication," Proc. I.E.E. (London), Vol. 99, Part III, p. 37, March, 1952. 4. Woodward, P. M., and Davies, I. L., "A Theory of Radar Information," Phil. Mag., Vol. 41, p. 1001, 1950. 5. Woodward, P. M., "Information Theory and the Design of Radar Receivers," Proc. I.R.E., Vol. 39, p. 1521. Woodward and Davies have introduced the idea of a receiver having a posteriori probability as its output, and they point out that such a receiver gives a maximum amount of information. They have handled the case of an arbitrary signal function known exactly or known except for phase with no more difficulty than other authors have had with a sine wave signal. Their methods serve as a basis for the second part of this report. 6. Reich, E., and Swerling, P., "The Detection of a Sine Wave in Gaussian Noise," Jour, App. phys., Vol. 24, p. 289, March, 1953. This paper considers the problem of finding an optimum criterion of the second type presented in this report for the case of a sine wave of limited duration, known amplitude and frequency, but unknown phase in the presence of Gaussian noise of arbitrary autocorrelation. The method probably could be extended to more general problems. On the other hand, the methods of this report can be applied if the signals are band limited even in the case of non-uniform noise by putting the signals and noise through an imaginary filter to make the noise uniform before applying the theory. See The Theory of Signal Detectability, Part II, Section 3. 48

7. Middleton, D., "Statistical Criteria for the Detection of Pulsed Carriers in Noise, Jour. Appl. Phys., Vol. 24, p. 371, April, 1953o A thorough discussion is given of the problem of detecting pulses (of unknown phase) in Gaussian noise. Both types of optimun criteria are discussed, but not in their full generality. The sequential type of test is discussed also. 8. Slattery, T. G., "The Detection of a Sine Wave in Noise by the Use of a Non-Linear Filter," Proc. I.R.E., Vol. 40, p. 1232, October, 1952. This article considers the problem of detecting a sine wave of known duration, amplitude, and frequency, but unknown phase in uniform Gaussian noise. The article contains several errors, and although the results appear to be correct, they are not clearly presented. 9. Hanse, H., "The Optimization and Analysis of Systems for the Detection of Pulsed Signals in Random Noise," Doctoral Dissertation (MIT), January, 1951. 10. Schwartz, M., "A Statistical Approach to the Automatic Search Problem," Doctoral Dissertation (Harvard), June, 1951. These dissertations both consider the problem of finding the optimum receiver of the criterion type for radar type signals. 11. North, D. O., "An Analysis of the Factors w~hich Determine Signal-Noise Discrimination in Pulsed Carrier Systems," RCA Laboratory Report PTR-6C, 1943, The ideas of false alarm probability and probability of detection are introduced. North argues that these probabilities will be most favorable when peak signal to average noise ratio is largest. The ideal filter, which maximizes this ratio, is derived. (This commentary is based on second-hand knowledge of the report.) 12. Kaplan, S. M., and Fall, R. W., "Tlhe Statistical Properties of Noise Applied to Radar Range Performance," Proc. I.R.E., Vol. 39, p. 56, January, 1951. The ideas of false alarm probability and probability of detection are introduced and an example of their application to a radar receiver is given. On Statistics: 13. Neyman, J., and Pearson, E. S., "On the Problem of the Most Efficient Tests of Statistical Hypotheses," Phil. Trans. Roy. Soc., Vol. 231, Series A, p. 289, 19533.. 49

14. Cramer, H., Mathematical Methods of Statistics, Princeton University Press, Princeton, 1951. On Related Topics: 15. Dwork, B. M., "Detection of a Pulse Superimposed on Fluctuation Noise," Proc. I.R.E., Vol. 38, p. 771, July, 1950. 16. Harrington, J. V., and Rogers, T. F., "Signal-to-Noise Improvement Through Integration in a Storage Tube," Proc. I.R.E., Vol. 38, p. 1197, October, 1950. 17. Harting, A. E., and Meade, J. E., "A Device for Computing Correlation Functions," Rev. Sci. Inst., Vol. 23, 347, 1952. 18. Lee, Y. W., Cheatham, T. P., Jr., and Wisner, J. B., "Applications of Correlation Analysis to the Detection of Periodic Signals in Noise," Proc. I.R.E., Vol. 38, p. 1165, October, 1950. 19. Levin, M. J., and Reintzes, J. F., "A Five Channel Electronic Analog Correlator," Proc. Nat. El. Conf., Vol. 8, 1952. 20. Rice S. O.,'IMathematical Analysis of Random Noise," B.S.T.J. Vol. 23, p. 282-332 and Vol. 24, p. 46-156, 1945-6. 21. Shannon, C. E., "Communication in the Presence of Noise," Proc. I.R.E., Vol. 37, PP. 10-21, January, 1949. 50

LIST OF SYUmQOIS A The event "The operator says there is signal plus noise present," or a criterion, i.e., the set of receiver inputs for which the operator says there is a signal present. Al(P) Any criterion A which maximizes PSN,(A) - P P N(A), i.e., an optimum criterion of the first type. A2(k) Any criterion A for which PN (A) k, and PS11(A) is maximum, i.e., an optimum criterion of the second type. CA The event "The operator says there is noise alone." d A parameter describing the ability of a receiver to detect signals. (See Section 5.1 and Fig. 5.1.) E, E(s) The signal energy. En The n-dimensional Euclidean space. fN(x) The probability density for points x in II if there is noise alone. fS11(X) The probability density for points x in R if there is signal plus noise. FN(t3), Fu(~) The complementary distribution function for likelihood ratio if there is noise alone, i.e., F (1) is the probability that the likelihood ratio will be greater than D if there is noise alone. FSN(P)IFSN(E) The complemnentary distribution function for likelihood ratio if there is signal plus noise. k fA symbol used primarily for the upper bound placed on false alarm probability PN(A) in the definition of the second kind of optimum criterion. fsN(x) X(x) The likelihood ratio for the receiver input x. C(x) = f ~ n The dimension of the space of receiver inputs. n = 2WT N The event "There is noise alone," or the noise power. INo The noise power per unit bandwidth. No = N/. P BN(A) The probability that the operator will say there is signal plus noise if there is noise alone, i.e., the false alarm probability.

PSN(A) The probability that the operator will say there is signal plus noise if there is signal plus noise, i.e., the probability of detection. Px(SN) The a posteriori probability that there is signal plus noise present. (See Sections 1.3 and 2.3.) Ps(s) The probability measure defined on R for the set of expected signals. The space of all receiver inputs. (The set of all possible signals is the same space.) s A signal s(t), which may also be considered as a point s in R with coordinates (s1l s2,'. " sn). SN The event "There is signal plus noise." t Time. T The duration of the observation. W The bandwidth of the receiver inputs. x A receiver input x(t), which may also be considered as a point x in R with coordinates (x1, x.2..., Xn) xB nA symbol usuallyr usecd for the likelihood ratio level of an optimum criterion. Iyt SN (Z) The mean of the random variable z if there is signal plus noise. /I.N(z) The mean of the random variable z if there is noise alone. 2, (Z) The variance of the random variable z if there is noise alone. I2 The variance of likelihood ratio if there is noise alone. 52

DISTRIBUTION LIST 1 copy Director, Electronic Research Laboratory Stanford University Stanford, California Attn: Dean Fred Ternan 1 copy Commanding Officer Signal Corps Electronic Warfare Center Fort Monmouth, New Jersey 1 copy Chief, Engineering and Technical Division Office of the Chief Signal Officer Department of the Army Washington 25, D. C. Attn: SIGGE-C 1 copy Chief, Plans and Operations Division Office of the Chief Signal Officer Washington 25, D. C. Attn: SIGOP-5 1 copy Countermeasures Laboratory Gilfillan Brothers, Inc. 1815 Venice Blvd. Los Angeles 6, California 1 copy Commanding Officer White Sands Signal Corps Agency White Sands Proving Ground Las Cruces, New Mexico Attn: SIGWS-CM 1 copy Signal Corps Resident Engineer Electronic Defense Laboratory P. 0. Box 205 Mountain View, California Attn: F. W. Morris, Jr. 75 copies Transportation Officer, SCEL Evans Signal Laboratory Building No. 42, Belmar, New Jersey For - Signal Property Officer Inspect at Destination File No. 25052-PH-51-91(1443) 53

1 copy W. G. Dow, Professor Dept. of Electrical Engineering University of Michigan Ann Arbor, Michigan 1 copy H. W. Welch, Jr. Engineering Research Institute University of Michigan Ann Arbor, Michigan 1 copy Document Room Willow Run Research Center University of Michigan Willow Run, Michigan 10 copies Electronic Defense Group Project File University of Michigan Ann Arbor, Michigan 1 copy Engineering Research Institute Project File University of Michigan Ann Arbor, Michigan 514

ENTGINI.ING RESEARCI IINSTWZUTEY UTIVEiSTTY OF MICHIGAN AITI ARBOR THE THIECY OF SIGNAL JDECTABEILTY PART II. APPLICATIONS WITH GAUSSIAN NIOISE ISSUED SEPARATELY: PART I. THE GENERAL THEORY Tecihnical Report No. 13 Electronic Defense Group Department of Electrical Engineering By: W. W. Peterson Approved by: /~ ~ @@ T. G. Birdsall H. W. Welch, Jr. Project Engineer Project M970 TASK ORDER NO. EDG-3 CO0ACTr NO. DA-36-039 sc-15358 SIGNAL CORPS, DEPARTME OF TIMfE ARIC DEPARTIf~I.CT OF ARMI PROJECT IO. 3-99-041-042 SIGNAL CORPS PROJTWCT NO. 29-1914B-0 July, 1953

TABILE COF CONTET'S PART II. APPLICATIONS WITH GAUSSIAN NOISE Page LIST CF ILLUSTRATIONIS ABSTRACT ACItTO1OLEDGIENTS 3. IITRODUCTION AID GAUSSIAN NOISE 1 3.1 Introduction 1 3.2 Gaussian Noise 4 3.3 Likelihood Ratio with Gaussian Noise 7 4. LIKELMOOD RATIO AND ITS DISTRIBUTION FOR SPECIAL CASES 9 4.1 Introduction 9 4.2 Signal Known Exactly 9 4.3 Signal Known Except for Carrier Phase 17 4.4 Signal Consisting of a Sample of White Gaussian Noise 22 4.5 Video Design of a Broad Band Receiver 27 4.6 A Radar Case 38 4.7 Approximate Evaluation of an Optimum Receiver 44 4.8 Signal Which is One of M Orthogonal Signals 47 4.9 Signal Which is One of M Orthogonal Signals with Unknmown Carrier Phase 49 5. DISCUSSION OF THE SPECIAL CASES 55 5.1 Receiver Evaluation 55 5.1.1 Introduction 55 5.1.2 Ccmparison of the Simple Cases 55 5.1.5 An Approximate Evaluation of Optimum Receivers 59 5.1.4 Signal One of M Orthogonal Signals 60 5.1.5 The Broad Band Receivers and the Ideal Receiver 61 5.1.6 Uncertainty and Signal Detectability 62 5.2 Receiver Design 64 5.3 Conclusions 68 APPENDIX D 72 APPEMIX E 78 APPENDIX F 81 BIBLIOGRAPHY 84 LIST OF SYmvtBOLS 87 DISTRIBUTION LIST 89 ii

TABLE OF CON11TETS (Cont. ) PART' I. THE CERAL TEEORY (Issued Separately) Page ABSTRACT v ACIKNOUJLED.IM4 EITS vi 1. CONCEPTS AID TEEOI01ICAL RESULTS 1 1.1 Introduction 1 1.2 Detectability Criteria 3 1.3 A Posteriori Probability and Signal Detectability 1. 4 Optitmum Criteria 6 1.5 Theoretical Results 7 1.6 Receiver Evaluation 8 2. MIATEMITICAL THEORY 12 2.1 Introduction 12 2.2 Mathematical Description of Signals and Noise 12 2.3 A Posteriori Probability 14 2.4 Criteria and Optimum Criteria 15 2.4.1 Definitions 15 2.4.2 Theorems on Optimum Criteria 16 2.5 Evaluaticn of Optimum Receivers 26 2.5.1 Introduction 26 2.5.2 Evaluation of Criterion Type Receivers 26 2.5.3 Evaluation of A Posteriori Probability Woodward and Davies Type Receivers 29 2.6 Conclusions 30 APPENDIX A 31 APPENDIX B 33 APPENDIX C 42 BIBLIOGRAPHY 48 LIST OF SYMBOLS 51 DISTRIBUTION LIST 53 iii

LIST COF ILLUSiTRATIONS Fig. No. Title Page 4.1 Receiver Operating Characteristic (Ln I is a Normal Deviate) 13 4.2 Receiver Operating Level (Ln I is a Normal Deviate) 14 1K.3 Receiver Operating Characteristic (Ln i is a Normal Deviate) 15 4.4 Receiver Operating Level (Ln I is a Normal Deviate) 16 4.5 Receiver Operating Characteristic (Signal Known Except for Phase) 21 4.6 Receiver Operating Characteristic (Signal a Sample of White Gaussian Noise) 24 4.7 Receiver Operating Characteristic (Signal a Sample of White Gaussian Noise) 25 4.8 Block Diagram of a Broad Band Receiver 27 4.9 Graph of Ln Io(x) 33 4.10 Probability Density of Ln I vs. E Ln 39 4.11 Receiver Operating Characteristic (Broad Band Receiver with Optimum Video Design, M = 16) 40o 4.12 Signal Energy as a Function of M and d (Signal One of M Orthogonal Signals) 50 4.13 Signal Energy as a Function of M and d (Signal One of M Orthogonal Signals Known Except for Phase) 54 5.1 Receiver Operating Characteristic (Ln ~ is a Normal Deviate) 56 5.2 Receiver Operating Characteristic (Signal Known Except for Phase) 57 5.3 Comparison of Ideal and Broad Band Receivers 63 F.1 Maximum Response of RC Filter to a Rectangular Pulse as a Function of Filter Time Constant 83 iv

AB;STIRICT PART I The several statistical approaches to the problem of signal detectability which have appeared in the literature are shoun to be essentially equivalent. A general theory based on likelihood ratio embraces the criterion approach, for either restricted false alarm probability or minimum weighted error type optimum, and the a posteriori probability approach. Receiver reliability is shmwn to be a function of the distribution functions of likelihood ratio. The existence and uniqueness of solutions for the various approaches is proved under general hypothesis. PART II The full power of the theory of signal detectability can be applied to detection in Gaussian noise, and several general results are given. Six special cases are considered, and the expressions for likelihood ratio are derived. The resulting optimum receivers are evaluated by the distribution functions of the likelihood ratio. In two of the special cases studied, the uncertainty of the signal ensemble can be varied, throwing some light on the effect of uncertainty on probability of detection. V

ACETI(LEDGiAENMTS In the work reported here, the authors have been influenced greatly by their association with the other members of the Electronic Defense Group. In particular, Mr H. H W. Batten contributed much to the early phases of the work on signal detectability. Mr. W. C. Fox assisted in the calculations. The authors are indebted to Dr. A. B. Miacnee and Dr. J. L. Stewart for the many suggestions resulting from their reading the report. The authors also wish to acknowledge their indebtedness to Geraldine L. Preston and Jenny-Lea E. IMesler for their assistance in the preparation of the text. vi

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TIE, THEORY OF SIGNAL DETECTABILITY Part II. APPLICATIONS WITH GAtSBSIAN NOISE ISSUED SEPARATELY: Part I. TEE GENERAL TBEGRY 3. IITRODUCTION AND GAUSSIAN NOISE 3.1 Introduction The chief conclusion obtained from the general theory of signal detectability presented in Part I is that a receiver which calculates the likelihood ratio for each receiver input is the optimum receiver. The receiver can be evaluated (e.g., false alarm probability and probability of detection can be found) if the distribution functions for likelihood ratio are known. It is the purpose of Part II to consider a number of different ensembles of signals with Gaussian noise. For each case, a possible receiver design is discussed. The primary emphasis, however, is on obtaining the distribution functions for likelihood ratio, and hence on estimates of receiver performance for the various cases. The special cases which are presented were chosen from the simplest problems in sigal detection which closely represent practical situations. They are listed in Table I along with examples of engineering problems in which they find application.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE I Description of Section Signal Ensemble Application 4.2 Signal Known Exactly1 Coherent radar with a target of knomw range and character 4.3 Signal Known Except for Ordinary pulse radar with no intePhasel gration and with a target of nmown range and character. 4.4 Signal a Sample of White Detection of noise-like signals; Gaussian Noise detection of speech sounds in Gaussian noise. 4.5 Video Design of a Broad Detecting a pulse of known startBand Receiver ing time (such as a pulse from a radar beacon) with a crystal-video or other type broad band receiver. 4.6 |A Radar Case (A train of Ordinary pulse radar with intepulses with incoherent gration and with a target of known phase)2 range and character. 4.8 | Signal One of M Orthogo- Coherent radar where the target is nal Signals at one of a finite number of nonoverlapping positions. 4.9 Signal One of M Orthogo- Ordinary pulse radar with no intenal Signals Known Except gration and with a target which for Phase may appear at one of a finite number of non-overlapping positions. Our treatment of these two fundamental cases is based upon Woodward and Davies work, but here they are treated in terms of likelihood ratio, and hence apply to criterion type receivers as well as to a posteriori probability type receivers. 2This is essentially the case treated by Middleton in Ref. 7. 2

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN In the last two cases the uncertainty in the signal can be varied, and some light is thrown on the relationship between uncertainty and the ability to detect signals.l The variety of examples presented should serve to suggest methods for attacking other simple signal detection problems and to give insight into problems too complicated to allow a direct solution. It should be borne in mind that this report discusses the detection of signals in noise; the problem of obtaining information from signals or about signals, except as to whether or not they are present, is not discussed. Furthermore, in treating the special cases, the noise was assumed to be Gaussian. The reader will probably find the discussion of likelihood ratio and its distribution easier to follow if he keeps in mind the connection between a criterion type receiver and likelihood ratio. In an optimum criterion type system, the operator will say that a signal is present whenever the likelihood ratio is above a certain level 3. He will say that only noise is present when the likelihood ratio is below P. For each operating level P, there is a false alarm probability and a probability of detection. The false alarm probability is the probability that the likelihood ratio o (x) will be greater than P if no signal is sent; this is by definition the complementary distribution function FN(P). Likewise, the complementary distribution FSN(p) is the probability that C (x) will be greater than B if there is signal plus noise, and hence FSIN(P) is the probability of detection if a signal is sent. The only discussion in the literature on the effect of uncertainty on signal detectability which has come to our attention is in Davies, Ref. 2, where the effect upon signal detectability of not knIwing carrier phase is shown quantitatively. 2See the footnote on page 4 with reference to the spectrum of the assumed noise_ _ _-_

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 3.2 Gaussian Noise Throughout Part II, receiver input voltages, which are functions of time, are assumed to be defined for all times t in an observation interval, O < t < T. They are also assumed to be limited to a band of frequencies of width W. By the sampling theorem,1 each receiver input can be thought of as a point in a 2V1T dimensional space, the coordinates of the point being the value of the function at the "sample points" t =, for 1 i _ 21T. The notation x(t), or simply x denotes a receiver input, and x. denotes the ith sample value or coordinate. The signal as it would appear at the receiver input in the absence of noise is denoted by s(t), or simply s, and the coordinates, or sample values, of s are denoted by si. The receiver input, which may be due to noise alone or to signal plus noise, is random because of the presence of noise. Therefore, only the probability distribution for the receiver inputs x(t) can be specified. The distribution must be given for the receiver inputs both when there is noise alone and when there is signal plus noise. The probability distributions are described in this report by giving the probability density function fSN(x) and fN(x) for the receiver inputs x in the 21T dimensional space. The noise considered in Part II is always Gaussian noise limited to the bandwidth W, and having a uniform spectrum over the band.2 This is ordinarily called white Gaussian noise. The probability density function for white Gaussian noise, and hence for the receiver inputs when there is noise alone, is: SIX;(x) /= er 2Nm, or (3.1) 1See Appendix D.'If the noise spectrwu is band limited, but not uniform, the noise and si grnls can be put through a filter which malkes the noise uniform, and then the theory can be applied. See IH. W. Bode and C. E. Shannon, "A Simplified Derivation of Linear Least Square Smoothing and Prediction Theory," Proc. I.R.E., Vol. 38, p. 417, April 1950.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN fN(x) = ( 21 exp [ 2 (3.2a) where n is the dimension of the space, i.e., 2WT,2 and N is the noise power.3 It can be shamc that this ensemble of noise functions has a Gaussian distribution at every time and that its spectra is uniform. By the sampling theorem,4 T X.2 = 2W f [x(t)2 it. (3.3) 0 Therefore n f (X) ( I) exp f x(t) dt], (5.2b) N whiere No = is the noise power per unit bandwidth.5 If the silals and their probabilities are Imown, then the sigal plus noise probability density function, fSI(x), can be found by the convolution integral, as described in Section 2.6 kUnless otherwise indicated, the limits on the sum are i = 1 to i = n = 2WT. 2 i 2If exp - 2 is called fjl(xi), then fN(x) = fN( ie. the x. are L- J ni=l independent and each has fYi(xi) for its probability density function. For a discussion of "independen-t," see Cramer, fRef. 14, p. 159. 3This assumes the circuit impedance is normalized to one ohm. See Appendix D. 5This form of the expression for fN(x), and the corresponding forms of the equations for fsN(x) and ~ (x) were first derived by Woodward. See Woodward and Davies, Refs. 2 and 3. 6See page 13 of Part I. " 5....

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN n SSN (x) Nx-s)d2S (s) (12 x (X si) s R R i=l n (3.4a) in nn n fs() = (x-s) dP() 2 xp [ [x(t)s(t)] dPS(s) 0 0 n (3.4b) = (2jftN exp[- xt dt] XP to] o R The factor xp X (t) dt exp [ ] can be brought out of 0 the integral since it does not depend on s, the variable of integration. Iote that the integral dt= 21 i E (3.5) is the energy of the eaxpected signal, while f x(t) s(t) dt = - is (3.6) is the crioss correlation between the expected signal and the receiver input. See footnote 3 on page 5.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 3.3 Likelihood Ratio with Gaussian liNoise Likelihood ratio is defined as the ratio of the probability density flnctions fsN(x) and f (x). With white Gaussian noise it is obtained by dividing Eq (3.4) by Eq (3.2). ~ (x) = xp E) exp 1 xisi dPS(s), or (3.7a) X(x:) = exp L- i exp[N f x(t) s(t) dt dPS(s) (3.7b) 0oo 0 If the signal is Ionan exactly or completely specified, the probability for that signa,6 or point s, is unity, and the probability for any set of points not containing s is zero. Then the likelihood ratio becomes Is(x) = exp [ _ E(s) exp N xisi or (3.8a) = exp! -o exp 2 T exp [ N exP [ No T x(t) s(t) dt. (3.8b) Thus the general forrmulas (3.7a) and (3.7b) for likelihood ratio state that l(x) is the weighted average of Cs(x) over the set of all signals, i.e., (x) S= (x) d P() * (3.9) If the distribution function P (s) depends on various parameters such as carrier phase, signal energy, or carrier frequency, and if the distributions 7

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN in these parameters are independent,1 the expression for likelihood ratio can be simplified somewhat. If these parameters are indicated by rl, r2,..., r, and the associated probability density functions are denoted by f (rl), f2 (r2)... J f (rn), then d Ps(s) = fl(rl) fn(rn) d'rl dr The likelihood ratio becomes f... (X ) f (rl). f (rn ) dr drn [r( * [Jfil(rl).iv(~x ) dr1] jdxn (3.10) Thus the likelihood ratio can be found by averaging ~s (x) with respect to the parameters. 1Cramer, Ref 14, p. 159.,. __. _-_- __................... _ ___ - 8

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 4. LIKELIHOOD RATIO AND ITS DSTIRIBUTION FOR SPECIAL CASES 4.1 Introduction The purpose of this section is to derive expressions or approximate expressions for likelihood ratio and its distribution functions for a number of special signals in the presence of Gaussian noise. The results obtained in this section are summarized and discussed in Section 5. 4.2 The Case of a Si a Known Exactly The likelihood ratio for the case when the signal is known exactly has already been presented in Section 3.3, Eq (3.8). E 1 T(x) = exp [ J exp ZN iF3i (4.1a) NO N i=l (x) = exp exp - x(t) s(t) dt (4.lb) As the first step in finding the distribution functions for i(x), it is convenient to find the distribution for N E xisi when there is noise alone. Then the input x = (xi, x2,..., xi) is due to white Gaussian noise. It can be seen from Eq (3.1) that each x. has a normal distribution with zero mean and variance N = WNo and that the xi are independent. Because the si are constants depending on the signal to be detected, s = (sl 822... s8n), each summand 1 sf N (xisi) has a normal distribution with mean 2. times the mean of xi, and varisi2 2 2 ance I times the variance of xi zero and. N = respectively. Nq2 i- NP- N Because the xi are independent, the summands sixi are independent, each with normal distributicxns, and therefore their sum has a normal distribution with 9

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN mean the sum of the means -- i.e., zero -- and variance the sum of the variances 1 2 i 2WE(s) 2E 2X Signal Enery (42) N N No Noise Power Per Unit Bandwidth ( The distribution for 1 xisi with noise alone is thus normal with zero mean and variance. Recalling (4.1a) N0 -g(x) = exp N(4.1a it is seen that the distribution for xsi can be used directly by introducing C defined by = e-p + a ora = + n (4.3) co FN() = 0 exp -dy (4.4) The distribution for the case of signal plus noise can be found by using Theorem 8, which states that2 dr (%) N = ~ dN() (4.5) Cramere Ref. 14, p. 212. 2See Part I, pp. 24 and 27. 10

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Differentiating Eq (4.4j, dFN(P) = - exp (41.6) and combining (4.3), (4.5), and (.6), N 2 dESN(P) A~ eXP|- NE. [ " (4.7) Thus 0GFsz(P) = A J exp - 0) dy (4.8) In summary, C., and therefore in (, has a normal distribution with signal plus noise as well as with noise alone; the variance of both distributions is I, and the difference of the means is 2E The receiver operating characteristic curves in Fig. 4.1 are plotted for any case in which In i has a normal distribution with the same variance both with noise alone and with signal plus noise. The parameter d in this figure is equal to the square of the difference of the means, divided by the variance. These receiver operating characteristic curves apply to the case of 2E the signal known exactly, with d = Eq (4.lb) describes what the ideal receiver should do for this case. T The essential operation in the receiver is obtaining the correlation, f s(t)x(t)dt. 0 lThe change in sign appears because the distribution functions FS((P3) and F1N(P) are probabilities that i(x) will lie between ( and 00, not - CO and ( as is usually the case. If the density function for FSN(P) is called g(p), then ___(= -= g(3),' and FSN(D) = fg(D) d3.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN The other operations, nmultiplying by a constant, adding a constant, and taking the exponential function, can be taken care of simply in the calibration of fthe receiver output. Electronic means of obtaining cross correlation have been developed recently.1 If the form of the signal is simple, there is a simple way to obtain this cross correlation.2 Suppose h(t) is the impulse response of a filter. The response e (t) of the filter to a voltage x(t) is3 eo(t) = - x(t ) h(t- r) d (4.9) If a filter can be synthesized so that h(t) = s(T-t) 0 t; < T h(t) = 0 otherwise, (4.10) then eO(T) = f X(T) s(t) dr, (4.11) 0 so that the response of this filter at time T is the cross correlation required. Thus, the ideal receiver consists simply of a filter and amplifiers. It should be noted that this filter is the same, except for a constant factor, as that specified when one asks for the filter which maximizes peak signal to average noise power ratio.4 1Harrington and Rogers, Ref. 16; Tarting and Meade, Ref. 17; Lee, Cheattinm, and Wiesner, nef. 18; Levin and Reintzes, Ref. 19. This appears to be due to Woodward. See Woodward, Ref. 5, and Woodward and Davies, Ref. 3. 3S. Goldman, Transformation Calculus and Electrical Transionts, Prentice Hall, NTew York, 1949, p. 112. Lawson and Uhlenbeck, Ref. 1, p. 206; Worth, Ref. ll. 12

~g-lI-L w3P EZ-29-V OL6-W 6I T liIII1TI lTI1Vl l~~~,~ _I /d8________t_ ______- _... —-< _~...,,'"J 1 / -'/d = - - 4 -" -/" -"/ -7" ='/ -— o= _ I - _, _.T,/ _ I TI_________7____7__/ " T /I 1 0.8 0___ II__III _/' _ r___7 /.,I.I.I 1/ 1I dct +16 e —< -O 1 T+, _e I 8 1 _ 1 1 1000 00 I 2 O.- I"TT- L.0 Z 0IT1T0 -11T Lv50t 000- j000 00. 0.0,!~~~~;0 /.- /~ /.' / 04 _ /_ _ 7rr _ ~ TA 1- _ T I [/ IX T r _ _ _ _ T-I 1* TO, 11ITI _~~~ ~~ 2oZ L 4 i0 I.0 0 l1 - O*"!lllll 0._ I _, I I I I / / / I I I l I 1 rTTI I'l/v /, 1' /I I I',_ II II I 1' / 4 o 0.7 /" /O, _ A I A I I I / _I 0.5 L~~~~~~~~~~~~! iI A' I z /5 I /AO~" / I I A I FT I/1T_/I ] / 1/1 I III I I I I I I1 II - I IITITITTI1 11ITT 03 0.Y.... 6 07 Q.. lJII /11 / /JLL~L M_ [O11~I/ /leI I FI f 1- I I III 0.$!/ / /l IIII! I I I I I I! F N (g) FIG 4. 1 RECEIVER OPERATING CHARACTERISTIC. jen i IS A NORMAL DEVIATE WITH cN-2 - C5S (MS-MN) 2- d N

~9-LI-L 13r ~L-~9-V OL6-W 10 7 FIG. 4.2 6 _ l RECEIVER OPERATING LEVEL. 5 _ _.n I IS A NORMAL DEVIATE WITH 2 4 L 5 i I I | N S ( SN N) CTN 1.0 0.9 0.8 0.7'0.6 0.3 I I I I I I I I 0.2 OD509 1 I I u i I \r 1 I I \d~~~~~~~d= =2 0.1 0.09 0.08 0.06 0.05 0.04 o 4 0.03 0.02 d, 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

29-21-E I-L YY 3r u-9vOL6-VY 99.9 d 36.____ _ _ __ d 25 OF ~/ / J 99. 5 ~ ~ ~, - - __!! d=9Of9 97 Z 6 5 Ci II 1/ I/ / / 1r l 9e 95 / ~ ~ ~ ~ ~ ~ ~ ~ _ //_ — 4 9 94 91 929 1 //r /,90 (3,~/.,.~~o,O/ / 0oe /f- 7 - 0 - 0 100 -n~~~~00FN~ ~'!"~~~~~ ~FG 4.3 RECEIVER OPRATING CHAACTERISTIC 50' 3 /1 / 3 -'' I / I I II I I/I I V I I II I I 1 40 oor ~~~~~~~~~~~~~~~~~~30 2 10~~~~~~~~~~~~~~~~~~~~~2./IIII / IIIII/ / I II lo Ole, 9o /, 1 ~.1 2.3.4.5 1 2 3 4 678910 20 30 4o 50 60 70 so 90 95; I I I!F i I G. 4.3 J ~ ~ ~ ECIE J'RAIN / — 3TEIS I " 0.1~~~~~~1

20 FIG. 4.4 RECEIVER OPERATING LEVEL. In ~ IS A NORMAL DEVIATE WITH 2 2 M-M2 d 2 s9 CA 1 % I ~1, M d 1 1. 10 8 0.9 0.7 0.4 0.2 0..01.05.1.2.5I 2 5 10 20 30 40 50 60 70 80 90 I00 FN(.,) 16

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 4.3 Signal Knowmn Except for Carrier Phase The signal ensemble considered in this section consists of all signals which differ from a given amplitude and frequency modulated signal only in their carrier phase, and all carrier phases are assumed equally likely. s(t) = f(t) cos(wt+0(t)-g). (4.12) Since the unknown phase angle @ has a uniform distribution, dPS(G) = d. (413) The likelihood ratio can be found by applying Eq (3.7), and since the signal energy E(s) is the same for all values of carrier phase 0,1 A(x) = exp [Ex p e xip dPs(s) (4.14) Expanding s into the coefficients of cos 0 and sin 9 will be helpful:2 s(t) = f(t) cos(wt+(t))cos g + f(t) sin(wt+0(t))sin g, (4.15) and 2xiSi =cos I 2Zxi f(ti) cos ( ti + 0(ti)) + sin g xi f(ti) sin (ti + 0(ti)) (4.16) Because we wish to integrate with respect to 9 to find the likelihood ratio, it is easiest to introduce parameters similar to polar coordinates (r, 90) such that 1For this to be rigorously true, it is sufficient that the signal be time limited and have its line spectrum zero at zero frequency and at all frequencies equal 2c to or greater than 2 2 ti denotes the ith sample point, i.e., ti = 17

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN r cogo Xxi f(ti) cos ( ti + (ti) 111 1' @ IT xi f(ti) sin (w ti + (ti)' (4.17) and therefore I1 ~ Xisi= cos (a - ro) (4.18) Using this form the likelihood ratio becomes 2it j(x) = exp [ ] j exip[ cos (G ~)j 1 =exp r ] I( (4.19) where Io is the Bessel function of zero order and pure imaginary argument. Io is a strictly monotone increasing function, and therefore the likelihood ratio will be greater than a value P if and only if - is greater than some value corresponding to P. The quantity r is defined by the Eq (4.17); N is the square root of the sums of the squares of the right-hand sides. The probability that N will exceed any certain value can be computed by observing that each of the right-hand sides is 2 times the cross correlation of x(t) with a fixed signal, either f(t) cos [ cot + 0(t)] or f(t) sin [w t + 0(t)] Therefore, the distribution of each can be found in the same manner as the distribution of 1 Z xisi was found for the case of the signal known exactly, and rr both r Cos 0o and r sin 00 have normal distributions with zero mean and variance 2NE Furthermore, f(t) cos (ct + (t)) and f(t) sin (c t + A(t)) are out of phase, or orthogonal, and therefore r cos Go0 and r sin 90 have independent distributions. See page 9. 2See footnote 1, p. 17. __ __ __ __ ____ ~18

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Because _N =E C gos ) 20 + (N sin 00)2, the probability that r will exceed any fixed value is given by the well-known chi-square distribution for two degrees of freedom, y' ( 1c2). The proper normalization yielding r No zero mean and unit variance requires that the variable be 2 / that is I; (2 2( = exp (4.20) If c is dcefined by the equation B = exp [m]'o (, (4.21) the distribution for i(x) in the presence of noise alone is in the simple form FN(f) = exp [-2] (4.22) Using Theorem 8 of Section 2, namely m dF (3) = dFSN(P3) (4.23) but making use of the parameter c~, we form first dFN( =) =- exp [ 2. ] dce, (4.24) and hence = - exp [ a~]ex[ ~ [e ] ( a ) (4.25) Integrate from ac to infinity. FSN(5) = ep exp [ ] I( a d. (4.26) Cramer, Ref. 14, p. 233, or Hoel, P. G., Introduction to Mathematical Statistics, Wiley, 1947, p. 134. 19

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN'iqs (4.22) and (4.26) yield the receiver operating characteristic in parametric form, and Eq (4.21) gives the associated operating levels.l These are graphed in Fig. 4.5 for the same values of signal energy to noise per unit bandwidth ratio as were used whlen the phase angle was known exactly, Fig. 4.1, so that the effect of knowing the phase can be easily seen. If the signal is sufficiently simple so that a filter could be synthesized to match the expected signal for a given carrier phase 9 as in the case of a signal known exactly, then there is a simple way to design a receiver to obtain likelihood ratio. For simplicity let us consider only amplitude modulated signals (0(t) = 0) in Eq. (4.12)). Let us also choose G = 0. (Any phase could have been chosen.) Then the filter has impulse response h(t) = f(T-t) cos [w(T-t)] 0 < t < T = 0 otherwise. (4.27) The output of the filter in response to x(t) is then eo(t) = x(r ) h(t-r) dr = f x(r ) (cos ( +T-t ) dr -0O t-T cos W(T-t) S x(r) f(r+T-t) cos c t dr t-T t - sin w (T-t) f x(r) f(r+T-t) sin co d T. (4.28) t-T 1Graphs of values of the integral (4.26) along with approximate expressions for small and for large values of a appear in Rice, Ref. 20. Tables of this function have been compiled by J. I. Marcum in an unpublished report of the Rand Corporation, "Table of Q-Functions," Project Rand Report lRM-399..... O20

~2-LI-L W3r 9L-~9-V 0L6-W 1.0 NO 0.9 "'""" 0.8/ /. 0.7 2E 0.6 0o.4 0.3 0.4I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 FN (x) O~~~~~~e~3~ 0.4 I 2 E O.2 w 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 FN (~) a. ~~~~~21. e-%~~~~~~~~~ &00' -

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The envelope of the filter output will be the square root of the sum of the squares of the integrals,1 and the envelope at time T will be proporr tional to N t since (~)2W) = [j x(t) f(r) cos r T d + [ x(-r ) f(t-) sin wr T d] (4.29) = Square of the envelope, at time T, of eo(t). If the input x (t) passes through the filter with an impulse response given by No r Eq (4.27), then through a linear detector, the output will be -2 I at time T. r Because the likelihood ratio, Eq (4.19), is a known monotone function of' l the output can be calibrated to read the likelihood ratio of the input. 4.4 Signal Consisting of a Sample of White Gaussian Noise Suppose the values of the signal voltage at the sample points are independent Gaussian random variables with zero mean and variance S, the signal power. The probability density due to signal plus noise is also Gaussian, since signal plus noise is the sum of two Gaussian random variables: fsN(X): (+) 2 N+Sl The likelihood ratio is n (x) e IS exp - x] (4-31) If the line spectrum of x(t) is zero at zero frequency and at all frequencies 2w equal to or greater than 2- then it can be shown that these integrals contain no frequencies as high as 2Cramer, Ref. 14: p. 212......_ __ 2....22

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN In solving for the distribution functions for i, it is convenient to introduce the parameter a, defined by the equation n i 2 2 Then the condition e(x) > p is equivalent to the condition that x Z xi> a 2 In the presence of noise alone the random variables( have zero mean and unit variance, and they are independent. Therefore, the probability that the sum of the squares of these variables will exceed a2 is the chi-square distribution 1 with n degrees of freedom, i.e., FN(f) = E (C2) (4.33) Similarly, in the presence of signal plus noise the random variables - 1, 2 have zero mean and unit variance. The condition E X > a' is the same as requiring that i1 x > 2 N 2 and again making use of the chi-square N+S i - N+S distribution, FSN(P) = ( N (4.34) Receiver operating characteristic curves are presented in Figs. 4.6 and 4.7 for four possible choices of n (102, 103, 10 4 105), and in each case for three values of signal to noise ratio three db apart. For large values of n, the chi-square distribution is approximately 2 2 normal over the center portion; more precisely, for a > > 0 1Cramer, Ref. 14, p. 233. Tables of It(ca2) can be found in most books on statistics. Extensive tables are listed in the bibliography of Ref. 14, p. 570. 2 P. G. Noel, Introduction to Mathematical Statistics, New York: Wiley, 1947, p. 246. 23

1.0 IC _ _ _ _ _ _ _ 0.9 0.9 ) 0.8 N:O~=.098 0.8 0.8 07 0.7w 0.6 ~~~~~~~~~~~~~~~0.6 z 0.5 0.5 04 0.4 0.3 0.3 0.2 0.2 0.1 10.1 0 _0__ _ 0 0.1 0. 2 Q.3 0.4 N(C 0.6 0.7 0.8 0.9 1.0 0 0.1 0.2 0.3 0.4 FN (i) 0.6 0.7 08 0.. I10 I10 QL_ 5 5__ __ _ 0. 5 S'0.5 CL 68~~~INLA APEO WIEGUSINNIE

1.058 1.0.0-l 1. 00 ( 0.05804 __ I 0.9 0.9, 0.0 0902 t N cu~~~~~~~~ ~~~o9 r./~ 0.8 ooo=0.0291 0.8 OF 0.7 0.7 0 = 0.0145 N 0 00451 0.8o, 0.4 4 r r n 0.3 0.6 0.6 ~~z 0.5 0.5~N=1 N I 04 ~~ ~ ~0 0.4 I / I I I 1.0 0.3 0.3 U1 ~0.1 __ N N:I0 0 0 0 0.1 0.2 0.3 0.4 FN (2) 0.6 0.7 0.8 0.9 1.0 0 0.1 0.2 0.3 0.4 FN (2) 0.6 0.7 0.8 0.9 1.0 10 10 5 5 LI LI 08 = 0.~~~S=0145 z N Z ~ i0.0582 Q50.00451 c-o., / /. o.//, /,5,ools 0:: 0.0291 (I I I I \ 0.0 0 0N0!2 0 0.1 0. FIG. 4.7. RECEIVER OPERATING CHARACTERISTIC. SIGNAL A SAMPLE OF WHITE GAUSSIAN NOISE.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN c0 Kn(c22) i I exp (4 35) 2Y2.,_ V2n-1 and F(n(2 cx2) 1 exp y2 dy If the signal energy is small compared to that of the noise, is nearly unity and both distributions have nearly the same variance. Then Fig. 4.1 applies to this case too, with the value of d given by d = (2n-1) l - (4.37) For these small signal to noise ratios and large sairples, there is simple relation between signal to noise ratio, the number of samples, and the detection index d. I _1S S 1- Jilj:-t~- forf << 1 2 d — nS- (4.38) 2N2 operatin characteristic if the corresponding numbers of sa and n2a, satisfy 2 nL ( (I)2 26

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN This can be verified for the three curves of Fig. 4.7 for n = 10, compared with Fig. 4.1 for d = 1, 4, 16. The receiver specified is any device that produces the likelihood ratio of its input, n i(x) 2+S Z x 2] (4.31) An energy detector has as its output T e(t) = T x(t)2 dt 2 (4.40) 0 and this receiver can be calibrated so that its output at the end of the observation time, eo(t), vwill be read as n o (X) = (1+S) eiTp [ S o() 3 (4.41) 4.5 Vidco DesiCi of a Broad Band Receiver The problem considered in this section is represented schematically in Fig. 4.8. The signals and noise are assumed to have passed through a band INPUT FROM BAND PASS LINEAR VIDEO ANTENNA. 7 FILTER DETECTOR AMPLIFIER OR MIXER POINT A POINT B FIG. 4.8 BLOCK DIAGRAM OF A BROAD BAND RECEIVER. pass filter, and at the output of the filter, point A on the diagram, they are assumed. to be limited in spectrum to a band. of wid~th WS and center frequency 27

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN W > I. The noise is assumed to be Gaussian noise with a uniform spectrum 2> 2 over the band. The signals and noise then pass through a linear detector. The output of the detector is the envelope of the sigals and noise as they appeared at point A; all knownledge of the phase of the receiver input is lost at point B. The signals and noise as they appear at point B are considered receiver inputs, and the theory of signal detectability is applied to these video inputs to ascertain the best video design and the performance of such a system. The mathematical description of the signals and noise will be given for the signals and noise as they appear at point A. The envelope functions, which appear at point B, will be derived, and the likelihood ratio and its distribution will be found for these envelope iunctions. The only case which will be considered here is the case in which the amplitude of the signal as it would appear at point A is a known function of time. Any function at point A will be band limited to a band of width ~W and center frequency @L> - -. Then the alternate form of the sampling theorem 2ic 2 can be used.1 Any such function f(t) can be expanded as follows: f(t) = x(t) cos ct + y(t) sin t (4.42) W where x(t) and y(t) are band limited to frequencies no higher than 2-, and hence can themselves be expanded by the sampling theorem: f(t) (= Z [x(w)i(t) cos w t + y( )fi(t) sin wt] (4.43) The function can be thought of as a point in a space of n = 2WT dimensions with coordinates x( ) = xi and y() = i. This is a rectangular coordinate 1See Appendix D..... 28........28

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN system, since the family of functions 1i(t) cos Wt and ik(t) sin wt form an orthogonal system. The amplitude of the function f (t) is r(t) = [x(t)] + [y(t)]2 (4.44) and thus the amplitude at the ith sampling point is r(' ) = r. = 2+ y2 (4.45) The angle Yi Xi. = arctan = arccos -- (4.46) might be considered the phase of f(t) at the ith sampling point. The function f(t) then might be described by giving the ri and.i rather than the xi and yi. The ri and gi are sample values of amplitude and phase, and forma sort of polar coordinate system in the space associated with the set of functions. Let us denote by xi f,i, or ri. Gi. the coordinates or sample values for a receiver input after the filter (i.e., at point A in Fig. 4.8). Let ai, bi, or ip i denote the coordinates for the signal as it would appear at point A if there were no noise. The envelope of the signal, hence the coordinates fit are assumed known. Let us denote by FS(l, 02' 2'' On) the distribution function of the phase coordinates Oi. The probability density function for the coordinates xi., yi when there is white Gaussian noise and no signal is n fN(x Y) ( ~1 [ 1 n/2 n/2 1 fT (X Iy) = (I) exp 2 2N Xi + i ) (4.47) i=l i=l 29

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN and for signal plus noise an 1 1 n/2 2 n/2:SN(x y) = ( ex[ n )2]d 2- iS=l (xi-ai + y Y-bi Ps(aibi) \ -N(i1 \i=l (4.48) Changing to the polar coordinates, (') i n (449) andl n fsN3fS'd Rr,O)x2n 2 n12 2 2 S= ( ) n/ r exP [- 2 {r.i -2rifi cos (Gi-,i i=l' = S n/21 *** *50) The factors l ri are introduced because they are the Jacobinn of the i=l 1, 2 transformation from rectangular to polar coordinates.' The probability density function for r alone, i.e., the density function for the output of the detector, is obtained by simply integrating the density functions for r and 9 with respect to 9.3 2x 2x 2c fN(r)= rJ J f (ri i) dG dG2 oedgn 0 0 0 2 n 2 n/2 n/2 N) ri exPL 2 ri2] (4.51) i=l i=l Cramer, Ref. 14, page 292. For excmple, in two dimensions, fT(x, y)cdi dy = fN(r, 0) r dr d0. JCramer, Ref. 14, page 291. 30

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN and 2 2x 2x 2 f %(r, qj)-, % d9... SNf(r).= S... (rini)d d 0 0 0 2 n /!2 /2[ n/2 (r f.\ (1) n2 ri P[ 2 (ri ~f+i ) I 2IoK )d(Il2 "on) 2 r n/2 n/2 =(Iri ( ) exp [ - e 2 (ri2fi2) (4.52) Niotice that the probability density for r is completely indenendent of the distribution which the Oi had; all inforrmation about the phase of tihe signals has been lost. The likelihood ratio for a video input is j) S(r) [ n/2 n/2 rf fSN (r) J rZfI (r)= [ 2ed C r.x 2] f II. (4 53) i=l i=l Again it is more convenienxt to work with the logarithm of the likelihood ratio. 21 il 2 - = J f(t)]2 dt = E and (4.54) 2~-0 i=l i =l Pn e (r) = - N7 ~ —) (4 55) which is approximately T in 2(r(t)) = - + W n r(t) f (t) dt. (4.56) 0 ~ 3~~~~~~~~~1

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN The function.1n Io(x) is plotted as a function of x in Fig. 4.9. This function is very nearly the parabola T for small values of x and is approximately linear for large values of x. Thus, the expression for likelihood ratio might be approximated by T,en( (t)) = o 4 [rW(t)]2 [f(t)]2 dt (4.57) 0 for small s ignals, and by T ini(r(t)) = C1 + C2 r(t) f(t) t (dt.58) 0 for large signals, where C1 and C2 are chosen to approximate An Io best in the desired range. The integrals in Eqs (4.57) and (4.58) can be interpreted as cross correlation. Thus the optimum receiver for weak signals is a square law dcetector, followed by a correlator which finds the cross correlation between the detector output and (f(t))2, the square of the envelope of the expected signal. For the case of large signal to noise ratio, the optimum receiver is a linear detector, followed by a correlator which has for its output the cross correlation of the detector output and f(t), the amplitude of the expected signal. The distribution function for.(r) cannot be found easily in this case. The approximation developed here will apply to the receiver designed for low signal to noise ratio, since this is the case of most interest in threshold studies. An analogous approximation for the large signal to noise ratios would be even easier to derive. First we shall find the mean and standard deviation for the distribution of the logarithm of the likelihood ratio:

tSg-ZZ —t W3r zI-~9-gv Oz61 3.5 3.0 2.5 2.0 PARABOLIC AP PROXIM O L ATI O N 1.0 0.5.. /.. GRAPH OF LN IC(X).

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN n/2 In ~(r) - +- Z ri2 fi2 (4.9) i=l for the case of small signal to noise ratio. The probability density functions for each ri are -ri ri2+fi2 r]f1 ad g6NJ(ri) = I exp q 11'r 2N- I0 IT~, and gN (ri) = I 2 (4.60) The notation gj(ri) and GSN(ri) is used to distinguish these from the joint distributions of all the ri which were previously called fN(r) and fSN(r). The ri fi2 mean of each term i 1 in the sum in Eq (4.59) is T2 2 (ri2fi2 f 1- exp - 2N' j Io /ri 0 co 0 CO O ei u (r-) dri = exp[ The second moment of each te ri is 0 4 o - 4 r5 (ri2+rif2) rfi

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN oO ri4fi i ri r 4 16i(/ 1612 u 0 co iexp ['i ri2 ](4.62) Thle integrals for the case of noise alone can be evaluated easily: (ri2fi2 f2 16 2N2 The integrals for the case of signal plus noise can be evaluated in terms of the confluent hypergeometric function, which turns out for the cases above to reduce to a simple polynomial. The required formulas are collected in convenient fonr in the book, Threshold Sigals by Lawson and UThlenbeck.1 The results are rif i f 1 fi fi ISN\ 4I22 f2 ( /2 r i4fi4 f14 f 2 fi4 4 )SN 816N 4 (4.64) Since (Z) = (Z () (4.65) (z) = (Z2) - [/(z)J, ([.65) ri2f.2 the variance of 2 is N2 Iief. 1., p. 174 255

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN (2f)2 f (4 f 2 2 ri2fi _ 2. N 4N2 412 (4.66) For the sum of independent random variables, the mean is the sum of the means of the terms and the variance is the sum of the variances. mean of In t(x) is n/2 n/2 f 2 f4 n/2 f4 z, 2 z i I - i SN (in A(r)) = E-' ~ 2fi + j + ]= 2 l i=l i=lI N i=l n/2 n2 2 / nN (in)r)) = - + i i;~I 2 N~ ((. 67) i=l i=l and the variance of' in (r) is 6 n/2 f 4 f C 2 (4.68) If the distribution functions -n L(x) can be assumed to be normal, deviation of the distribution. In some cases the normal distribution is a good approximation to the actual distribution. 36..

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Let us consider the case in which the incoming signal is a rectangular pulse which is - seconds long. The energy of the pulse is half its duration times the amplitude of its envelope, and therefore the amplitude has the value fi = / (4.69) where E is the pulse energy. It has this value on M sample points and is zero at all others. For this case I s, ~n ec,) =1 E2 /.LSN (.e n ~(r)) M No2l T7 (1n A(r)):= 0 (~ n 1(r)) 2 (1 + - ) ~~~2 B'2 Cr2 (n (r) = (4.70) 112102 Also, for this case, the distribution of in i(x) is approximately normal, if M is much larger than one. Since it is the sum of M independent random variables, all having the same distribution, it must, by the central limit theorem,2 approach the normal distribution as M becomes large. The actual distribution for the case of noise alone can be calculated in this case) since the convolution integral3 for the gi(ri) with itself any number of times can be 1The problem of finding the distribution for the sum of 14 independent random variables, each with a probability density ifnction f (x) = x exp [- (x2.c2)] I(C) arises in the unpublished report by J. I. M4arcum, A statistical Theory Of T:ar Detection by Pulsed Radar: Mathematical Appendix, Project Rand Report R-113. Marcum gives an exact expression for this distribution which is useful only for small values of M., and an approximation in Gran-Clarlier series which is more accurate than the normal approximation given here. Marcum's expressions could be used in this case, and in the case presented in Section 4.6. Camer, Ref. 14, p. 213 and 316. 3Cramer, Ref. 14, p. 188-9. 37

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN expressed in closed form. The density function for this distribution is plotted in Fig. 4.10 for several relatively small values of M. The distribution of -n A(x) for signal plus noise is more nearly normal than the distribution for noise alone, since the distributions gsN(ri) are more nearly nozrml than gN1(ri) ~ The receiver operating characteristic for the case M = 16 is plotted in Fig. 4.11 using the normal distribution as approximation to the true distribution. In many cases it will be found that 1 2E << 1. (4.71) 0 In such a case the distributions have approximately the same variance. Assuming normal distribution then leads to the curves of Fig. 4.1, with d 1 (? i (4272) 4.6 A Radar Case This section deals with detecting a radar target at a given range. That is, we shall assume that the signal, if it occurs, consists of a train of M pulses whose time of occurrence and envelope slope are known. The carrier phase will be assumed to have a uniform distribution for each pulse independent of all others, i.e., the pulses are incoherent. The set of signals can be described as follows: M-1 s(t) = 2 f(t+mr) cos (wt+Gi) (4.73) m=O0 where the M angles Gi have independent uniform distributions, and the function f, which is the envelope of a single pulse, has the property that 38

M=$ M=9 M-I? 1.0 10~~~~~~~~~~~~~~~~~~0 I 0.9 0.8 1 I r I n I I n II I 1\1\ I 1 I I I. /~~~~~~~~~~~~~~~~ 0.8 ___ G(9 0.7 0.6 0.5 04 0.2 0.1 0~ -I -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 N0.LN FIG. L4.10 PROBABLE DENSITY O LN VS. LN MAX. PROBABLE DENSITY E

~-IZ-L W 3p 08og-~-v OLZ699.9:i0 t 8/t- ~ /' 0000 99,5 2 E = 36 PI / 00. /./ 98 No 00 9 OAN3-<or~~ 0<CA'. 7t-0, 97 XX /22-/"' / Z= I -'/ /E Z _ I I I _ _ _ _ = = ==o95 J' /''/'- /~:694 I tLC 1 1 11 I I 11~~- 00= 1.10 r 1100 1 2 E / 3.,',' ~ 92 /f'1~~~ = /'/'.//,// ~~~~~~~~~~~~91 // 2E-I 4 1 9 31 No 1000N0(~ d// 00<'100 11.080./~~~~~~~~~~~~~~~? I z / IIII I 70.o/ / ~~~~~~~~ 50 RC/ / R I 6C0T I I/O 0/,DEO D0 M1 40 000,~ ~ ~ ~ ~~~~~~3 /o /,,/. 20 //I? 4 70 ~.. 3 0. 5 ~~~~~ior.2.3.4.5 I 2 3 4 5 678910 20 30 40 50 60 70 80 90 95 I00 FN(E) FIG. 4,11 RECEIVER OPERATING CHARACTERISTIC, BROAD BAND RECEIVER WITH OPTIMUM VIDEO DESI6N, M=16. 40

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN fT (t+i T ) f(t+j T ) dt -= (4-74) 0 where 8. is the 1ironecker delta function, which is zero if i f j, and unity if i = j. The time r is the interval between pulses. Eq (4.74) states that the pulses are spaced far enough so that they are orthogonal, and that the total signal energy is E.1 The function f(t) is also assumed to have no frequency components as high as 2x - The likelihood ratio can be obtained by applying Eq (3.7). 0 I-T[ 2xA 2g - T M-l = exp[- j f1 -.Ef exp[2 f X f(t+mr)x(t)cos (t+@m)dt d.o~.. 11 N~o N 0 0 m=o dO (4.76) The integral can be evaluated, as in Section 4.3, and -i(x) = exp J I( (4.77) M[-1 where 2 T T2 = N = ff(t+m r)x(t)cos 0 tdt + - of(t+mr) x(t)sinw tdt (4.78) 0 0 This quantity r is almost identical with the quantity r which appeared m in the discussion of the case of the siglal known except for carrier phase, Section 4.3. In fact, each rm could be obtained in a receiver in the manner 1The factor 2 appears in (4.74) because f(t) is the pulse envelope; the factor M appears because the total energy E is M4 times the energy of a single pulse. I ___ ____ ___ ____ ____ ___ ____ ____ 41 _ ___I

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN described in that section. The quantity ro is connected with the first pulse; it could be obtained by designing an ideal filter for the signal s (t) = f(t) cos (wt+9) (4.79) for any value of the phase angle G and putting the output through a linear detector. The output will be No ro at some instant of time to which is deter2 N mined by the time delay of the filter. The other quantities rm differ only in that they are associated with the pulses which come later. The output of the filter at time to + mr will be 2~ rm. 2 N It is convenient to have the receiver calculate the logarithm of the likelihood ratio, M-1 In I(x) - + in Io( (4.80) N0 m=o Thus the In Io (r) must be found for each rm, and these M quantities must be added. As in the previous section, rm will usually be small enough so that in Io(x) can be approximated by xv 1 The quantities1 (I;) can be found by using a square law detector rather than a linear detector, and the outputs of the square law detector at times to, to tT+..., to + (M-1)r then must be added. The ideal system thus consists of an i.f. amplifier with its passband matched to a single pulse,2 a square law detector (for the threshold signal case), and an integrating device. We shall find normal approximations for the distribution functions of the logarithm of the likelihood ratio using the approximation n I rm)i n (4.81) 1See Fig. 4.9. 2It is usually most convenient to make the ideal filter (or an approximation to it) a part of the i.f. amplifier. 112

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN which is valid for small values of 1 N' E D M 1 I rm )2 (4 ) n~+ N (4.82) 0 n=o The distributions for the quantities rm are independent; this follows from the fact that the individual pulse functions f(t+m-r) cos (w t+Gm) are orthogonal. The distribution for each is the same as the distribution for the quantity r which appears in the discussion of the signal known except for phase; the same analysis applies to both cases. Thus, by Eq (4.22)2 P( ( N- ) = exp- 2 or r2N (4.83) r a p\- ) a - expn J and by (4.26), 0 rm > NoM E a2N SN N = e a exp [ IO(a) da (4.84) The density functions can be obtained by differentiating (4.83) and (4.84): CJ( N) = O( N) exp [( ) ( )] N (in) ~ )e]p [ ( ) (- ). (4.85) rm = o m exp [- rm I ( 1See footnote 1, p.37. 2The M appears in the following equations because the energy of a single pulse is rather than E.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN This is the same situation, mathematically, as appeared in the previous section on page 34. The standard deviation and the mean for the logarithm of the likelihood ratio can be found in the same manner, and they are pSN (In ) -= E2 bL N (In l) = 0 CS.N (9 n ) = E 1 + = E2 (nIN) = E 2 (4.86) If the distributions can be assumed normal, they are completely determined by their means and variances. These formulas are identical with the formulas (4.70) on page 37 of the previous section. The problem is the same., mathematically, and the discussion and receiver operating characteristic curves at the end of Section 4.6 apply to both cases. 4.7 Approximate Evaluation of an Optinum Receiver In order to obtain approximate results for the remaining two cases, the assumption is made that in these cases the receiver operating characteristic can be approximated by the curves of Fig. 4.1, i.e., that the logarithm of the likelihood ratio is approximately normal. This section discusses the approximat ion and a method for fitting the receiver operating characteristic to the curves of Fig. 4.1. It was pointed out in Section 2.5.1 of Part I of this report that FSN(t ) can be calculated if FN(fe) is known. It was further pointed out that the nth moment of the distribution FN(.t) is the (n-l)th moment of the distribution FSN(le). Hence, the mean of the likelihood ratio with noise alone is L.....

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 2 unity, and if the variance of the likelihood ratio with noise alone is cN, the second moment with noise alone, and hence the mean with signal plus noise is 1 + 0~. 2Thus the difference between the means, and the variance with noise 2 alone are the same number ONT This number probably characterizes the receiver reliability better than any other single number. Suppose the logarithm of the likelihood ratio has a normal distribution with noise alone, i.e., co -N~i L = ep 2d X, (4.87) where m is the mean and d the variance of the logarithm of the likelihood ratio. The nth moment of the likelihood ratio can be found as follows: co co ZN(Ln) = n dFn() = 1 exp[nx]exp 2 x (4.88) ~0 2c1 -JX 2:CO where the substitution I= exp x has been made. The integral can be evaluated by completing the square in the exponent and using the fact that exp[ _x x ja=v i f-~r 22d -00 N(L) = ex[ 2-+mn]. (4.89) In particular, the mean of >I(x), which must be unity, is yN(%) e = 1 = exp[ + m,, (4.90) and therefore m = -i, (4.91) The variance of ~(x) with noise alone is 2C2 and therefore the second moment 45

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN of ~(x) is Ix I2) /~N(_ 22 2(N) l+ o -,2(: ) (.92) and this must agree with (4.89). /LLt(2) =1+ N2 exp [2d + = exp[d] (4.93) and therefore d = in (1 + o-) (4.94) The distribution of likelihood ratio with sigmal plus noise can be found by applying Theorem o.1 d.si(- ) = iF(I ) ) CO = f d~~~~~~ ~(4.95) FSN(Z) = - J' J N( Substituting for FI () from (4.87), and letting I = exp x yields II Fs(- ) J exp[xep 2d dx Thus the distribution of veni is normal also when there is signal plus noise, in d. this case with mean - and variance d. In summary, the variance cr-2 of the likelihood ratio probably measures the receiver reliability better than any other single nugmber. If the logarithm of the likelihood ratio has a normal distribution, then this distribution, and 1See Part I, Section 2.4. 46

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN hence the signal plus noise distribution, are completely determined if CO is given. Both distributions of -in A(x) are normal with the same variance d, and the difference of the means is d. The receiver operating characteristic curves are those plotted in Fig. 4.1, with the parameter d related to - N2 by the equation d = in (1 + 2) (4.94) In the case of a signal known exactly, this is the distribution which occurs. In the cases of Section 4.41 Section 4.5, and Section 4.6 this distribution is found to be the limiting distribution when the number of sample points is large. Certainly in most cases the distribution has this general form. Thus it seems reasonable that useful approximate results could be obtained by calculating only o 2 for a given case and assuming that the receiver reliability is N approximately the same as if the logarithm of the likelihood ratio had a normal distribution. On this basis, oN2 ( ) is calculated in the following sections for two cases, and the assertion is made that the receiver reliability is given approximately by the receiver operating characteristic curves of Fig. 4.1.l with d = An (1 + N2). 4.8 Signal Which is One of M Orthogonal Signals The following case has several applications, which will be discussed in Section 5.3. The importance of this case, and the one which follows it, lies in the fact that the uncertainty of the signal distribution can be varied by changing the parameter M. Suppose that the set of expected signals includes just M orthogonal functions sk(t), all of which have the same probability, the same energy E, and 47

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN are orthogonal. That is, T sk(t) Sq(t) dt = E kq (4 0 Then the likelihood ratio can be found from Eq (3.7) to be 1(x) = E M exp [ - N l= kl ep E xi9s ] 1M i k=l o i=l M n M 1 N 1N (4.98) where ski are the sample values of the function sk(t). It should be clear that with noise alone, the terms xki n 2 have a Gaussian distribution with mean zero and variance = ki' N N0 ni=l Furthermore, the M different quantities xii are independent, since the i=l functions sk(t) are orthogonal. It follows that the terms exp ~ xiSki- N i=l are independent. Since the logarithm of each term Z = exp sJ has a normal distribution with mean - I; and variance the moments of the distriITO N' th bution can be found from Eq (4.89). The n moment is P/ (Zn) = exp [n(n-1) 1 (4.99) 1The reasoning is the same as that on page 9. h8

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN It follows that the mean of each term is unity, and the variance is 2(z) = L(z2) - [iil(Z) 21 e 2ETx[] l. (4.lOo) The variance of a sun of independent random variables is the sum of the variances of the terns. Therefore 2 (M ~= [M ( [ x( ) - 1] (4.101) and it follars that the variance of the likelihood ratio is 02() = 4ex(2) _l]. (4.102) It was pointed out in Section 4.7, page 47 that the receiver operating characteristic curves are approximately those of Figure 4.1, withf d I On (1 + M 2)= n - + ) exp ) (4.103) This equation can be solved for -i: N0 -Eo n 1 + M4 (ed 1). (4.104) Curves of To for constant d are plotted in Fig. 4.12. They show how much the signal energy must be increased when the number of possible signals increases.!4.9 Signal Which is One of M Orthogonal Sigals with Unknown Carrier Phase Consider the case in which the set of expected signals includes just M different amplitude modulated signals which are known except for carrier phase. Denote the signals by sk(t) = 4k(t) cos (wt + 6) (4.105)

W I oI I I I I I I I I = 4 2 3 4 5 6 7 8 9 10 20 30 40 50 FIG. 4.12. SIGNAL ENERGY AS A FUNCTION OF M AND d. SIGNAL ONE OF M ORTHOGONAL SIGNALS.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN It will be assumed further that the functions fv(t) all have the sane energy E and. are orthogonal, i.e., rp I fk(t) fiq(t) dt = 2Bkq, (4.106) twhlere the 2 is introduced because the f's are the signal nmplitudes, not the actual signal functions. Also, let the fk(t) be band-limited to contain no frequencies as high as W. Then it follows that any two signal functions with different envelope functions will be orthogonal. Let us assume also that the distribution of phase G is uniform, and that the probability for each envelope function is. With these assumptions, the likelihood ratio can be obtained from Eq (3.7), and it is M 2~c 14~~~iz where ski are the sample values of sk(t), and hence depend upon the phase 0. The int;eration is the same as in the case of the signal knaon except for phase, and the result can be obtained from Eq (4.19) 2(x) = exp [ 1 NE ) (4.108) k=l [ where r = ( XXi fk(ti) t (2 f )t sin wt)2. (4.109) rk i ) cosw +t (t skino Now the problem is to find crN (2). The variance of each term in the sum in Eq (4.108) can be found, since the distribution function with noise alone can be found as in Section 4.3. Since the fk(t) are orthogonal, the 51

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN distributions of the rk are independent, and the terms in the sum in Eq (4.107) are independent. Then the variance of the likelihood ratio, 0N2 ( ) is the sum of the variances of the ternms, divided by 142. 1 The distribution function for each term exp NE ( ) s given in Section 4.3 by Eq (4.21) and (4.22). If a is defined by the equation = exp [ E I (4.10) then the distribution function in the presence of noise for each term in Eq (4.108) is FN )(k ) = exp a[- ]. (4.111) The mean value of each term is (k) = PdP (k) = Ia exp E l ( @) N ITO-,0) exp jj j o 2 1ex{ This can be evaluated,2 and the result is that () = 1. The second moment of each term is CD (k)(P2) =2 (k)@) 0 2E C? 3[.x - I a a )] p - ] (4-e113) 1Cramer, Ref. 14, p. 188. 2Lawson and Uhlenbeck, Ref. 1, p. 174. 52

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The integral is evaluated in Appendix E, and the result is |" fiLN ( ~( I0 )2 (4.14) 0 The variance is [LNn(k) ] 2 i - 2i ~= I [<J (N) (13t EL~k) 2) [1 ELk) (P)] = I ( 213 )_ 1 (4.1 5) 0 It follows that the variance of M ~ is 0o (Mi) M (I - 1, and (4,116) 2 0- ( )i( N) I]' (4.117) ~N (' M oJ since the variance for the sum of independent ralndoam variables is the sum of the variances. If the approximation described in Section 4.7 is used, the receiver operating characteristic curves are approximately those of Fig. 4.1, with d: 1n (1 + M2 In - N+E ))I (4.118) Curves of E vs M for constant d are plotted in Fig. 4.13. N3

d=4 5 w 00000~~~~~~~~~~ CDf' 51~~~~~~~~~~__ r 1 0 d~l 4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ d=~.50~~0 WI? 3 0000.25 2 I 2 3 4 5 6 7 8 9 10 20 30 40 50 M FIG. 4.13. SIGNAL ENERGY AS A FUNCTION OF M AND d. SIGNAL ONE OF M ORTHOGONAL SIGNALS KNOWN EXCEPT FOR PHASE

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 5. DISCUSSION F TF SPECIAL CASES 5.1 Receiver Evaluation 5.1.1 Introduction. In Section 2.5 it was shown that the receiver reliability can be determined from the distribution functions for likelihood ratio. In particular an optin criterion receiver operating at the level. of likelihood ratio has false alarm probability P (A) = F ((), and probability of detection PsI(A) = FSW(3). The functions F (() and FN(B) are calculated in N FSN(a a Section 4 for a number of special cases. For the purpose of discussing receiver reliability it is sufficient to have the receiver operating characteristic in which FSN(() is plotted as a function of FN(3). In this discussion B plays only a secondary role. The receiver operating characteristic shown in Figure 5.1 applies to several cases. Among them is the case of the signal known exactly, with the parameter d equal to NE twice the ratio of signal energy to noise power per unit bandwidth.l Thus, for example, if the signal is a voltage which is a known function of time, and if the signal energy is twice the noise power per unit bandwidth, theoretically a receiver could be built with false alarm probability of 0.25 and a probability of detection 0.90. If the false alarm probability is required to be no greater than 0.10, the probability of detection can be made no greater than 0.76. If the false alarm probability is required to be no greater than 0.025 and the probability of detection is to be at least 0.98, the signal energy must be at least eight times the noise power per unit bandwidth. 5.1.2 Comparison of the Simple Cases. Several curves for the case of a signal known except for phase are shown in Fig. 5.2 for some of the same values 1See Section 4.2.......... -55....

~S-Ll-L W3r ZL-~9-tV OL6 -W 1.0 d1 I 1.-.. 1.~ t "I/ w 7; < > 6-7 0.9 I 0.7.../ #~~~~~~o lo I 11 1 O 4____ _r"_ l _ ___ I I I00 1 1 1 11 1 1I I, _,. I I 1 1 1r/11101/ I/d:I14 Ffe w lI HI/ I II I II " I 1/1 1/,,,/ _, _ I//_II' dl__I -_ 0.8! /'d=2 0.7 /." I/Tll__ /1 _/ / III 4- I// -------— I' 0.6 Or' 1 0 2 0.3 L4HIIII1MS111111 111111 / /' /-./ /OF' / oz /5, OFa. // 0 1 Is 1 1ORMAL DEVI I I ITH / I I:.L I II I~~ /N J / 0.6 I / Yl' I /' I %0. 5,/ 04 Ii [JL~q[1/1 1-I,,1/1! lI I 1/I! II IIIIIIIIIII i I 0I2 l/ O 1Z. 1 I/0' I ___ 1J 1 I~l IA I 1} tI I I I! I A l! /1 I X l l l l I / 1.I:tg! /;'X,~ / 1///1I V' A I~0r& A A _% 11 I I I A -I Ir _ - _s _a__ll__, I I 2 2 If IL A IL Is II -' 1t I r0X I /i Ii / I' / l|_IIIIIIIIIIT - I I I I / I O 0. 1 0.2 /. /.. 6 07 Q.. -.~ ~~~~~FG 5.1 REEIE OPRTNGCAATEITC /n. i ANRA DEIT WIHF=CS, MNM)=d N2 / I~~~

~S-LZ-L W 3rP S-~9-tv 0oL6 -W 1.0 I I I 0.9 No ~~~~~~~~~0.70~~~~~~~~~~~~.., e * r 1iri n en 2.,"r E'.f /ff j' X< 0.8 0 0.1~~~~~~~~~~~~~~~~~~0 PI I 1,00 1,000" S ~2E F. 4 i. N o'/ J" / /' 0.7RCEVE OPRTN HRACEITC i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~2~~~~~~~~~~~10 0, 7 /'' / 2 E 0.6~ ~ ~ ~~~~ -/ Ii I/'.......: SGNAL KNOWN EXCEPT FOR PHASE. 5,'7 o z~~ ~ I,f j, o 0.6~~~~~ i ~, F I' -' I/, LIT ~ I I,OF 2e1 /' / 0. 5 iOF'' /' / i 0"Z,I/,0'' ii....... 0'4~~~~~~~ I G.5. RECEIVER —/' — OPRAIN CA CTRSI....] -J..J'... f ~IGA KNW EXEP FO PHASE... _

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN of the ratio f as appear in Fig. 5.1. The curves for a given energy lie below those for the case of the signal Imowm exactly; with a given false alarm probability and a given value of E one cannot achieve as high a probability of detection if the carrier phase of the signal is unknown. It was found that in several cases the distribution of in I(x) approached a normal distribution as a limiting case, and that in the limit the variance with signal plus noise and the variance with noise alone are equal. In any such case the curves of Fig. 5.1 apply, and a comparison of these cases is simplified. For example, in the case of a signal which is a sample of white Gaussian noise, it was found that if the numiber of sample points is large and the signal to noise ratio is small, then this approximation applies, with d = (2n-1) (1 - T r n 2 (437) Other curves for this case, sane with small sample number and moderate signal to noise ratio, are given in Figs. 4.6 and 4.7. The exact equations for the distribution are Eqs (4.33) and (4.34). The following two cases lead to the same receiver operating characteristic in the approximation considered in Sections 4.5 and 4.6: (1) the broad band receiver with optimum video design, with a pulse signal, and (2) the optimum receiver for a train of pulses with incoherent phase. In the first case the parameter M was taken as the product of the total bandwidth of the receiver and the pulse width of the signal. In the case of the train of pulses, M is the number of pulses. In each case E is the total energy of the signals. Approximate receiver operating characteristics are plotted in Fig. 4.10. Small signal to noise ratio and large M lead to the distributions for which Fig. 5.1 is

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN plotted, this time with d 1c2 )= (4.71) 0 5.1.3 An Approximate Evaluation of Optimum Receivers. Some simpler evaluation of receivers was needed because of the difficulty in solving directly for the distribution function of likelihood ratio in any cases more complicated than the ones already mentioned. It seemed reasonable to approximate the actual receiver operating characteristic by the curves given in Fig. 5.1, finding in scme manner the value of the detection index d which leads to the best fit of the approximate curve to the real curve. This is suggested by the occurrence of the curves of Fig. 5.1 in four of the five cases already discussed. Also, any receiver operating characteristic must have in conmmon with the curves of Fig. 5.1 that its slope is positive and its second derivative is negative, and that it must start at the lower left hand corner and end at the upper right Luand corner of the graph. It is shomn in Section 2.5.2 that the variance CN2 of the likelihood ratio when there is noise alone is the same as the difference of the means of likelihood ratio with noise alone and with signal plus noise. This paraemeter aN2 seems to characterize signal detectability better than any other single number. In Section 4.7, it is shoam that if aI'2 is given and the logarithm of the likelihood ratio is assumed to have a normal distribution with noise alone, then it follows that the logrithmr of the likelihood ratio with signal plls noise also has a normal distribution with the sane variance, and thus the receiver operating characteristic is that of Fig. 5.1. The index d is given by d = n (1+ 2). (4.94) 59

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN It seems reasonable that the curves be fitted on this basis, i.e., that oIT be determined for tile actual situation ald the approximate receiver operating characteristic graph be taken as the curve of Fig. 5.1 with index d given by the above Eq (4.94). 5.1.4 The Signal One of'M Orthogonal Signals. The methods of the previous section have been applied to the case where tihe operator knows that the signal, if it occurs, will be one of M orthogonal functions of equal energy. Orthogonal, of course, means that the functions have zero cross correlation, i.e., f(t) and g(t) are orthogonal if T f f(t) g(t) dt 0 (5.1) where the integration is over the observation interval. The value obtained for 2 1 cIT is 2 22 C, e ITxp ) _ (4.102); 1 [ ( a)_ 1], and hence (4.117) 6d = n l1 - M+ M ) 2.E (4.118) See Section 4.6. 2See Section 4.7. 60

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN These two cases are the basis for the best approximation available to the problem of a signal of unilmnown t-ime origin or a signal of unknowm frequency or both. Wor example, wre have been unable to find the distribution of likelihood ratio for the case of a signal which is a pulse of uninown carrier phase if the starting- time is random and distributed uniformly over a tire interval. IHowever, if the problem is clhnged sligitly, so that the starting time is restricted to tines spaced approximzately a pulse width apart, then pulses starting at differelnt timnes woutld be approxinately orthogonal, and the case of the si ial one of 14 orthogonal signals knoim except for phase could be applied. Eq (4.118) should be used with 14 equal to the ratio of observation tinle to pulse width. A similar argument applies to the case in which a signal is a pulse kmnown except for phase and center frequency. Eq (4.118) should be used with 1M taken as the ratio of total bandwidth to signal bandwidth. It should be pointed out that it is not the sane to assume that the signal can appear in only a finite number of different positions, even thougl the positions are close to each other, as to say that the signal can appear anywhere in an interval. There is more uncertainty in the latter case, and the signal cannot be detected as easily. 5.1.5 The Broad Band Receiver and the Ideal Receiver. One common method of detecting pulse signals in a frequency band is to build a receiver whose bandwidth is the entire frequency band. The receiver operating characteristic for such a receiver with a pulse signal of known starting time is calculated in Section 4.4. This is not a truly ideal receiver, and it would be interesting to compare it with an ideal receiver. This can be done using the approximation of the preceding paragraph for the ideal receiver. Since the bandwidth of a pulse is approxir;ately the reciprocal of the pulse width, the parameter H of Section Is.4 and the parameter 1I in Eq (4.118) are both equal to 61

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN the ratio of total bandwidth to pulse bandwidth. Curves showing 1 as a function of d are given in Fig. 5.5 for the approximate ideal receiver and the broad band receiver for several values of M. The expression used for d is Eq (4.71) which holds for large values of M. 5.1.6 Uncertainty and Si Detectabilit. In the two cases discussed in Section 5.1.4, where the signal considered is one of M orthogonal signals, the uncertainty of the signal is a function of M. This gives us an opportuity to study the effect in these two cases of uncertainty on signal detectability. In the approximate evaluation of the receiver built to detect the presence of a signal when the signal is one of M orthogonal functions, the curves of Fig. 5.1 are used with the detection index d given by d = en [l - exp ()] (4.103) This equation can be solved for the signal energy. 21"= n [1- M + Med] i V n 1M +*- n (del) (5.2) 2E 1 the approximation holding for large. From this equation it can be seen that the signal energy is approximately a linear function of In MI awhen the detection index d, and hence the ability to detect signals, is kept constant. 1If 21 > 53 the orror is less than 10P7. 2It might be suspected that 2~ is a linear function of the entropy = - E pinpi, where Pi is the probability of the ith orthogonal sinal. This is not the caseo except when all the Pi are equal. The expression which occurs in this more general case is: I2 as - in [ pi2]+fn (d-.l) _ 62

~g-liz-L W3P ~8-~9-V O.6-W 100 __-_ 90 _ 80 200 70 50 BROAD BAND 40 WIlz 30 20. IDEAL M = 200. I0 9 8 f 6 4 2 4 6 8 10 12 14 16 18 20 22 24 d

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 5.2 Receiver Design. There are a few cases when the receiver design is simple to specify if the noise is Gaussian. If, for example, not only the noise, but also the signal are Gaussian, and both have a uniform spectrum over their bandwidth, then the optimum receiver simply measures the energy which comes in during the observation period. The simple relation between energy and lilelihood ratio is given by Eq (4.41) of Section 4.4. The simplest remaining case is that in which the signal is kmown exactly. Then the theory specifies that the receiver find the cross correlation between the expected signal and the receiver input, i.e., f s(t) x(t) dt, (5.-) 0 where s(t) is the expected signal and x(t) is the receiver input, and the observation interval is from t = 0 to t = T. The ratio of this cross correlation to the noise power per unit bandwidth is one-half the natural logarithm of the likelihood ratio.1 Several elaborate correlating devices have been built recently.2 There is, in this case, a simple means of obtaining the correlation, if the signal is simple in form, for example, a pulse. If a filter can be designed with impulse response h(t) = s(T-t) if O t< T, = 0 otherwise, (4.10) and the receiver input applied to the filter, then the output at time T will be 1See page 9, Eq (4.lb). 2Harrington and Rogers, Ref. 16; Harting and Meade, Ref. 17; Lee Cheathan, and Wiesner, Ref. 18; Levin and Reintzes Ref. 19. 1 61p~~~~~~~Q

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN T T x (T-) h(T-T) dT = f x(-) s (T) d-, (4.11) -CO 0 which is the required correlation. It turns out that this is the same filter specified by Middleton, Van Vleck, Wiener, North, and Hansen as the filter which maximizes signal-to-noise ratio.1 If the signal being sought is an amplitude modulated signal known except for carrier phase, then the ideal receiver has a filter like the one specified in the previous paragraph designed for any particular phase. The receiver input is applied to this filter, and the output is an rf (or more likely, if) voltage. It turns out that the envelope of this voltage is the required quantity. Its relation to likelihood ratio is derived in Section 4.3 and presented in Eqs (4.19) and (4.29). A look at the general equation for likelihood ratio j~(x) = p [ ( exp 2 x(t) s(t) dt dPs(S) (3.o) 0 suggests the follaoing method for designing the optimum receiver for signal detection. First find the correlation as described above, between the receiver input and each possible expected signal. Next, divide each by N0, the noise per unit bandwidth, and find the exponential function of each. Finally, find the weighted average of all these quantities. The hard part is to find the cross correlation between each expected signal and the receiver input. This means that the ideal filter and associated amplifiers are needed for each expected signal, or essentially a separate receiver for each expected signal. In most Lawson and Uhlenbeck, Ref. 1, p. 206; NTorth, Ref. 11. 65

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN cases this is out of the question. In the cases studied in Sections 4.2, 4.3, 4.4, and 4.6, some peculiarity of the set of expected signals made a simpler ideal receiver possible. There is another noteworthy case. If the signal is known except for starting time, then it is sufficient to look at the same ideal filter at different times rather than to have a different filter for each starting time. For even a simple square pulse, it is impossible to synthesize the ideal filter exactly. Just how critical, then, is the design of the ideal filter? This can be answered by finding how well signal detection can be accomplished with an approximation to the ideal filter. For simplicity, consider the case of the signal known exactly. The results for this will follow with little modification for the other cases where the ideal filter is used. The theory specifies that the response of a certain filter to the receiver input be observed at a certain instant. Once it is known that the ideal receiver has this form, it is clear that this filter must be the one which maximizes the instantaneous signal output voltage (or power), the noise rms voltage (or average power) being kept fixed. This is the reason the filter which other authors have found maximizes signal-to-noise ratio is the one which is the absolute optimum for this case. If a filter can be built for which the output ratio of peak signal to rms noise is nearly the same as that obtained with an ideal filter, then this filter will give results nearly as good as the ideal filter. The noise power at the output of a filter with transfer function H(w) is equal to No OD N = 2 H(w) H() do (5.4) -cO 1 See Footnote, p. 65. 66

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN where No is the noise power per unit bandwidth of the input noise. By Parseval's theorem,l and the fact that h(t), the impulse response, is the Fourier transform of H(o). N0 j N = T f H(w) H(w) dw -00 2 h(th(t) (t) dt. (5.5) -00 In the case of the ideal filter, Eq. 4.10 can be applied, and the result is No T NE N 2 f s(T -r) d-r 2 (5.6) 0 where E is the signal energy. The peak voltage output if there is signal but no noise is T 2 I s(t)2 dt = E, (5.7) 0 and hence the peak signal power at the output is E2. The ratio of peak signal power to average noise power is thus for the ideal case. For the particular case of the signal consisting of a single rectangulan RC filter is used with time constant 80% of the pulse duration, the receiver operating characteristic will be the same as if the ideal filter were used and the signal reduced 0.90 db. This is derived in Appendix F. Several other pulse cases have been treated and the results for the best filter of each type are summarized in the following table: Titchmarsh, An Introduction to the Theory of Fourier Integrals, Oxford University Press, 1937, P. 50. 67

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN TABLE II Equivalent Loss Pulse Filter in Signal Strength Gaussian Rectangular Passband 0.98 dbz Rectangular Gaussian Passbandc Rectangular Rectangular Passband 0.83 dbl Rectangular Simple RC Filter (or Single Tuned Circuit) 0.90 db Reitangu impulse Rectangular response 0.51 db impulse response 1. 62 db Simple RC Filter (or (Exponential Decay) Single Tuned Circuit) 2.67 db The minimnu equivalent loss was obtained. by adjusting the bandwridth of the filter. Thus in detecting pulses the form of the filter passband is relatively unimportant. However, it is important to have the correct filter bandwidth. This is essentially the present-day attitude in building receivers for receiving pulses of known frequency. 5.5_ Conclusions Part II of The Theor' of SialDetectabii consists of the application of the theory presented in Part I to some special cases of signal detection problems in order to obtain information on (1) the design of optimum receivers for the detection of sirgnals, and (2) the performance of these receivers. These cases are derived in Lawson and Uhlenbeck, Ref. 1, p. 206. 68

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The special cases which are presented were chosen from the simplest problems in signal detection which closely represent practical situations. They are listed in Table I along with examples oFa engineering problems ian whrich they find application. TABLTE I Description of Section Signal Ensemble Application 4.2 Signal Known Exactly Coherent radar with a target of known range and character 4.3 Signal Known Except for Ordinary pulse radar with no intePhase gration and with a target of known range and character. 4.4 Signal a Sample of White Detection of noise-like signals; Gaussian Noise detection of speech sounds in Gaussian noise. 14.5 Video Design of a Broad Detecting a pulse of known startBand Receiver ing time (such as a pulse from a radar beacon) with a crystal-video or other type broad band receiver. 4.6 A Radar Case (A train of Ordinary pulse radar with intepulses with incoherent gratian and. with a target of known phase) range and character. 4.8 Signal One of M Orthogo- Coherent radar where the target is nal Signals at one of a finite number of nonoverlapping positions. 4.9 Signal One of M Orthogo- Ordinary pulse radar with no intenal Signals Known Except gration and with a target which for Phase may appear at one of a finite number of non-overlapping positions. ----- 69

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN In the last two cases the uncertainty in the signal can be varied, and some light is thrown on the relationship between uncertainty and the ability to detect signals. The variety of examples presented should serve to suggest methods for attacking other simple signal detection problems and to give insight into problems too complicated to allow a direct solution. It should be borne in rmind that this report discusses the detection of signals in noise; the problem of obtaining information from signals or about signals, except as to whether or not they are present, is not discussed. Furthermore, in treating the special cases, the noise was assumed to be Gaussian.1 In addition to general remarks on receiver design, most sections on special cases include specific information describing the simplest design for the optimum receiver for the case considered in those sections. For the simple cases, the design indicated corresponds closely to the design indicated by the type of analysis in which signal to noise ratio is maximized. For the more complicated cases, the design suggested is usually impractical. For some problems it may never be practical to attempt to build an optimum system. For others, however, engineers equipped with a good understanding of statistical methods and their application to the problem of signal detectability, and to communication theory in general, will undoubtedly invent systems which approach the optimum system. For each special case treated in this report, at least an approximation is given for the receiver performance. Receiver performance received primary emphasis because it has generally been slighted in previous work. It is 1See the footnote on page 4I with reference to the spectrum of the assumed noise. 2See Section 5.2. 70

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN important to Tnow the performance which could be obtained from an optimum receiver even if an optimum receiver cannot be built, since this gives an upper bound on the performance which can be obtained with any receiver in a given situation, and since this also gives an upper bound on what can possibly be accomplished by improvements in receiver design. 71

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN APPEEIDIX D The Sampling eorem__ Suppose f(t) is a measureable function which is defined for 0~ t - T. Then f(t) can be expanded in a Fourier series in this interval. The frequency of any term in the series is an integral multiple of 1/T. Suppose there are no terms 1 in the series with frequency above W. This makes the function band limited. Denote by 4m(t) the function sin [ (21WT) (t ) (D.l)' *M(t) (D.1) (2WT) sin[ itT 2(.. Then 2WT f(t) = f(2') m(t (D.2) m=l Furthermore, the functions /m are orthogonal on the interval 0 <t < T, T~ (D,(0.3) 4m(t) kk(t) dt = 2 0 and T qf4 m(t) dt = (D4) 0 where 8km is the Kronecker delta function, which is zero if k f m and unity if k = m. T-Te shall assume 2fT, is an odd integer. This equivalent to choosing the limit of the band half way between the frequency of the last non zero term in the Fourier series and the frequency of the next term (which, of course, has a zero coefficient). 7

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN It follows from Eq (D.2) and Eq (D.3) that T 2WT L S f(t) J dt 1 E i m [f (D()) and from Eqs (D.2) and Eq (D.4) T 2WT f(t) dt 2W ( 2 (D.6) Thus the 2WT functions *m have the same properties for the finite interval which Shannon's interpolation functions have on the infinite interval. It is interesting to note that when 2WT is large, these functions, except the ones near the ends of the interval, are approximately the same as Shannon's. The Fourier series for qJm(t) has no terms with frequency above W. It is, in exponential form, T- 12 Mlt) = /2WT Xexpl 2 mn123P 2 nt n= -(WT-) ( This can be shown by expressing the sine functions in Eq (D.1) as exponentials and using the algebraic identity n+l -n-1 n a -a k a ~- - Z a (D.8) k= -n Formula (D.4) can be proved by integrating Eq (D.7) directly. Note that the only term which contributes to the integral is the term for which n = 0. See Shanncn, Ref. 21. 73

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Formula (D.3) can best be proved also by using the Fourier series. T S m((t) -kk(t) dt 0 T WT-! WT-_ r 2 2 W1 1 - ~~2(2' _, w[ T)2 2' ex p [j'exp 2 ] dt 0 n -(wT. —) p =( WT- WT -T1 2I 2 i;7 (1rt( 2 )n — 2 (2WT)2 L 2WT+l T n2W)WT - 1-, ex [- e 2W d 2 n: -(WT- i _ (WT_. ) 2 T T 2exp[j r -2rn(m -k) (2WT)2n = -(T - ) If m = k, each of the 2WT terms in the sum is unity. If m / k, the terms in the sum are equally spaced around the unit circle in the complex plane and must sum to zero. Thus T f m(t) k(t) dt = 0 which was to be proved. 74

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN The validity of the expansion in equation (D.2) follows from the fact that the functions Am(t) are 2WT linearly independent linear combinations of the 2WT functions exp[j 2nt ] 2 WT n < T - which are used in the Fourier series expansion. Thus any function which can be expanded in a Fourier series with only the first 2W1T terms can also be elpanded in a unique way in terms of the functions 1m. There is an alternate form of the sampling theorem for band limited signals. With this form the signal function can be described by giving sample values of the envelope and phase of the signal, and hence this form is often convenient to use in describing rf signals. Suppose the function f(t), when expanded in a Fourier series on the interval 0 t T has only a finite number of terms in its expansion, and suppose they are included in the terms ranging from frequency fl to frequency 2.1 The bandwidth then could be defined as w f2 -fl + z (D.9) and the center frequency is f2 + fl - - (D.10) 2:z 2 Then the Fourier series can be written -m ft) = Z k cos( + t k) t] + bk sin + 2xk)t] (D.ll) f(t) ak -ibk)exp (co T )t] (D.12) where R means "the real part of", and m = 21 (j - 1). We shall assume Tf2 - Tfl is an even integer and that Tfl- 1 75

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN f(t) = R exp [iwt] (a ib) expi 2[]tk L LJ( k [ kexPi T -m f(t) = R expiwt akexpi 27tkt + ibkexo[ k T k T -m>7 = RJexp [iWt] (x(t) - iy(t))J = x(t) coswt + y(t) sin owt where m ak 2~kt ] x(t) = 2 exp i and -m (D.14) y(t) = - 2 exp 2n kt -m The functions x(t) and y(t) meet the conditions of the first form of the sampling theorem, for a signal with frequencies no higher than -. 2 They can be expressed therefore in the form WT x(t) = X x () k (t) k =1 WT y(t) = z (t)) (D.15) k=l Where the / functions are defined for a signal with no frequencies above W Thus the original function can be written as WT WT f(t) = Z x (qf)k (t) cos cot + Z y( )'k(t)sin wt, k=1 k=l (D.6) 76

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN and the function f(x) can be represented by giving the sample values x ( ) and y Since f(t) can be expressed in the form f(t) = x(t) cos ct + y(t) sincwt (D.13) and x(t) and y(t) are limited to frequencies less than WK which is less than 2 2 21C the envelope of f(t) is 2 2 r(t) = x(t) + y(t) (D17) The angle Q(t) defined by cos @(t) = r } y(t) (D.1s) sin @(t) = t) (D.18) can be considered as the phase of the signal, since f(t) = r(t) cos Q(t) cos wt - r(t) sin @ (t) sin wt r(t) cos wt + @(t)] (D.19) Note that the sample values xi and Yi can be obtained from sample values of r and Q, x (= x( r( )cos [@ ()]= r cos e (D.20) Yi = ri sin @. i 1 1 Thus the function f(t) may be represented by giving the sample values of its amplitude and phase at points spaced 1 apart through the observation interval. 77

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPEIDIX E The integral exp (-b2) () a, exp da (E.1) 0 is required. The integral Go 00 0 = n'2n exp F(-n, 1; - 2) 2 where F (-n, 1; - -2 ) is the confluent hypergeometric function.1 The function Io (bt) can be expanded in a power series 00 2n 2n Io(bCa) = 22n (E,-) n=O Then the integral (E.1) can be written 00 ep (-b2) fI ) 2 e [ ] o:= (E.4) 0 (Substituting (E.3) for I (bc)) 0 ~000 2n 2 n 2 exp (-b2) f b I I(bc) o exp[ ]da n=0 2 n.n. 1 Lawson and Uhlenbeck, Ref. 1, p. 174. 78

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN co ~exp (~b2) 2n 2n+1 e[a2]o = exp ( -b 2 - - a I (ba ep ] da (E.6) n=O 0 (Substituting from (E.2)) 0 2n 2 n 2k = exp.b) X 2. b n22n exp4 (kk k b (E.7) 2n n'n' n=0 k=O 2 00 n 2n+-2k = exp[- 2 b n=O k=O 2n+k(n-k)'k'k' (E.8) (Rearranging the terms in the double sum) b2 D Co 2n+2k exp - b b (E.9) k=O n=k 2n+k(n-k)'!kk: 2 O CoD 4k 2n - 2k: exp[- b C b k=O n=k 22k 2n-k k'k' (n-k)' (Letting m = n-k) 2 CO CO 4k 2m k=O m=0 22k k!km:2m( cOD 4k CD = exp _ E] b (E.12) k=O 22k k'k' m=0 m'2Tm = exp 2] I(b2) ep [_2]= Io (b2) (E.13) The steps in this derivation which must be justified are interchanging the order of integration and summation at step (E.6) and rearranging the double sum, at steps (E.9)and (E.12). It is easy to show that the integral (E.4) exists. The integrands in (E.6) are uniformly bounded by the integrand in (E.4). Thus 79

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN the integrals in (E.6) converge uniformly, and the order of integration and summation can be interchanged. As for rearranging double sums, this is possible since all the terms are positive, and hence the convergence is absolute. 8o

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX F Let us consider a simple case of approximating the ideal filter by some other filter. Suppose s(t) is a rectangular pulse of energy E and width d. Then < < Then ~~~~~s(t) = if 0 t d (F.1) = 0 otherwise Suppose the filter is made up of a single resistor and a single condenser, with an amplifier or attenuator, whichever is needed to make the NoE noise power at the output -- as in the ideal case. Then the impulse response t is of the form h(t) = he if h(t) = h e r if t _2 = 0 otherwise (F.2) where t is the time constant of the filter and ho is a constant depending on the gain of the amplifier or attenuator. The requirement that the noise power at the output be -N is,by (5.5), equivalent to requiring that GO 2 E = f [h(t)] dt, (F.3) -GO or CO 2 _ 2t ho2 E = f h e dt = (F.4) 0 o 2 which yields h2 2E,and (F.5) o t h(t) =E e r if t O (F.6) = 0 otherwise The response V(t) of this filter to the pulse s(t) is, by (4.11), T V(t) = J s( k ) h (t-x) dA. (F.7) -O 81

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Substitutions from (F.1) and (F.6) for s(t) and h(t) give t - (t-%: V(t) a e T dX if O < t d d (t-x) (F.8) -= X J Je dX if t > d. 0 These integrals can be evaluated easily, and V(t) = E i - e ) if O <t d (P.9) V(t) = E d (1 - e e if t > d V(t) increases with time if t < d and decreases with time if t > d, so it must have its maximum value at t = d. That maximum value is Vmax E d (F.10) d In Fig. F.1, Vmax/E is plotted as a function of _. It is seen that at d T = 0.8 approximately, Vmax has a maximum, and at this point Vmax is approxid mately 0.9E. For this particular case, if the RC filter with time constant t = O.8d is used in place of the ideal filter, the reliability of signal detection will be the same as if the ideal filter were used and the signal amplitude were reduced to ninety per cent, or 0.90 decibel. 82

1.0 0.9f - 1 0.8 d 0.7 - 0.6` 0 x Iw 0.5 0.4 0.3 0.2 0.1 O. Id 0.2d 0.4d 0.6d 0.8d d 2d 4d 6d 8d IOd FIG. F.I. MAXIMUM RESPONSE OF R.C. FILTER TO A RECTANGULAR PULSE AS A FUNCTION OF FILTER TIME CONSTANT.

BIBLIOGRAPHY On Statistical Approaches to the Signal Detectability Problem: 1. Lawson, J. L., and Uhlenbeck, G. E., Threshold Signals, McGraw-Hill, New York, 1950. This book is certainly the outstanding reference on threshold signals. It presents a great variety of both theoretical and experimental work. Chapter 7 presents a statistical approach of the criterion type for the signal detection problem, and the idea of a criterion which minimizes the probability of an error is introduced. (This is a special case of an optimum criterion of the first type.) 2. Davies, I. L., "On Determining the Presence of Signals in Noise," Proc. I.E.E. (London), Vol. 99, Part III, pp. 45-51, March, 1952. 3. Woodward, P. M., and Davies, I. L., "Information Theory and Inverse Probability in Telecommunication," Proc. I.E.E. (London), Vol. 99, Part III, p. 37, March, 1952. 4. Woodward, P. M., and Davies, I. L., "A Theory of Radar Information," Phil. Mag., Vol. 41, p. 1001, 1950. 5. Woodward, P. M., "Information Theory and the Design of Radar Receivers," Proc. I.R.E., Vol. 39, p. 1521. Woodward and Davies have introduced the idea of a receiver having a posteriori probability as its output, and they point out that such a receiver gives a maximum amount of information. They have handled the case of an arbitrary signal function known exactly or known except for phase with no more difficulty than other authors have had with a sine wave signal. Their methods serve as a basis for the second part of this report. 6. Reich, E., and Swerling, P., "The Detection of a Sine Wave in Gaussian Noise" ou. App. Phys., Vol. 24, p. 2&9, March, 19535. This paper considers the problem of finding an optimum criterion (of the second type presented in this report)for the case of a sine wave of limited duration, known amplitude and frequency, but unknown phase in the presence of Gaussian noise of arbitrary autocorrelation. The method probably could be extended to more general problems. On the other hand, the methods of this report can be applied if the signals are band limited even in the case of non-uniform noise by putting the signals and noise through an imaginary filter to make the noise uniform before applying the theory. See The Theory of Signal Detectability, Part II, Section 5.2. 84

7. Middleton, D., "Statistical Criteria for the Detection of Pulsed Carriers in Noise, Jour. Appl. Phys., Vol. 24, p. 371, April, 19553 A thorough discussion is given of the problem of detecting pulses (of unknown phase) in Gaussian noise. Both types of optimum criteria are discussed, but not in their full generality. The sequential type of test is discussed also. IMIiddleton's equation (6.1) does not hold for the sequential test, and as a result, his calculations for the minimum detectable signal with a sequential test are incorrect. The discussion of the tests is not clear. The comparison of the tests, which are designed to optimize differenzt; quantities, seems inappropriate; each test accomplishes its omm task in the best possible way. 8. Slattery, T. G.,'the Detection of a Sine Wave in Noise by the Use of a IJon-Linear Filter," Proc. I.R.E., Vol. 40, p. 1232, October, 1952. This article considers the problem of detecting a sine wave of known durat;ion, aLmplitude, and frequency, but unknown phase in uniform Gaussian noise. The article contains several errors, and the results are not clearly presented. 9. Hanse, H., "The Optimization and Analysis of Systems for the Detection of Pulsed Signals in Random. Noise," Doctoral Dissertation (MIT), January, 1951. 10. Schwartz, M., "A Statistical Approach to the Automatic Search Problem," Doctoral Dissertation (Harvard), June, 1951. These dissertations both consider the problem of finding the optimum receiver of the criterion type for radar type signals. 11. North, D. O., "An Analysis of the Factors which Determine Signal-Noise Discrimination in Pulsed Carrier Systems," RCA Laboratory Report PTR-6C, 1943. The ideas of false alarm probability and probability of detection are introduced. Norttl argues that these probabilities will be most favorable when peak signal to average noise ratio is largest. The ideal filter, which maximizes this ratio, is derived. (This commentary is based on second-hand knowledge of the report.) 12. Kaplan, S. M., and Fall, R. W., "Tthe Statistical Properties of Noise Applied to Radar Range Performance," Proc. I.R.E., Vol. 39, p. 56, January, 1951. The ideas of false alarm probability and probability of detection are introduced and an exaple of their application to a radar receiver is given.

'arcauni JI. I. Statistical Theory of Target Detection by Pulsed Radar Mathematical Appendix," Rand Corporation Report R-113, July i, l9198. This report contains a careful, thorough study of the mathematical problem which it considers. On Statistics: 13. Neyman, J., and Pearson, E. S., "On the Problem of the Most Efficient Tests of Statistical Hypotheses," Phil. Trans. Roy. Soc., Vol. 231, Series A, p. 289, 1933. 14. Cramer, H., Mathematical Methods of Statistics, Princeton University Press, Princeton, 1951. On Related Topics: 15. Dwork, B. M., "Detection of a Pulse Superimposed on Fluctuation Noise," Proc. I.R.E., Vol. 38, p. 771, July, 1950. 16. Harrington, J. V., and Rogers, T. F., "Signal-to-Noise Improvement Through Integration in a Storage Tube," Proc. I.R.E., Vol. 38, p. 1197, October, 1950. 17. Harting, A. E., and Meade, J. E., "A Device for Computing Correlation Functions," Rev. Sci. Inst., Vol. 23, 347, 1952. 18. Lee, Y. W., Cheatham, T. P., Jr., and Wiesner, J. B., "Applications of Correlation Analysis to the detection of Periodic Signals in Noise," Proc. I.R.E., Vol. 382 p. 1165, October, 1950. 19. Levin, M. J., and Reintzes, J. F., "A Five Channel Electronic Analog Correlator," Proc. Nat. El. Conf., Vol. 8, 1952. 20. Rice S. 0.'TMathematical Analysis of Random Noise," B.S.T.J., Vol. 23, p. 2$2-332 and Vol. 24, p. 46-156, 1945-6. 21. Shannon, C. E., "Communication in the Presence of Noise," Proc. I.R.E. Vol. 37, PP. 10-21, January, 1949. 86

LIST OF SYMBOLS Ai The event "Thle operator says there is signal plus noise present," or a criterion, i.e., the set of receiver inputs for which the operator says there is a signal present. A1() iny criterion A which maximizes PSN(A) - D PIT(A), i.e., an optimum criterion of the first type. A2(k) Any criterion A for which PN(A) -. k, and PS"(A) is maximum, i.e., an optmiumn criterion of the second type. CA The event "T'he operator says there is noise alone." dc A parameter describing the ability of a receiver to detect signals. (See Section 5.1 and Fig. 5.1.) 2 E (s) The signal energyr. AJ The n-dimensional Euclidean space. fiN(x) The probability density for points x in' if there is noise alone. fSIT() The probability density for points:: in R if there is signal plus noise. F ((), F1,(e) The complementary distribution function for likelihood ratio if there is noise alone, i.e., F (() is the probability that the likelihood ratio will be greater than D if there is noise alone. FSN(P)3s), sN() The complementary distribution function for likelihood ratio if there is signal plus noise. k IA sbymbol used primarily for the upper bound placed on false alarm probability P11(A) in thle definition of the second kind of optimum criterion. fsN(x) ~(x) The likelihood ratio for the receiver input x..C(x) = f(x) n The dimension of the space of receiver inputs. n = 2WT IN The event "There is noise alone," or the noise power. I~ The noise power per unit bandwidth. IT = /l. 0 0 P (A) The probability that the operator will say there is signal plus noise if there is noise alone, i.e., the false alarm probability.

PSJ(A) The probability that the operator will say there is signal plus noise if there is signal plus noise, i.e., the probability of detection. Px(SN) The a posteriori probability that there is signal plus noise present. (See Sections 1.3 and 2.3.) ps() The probability measure defined on R for the set of expected signals. R The space of all receiver inputs. (The set of all possible signals is the same space.) s A signal s(t), which may also be considered as a point s in R with coordimates (SL} s2i'.. Sn). SN The event "There is signal plus noise." t Time. T The duration of the observation. W The bandwidth of the receiver inputs. x A receiver input x(t), which may also be considered as a point x in R with coordinates ('x1 x2'...~ xn) n n~A symbol usually used for the likelihood ratio level of an optimum criterion.,UsN(Z) The mean of the random variable z if there is signal plus noise. HI'N(z) The mean of the random variable z if there is noise alone. ~CI (Z) The variance of the random variable z if there is noise alone. 2 The variance of likelihood ratio if there is noise alone. Note: The terms ormal distribution ad "Gaussian distribution" have been used interchangeably in this report. 88

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U NIVER SITy OF MICHIGAN 3 905 03695 4454 1 copy W. G. Dow, Professor Dept. of Electrical Engineering University of Michigan Ann Arbor, Michigan 1 copy H. W. Welch, Jr. Engineering Research Institute University of Michigan Ann Arbor, Michigan 1 copy Document Room Willow Run Research Center University of Michigan Willow Run, Michian 10 copies Electronic Defense Group Project File University of Michigan Ann Arbor, Michigan 1 copy Engineering Research Institute Project File University of Michigan Ann Arbor, Michiman 9o