Technical Report No. 209 036040-5-T THE FREE RUNNING DETECTION PROBLEM: THE DETECTION OF RANDOMLY OCCURRING PULSES AS THEY OCCUR by R. M. Heitmeyer Approved by:_ / _ Theodore G. Birdsall for COOLEY E LEC TRONIC SLABORATORY Department of Electrical and Computer Engineering The University of Michigan Ann Arbor, Michigan Contract No. N00014- 67-A- 0181- 0032 Office of Naval Research Department of the Navy Arlington, Va. 22217 September 1971 Approved for public release; distribution unlimited.

ABSTRACT This study provides an extension of classical detection theory by defining and analyzing two classes of decision devices. The common feature of these devices is that they both operate in real time to detect the presence of each pulse in a sequence of randomly occurring pulses, while that pulse is still present. These decision devices are distinguished by the fact that one, the respond-once device, seeks to detect each pulse once and only once, whereas the other device, the respond-and-hold device, seeks to respond at each instant of time at which a pulse is present. In this study, the problem of obtaining the decision devices that are optimum in the above setting is referred to as the free running detection problem. The basic assumptions are as follows. It is assumed that all of the pulses are identically shaped and completely known up to their arrival times. Moreover, we assume that it is possible to observe these pulses only through an observation that contains additive noise. Finally, it is assumed that the decision devices are allowed to make decisions on the presence or absence of pulses only at discrete, equally-spaced points in time. The mathematical model for the free running detection problem is constructed in terms of the general decision theory model. This provides the theoretical basis for describing the performance of both 1iii1

classes of decision devices and for obtaining decision devices that are optimum with respect to the Bayes criteria. The performance of the respond-once decision device is described in terms of the average number of pulses that are detected (detection rate), the average number of false alarms (false alarm rate), and the average number of times a pulse is detected more than once (extra detection rate). It is shown that the detection rate can be increased for the same false alarm rate by increasing the extra detection rate. The performance of the respond-and-hold decision devices is described in terms of the detection rate, a detection duty, and a false alarm duty. These last two quantities are a measure of the amount of time that the decision device responds when a pulse is present. For the respond-and-hold devices, it is seen that the detection duty can be increased for a fixed false alarm duty only by decreasing the detection rate. iv

FOREWORD In the past two decades, classical detection theory has been profitably applied to the design and evaluation of active surveillance systems. The applicability of detection theory is a result of the natural temporal structure inherent in these systems. That is, in the active surveillance system it is sufficient to make a decision at the end of an interval of time on the presence or absence of signal in the received waveform. In certain passive surveillance situations, however, the lack of temporal structure precludes the use of a "forced-response" detection theory as a realistic model for the systems detector. One such situation arises when targets are continually entering and departing from the field of view and it is desired to detect these targets while they are still present. In a situation such as this a "free-response" detection theory must be used to model the system detector. This report establishes the foundations of such a theory. A recursive relationship describing the optimum detector is developed and a scheme for evaluating the performance of any free-response detector is obtained in terms of an "ROC surface." Finally, some numerical results are presented as applications of the theory. v

TABLE OF CONTENTS Page ABSTRACT iii FOREWORD v LIST OF ILLUSTRATIONS viii LIST OF SYMBOLS xviii LIST OF APPENDICES xxiv CHAPTER I: THE FREE RUNNING DETECTION PROBLEM 1 1.1 The Continuous Action and the Discrete Action Pulse Monitoring Systems 1 1.2 The Performance Degradation Due to Noisy Inputs 5 1. 3 The Free Running Detection Problem 15 1.4 An Intuitive Look at the FRD Decision Device 23 1.5 Historical Remarks 34 1.6 Organization of This Study 35 CHAPTER II: THE MATHEMATICAL MODEL FOR THE FREE RUNNING DETECTION PROBLEM 37 2.1 Basic Elements of Decision Theory 38 2.2 The Action Space 42 2.3 Parameter Space and Prior Probabilities 47 2.4 The Observation Space and Real Time Decision Rules 64 2.5 The Loss Function 68 CHAPTER III: THE PERFORMANCE OF FRD DECISION DEVICES 80 3.1 A Review of Performance in Classical Detection Theory 81 3.2 Performance Sets for the FRD Decision Rules 84 vi

TABLE OF CONTENTS (Cont.) Page 3.3 Some General Properties of the Performance Sets 95 3.4 The Inhibit Rule and the RX = 0 Plane 107 3. 5 Optimality and ROC Surfaces 114 3. 6 Optimality and the Bayes Decision Rules 121 CHAPTER IV: THE BAYES DECISION RULES FOR THE FRD PROBLEM 138 4.1 The Bayes Decision Rule 138 4.2 Two Special Cases: The MDT Decision Device and the m = 1 Decision Device 151 4.3 The Updating Equation for Conditionally Independent Observations 160 4.4 The Bayes Decision Device for Conditionally Independent Observations 169 CHAPTER V: A NUMERICAL SOLUTION FOR THE OPTIMUM BAYES DECISION DEVICE AND ITS ROC SURFACE 178 5.1 The Basic Setting 178 5.2 The m - 1 Bayes Decision Device 188 5.3 The m = 2 Bayes Decision Device 194 5.4 The Matched Filter Decision Devices 220 5. 5 Performance 229 CHAPTER VI: SUMMARY AND CONCLUSIONS 250 REFERENCES 376 DISTRIBUTION LIST 377 vii

LIST OF ILLUSTRATIONS Figure Title Page 1. 1. General pulse monitoring system 2 1.2. The general pulse monitoring system as a decision device and an action device 3 1.3. The relation between the input pulse sequence and the response functions for the continuous action and the discrete action decision devices 4 1. 4. Typical inputs and outputs for a respond-andhold decision device 6 1. 5. A pulse sequence and two different response functions 8 1.6. A detected pulse 11 1. 7. Typical inputs and outputs for a respond-once decision device 12 1. 8. A typical pulse p(t) and signal s(t, a1, 2 "...) 17 1. 9. A pulse p(t) that is zero over part of the interval [0, T) 17 1. 10. The different possible decision outcomes 22 1.11. A pulse present at time t' 25 1. 12. An illustration of the most recent response time 26 viii

LIST OF ILLUSTRATIONS (CONT.) Figure Title Page 1.13. The form of the decision device 27 1. 14. The signals present in a respond-and-hold decision device with the configuration of Fig. 1. 13. 28 1. 15. The matched filter decision device (a)respond-andhold case, (b) respond-once case 34 2. 1. Typical response functions (a) respond-once response function (b) respond-and-hold response function 45 2. 2. The number of decision opportunities per pulse; (a) A = T /2 (b) A = T /4 46 2. 3. The density of the ith inter-arrival time 50 2. 4. Discrete inter-arrival time probability law 53 2. 5. The relation of 0. to a pulse arriving in the interval [t - Tp, ti) 55 2. 6. An illustration of 0 (a) a typical pulse sequence (b) the corresponding value of = (01...,N 57 2. 7. The definition of i* 58 ix

LIST OF ILLUSTRATIONS (CONT.) Figure Title Page 2. 8. Conditional cumulative distribution function F(c I Ck_1) (a) ck_1 E [ A), (b) ck1 e [A, Tp), (c) ck 1 Tp 60 2. 9. Conditional probability law P(Ok = 0[ Ok-1 = jv) (a) j e {0,..., s-1 I, (b) j e {s,..., q-1}, (c) j = q 62 2. 10. The definition of the most recent response function 70 2. 11. A previously detected pulse 70 2. 12. The condition that time t. is the last decision opportunity to detect a pulse present at time t. 72 2. 13. An illustration of the six basic events 74 2.14. The relation between TD, TF and NX, ND and NF 76 3. 1. A typical performance set for the classical detection problem 82 3. 2. The geometrical relation between the Bayes rule and the performance set 84 3.3. The region containing 9(m) and,2(m) 97 3. 4. The randomization between the performance points (0, 0, 0), (0, 1, 1) and (m-l, m, 1) 101 x

LIST OF ILLIJSTRATIONS (CONT.) Figure Title Page 3. 5. The chance plane for a(m) 103 3. 6. The set of performance points better than R(65") 115 3. 7. A typical respond-once ROC surface 116 3. 8. The set of performance points better than D (6") 118 3. 9. A typical respond-and-hold ROC surface 119 3. 10. The relation between the decision time for decision devices in 9(2m) and 9 (m) (a) the decision times associated with Jq(m) (b) the decision times associated with 1(2m) 121 3. 11. The Bayes risk plane for the respond-once case 124 3. 12. A section of the upper surface of 9,(m) at RF = RF 124 F F 3. 13. The Bayes risk line tangent to fR *)(WX) at the point RX = 0 127 3. 14. The intersection of the two Bayes risk lines with the section f (Rx) 129 F 3. 15. The Bayes risk plane for the respond-and-hold case 130 3. 16. The relation of the Bayes risk line with slope Wxm/ (1 + WX) to the section gD (DD) 132 3 17. The intersection of the Bayes risk lines with the section gD* (DD) 133 xi

LIST OF ILLUSTRATIONS (CONT ) Figure Title Page 3. 18. The relation of the losses WX, WF to the behavior of the Bayes decision rules 135 4.1. The relation of Ti. to T. when a. =0 and 1 i T. > A 140 4. 2. An example of the effect of the decision at time tk on the decision at time tk+ 142 4.3. The general Bayes decision device 148 4. 4. The MDT decision device 156 4. 5. The updating procedure for Ok(, Xk, Ok 1) 168 4. 6. The Bayes decision device for conditionally independent observations 177 5. 1. The temporal relation between, T, and xk 179 5. 2. The pulse configuration for the different values of k' (a) 0k = 0 (b) 9k=Tp/2 (c) k = T 181 5.3. The event k = T /2 185 k-1 p 5. 4. A pulse which satisfies the translation property 186 5. 5. The threshold function for the m = 1 Bayes decision device 190 5.6. The m = 1 Bayes decision device 190 xii

LIST OF ILLUSTRATIONS (CONT.) Figure Title Page 5.7. The temporal relation of the carry-over pulse, the previous carry-over odds Pkl-(Ikl) and the current likelihood ratio (xk) 192 5.8. The general form of the m = 2 Bayes decision device 196 5. 9. Threshold functions for Case 1; dT2 = 1,D =.3, W = 1, W = 200 X F 5.10. Threshold functions for Case 1; d2 = 1, D =.3, T p W 1, W =2 201 X F 5. 11. The function WX (WF) 203 5.12. Threshold functions for Case II; dT2 = D =.3 p W = 1, WF 1 204 X F 5.13. Threshold functions for Case II; d = 1, D = 3, T p W =.25, W 1 205 X F 5. 14. Threshold functions for Case II; d2 = 1, D =.3, T p W =. 1, W =2 206 X F 5. 15. Threshold functions for Case II; d2 = 1 D =.3 T p W =.25, WF =2 207 X F 5. 16. Threshold functions for Case III; d2T = 1, D =. 3 WX = 0, WF = 1 209 xiii F Xiii

LIST OF ILLUSTRATIONS (CONT.) Figure Title Page 5.17 Threshold functions for Case III; d 2 = 1, D = 3, T p W = 0, W =2 210 X F 5. 18. Threshold functions for Case IV; dT2 = 1, D =. 3, WX = -.5, WF =1 212 X F 5.19. Threshold functions for Case IV; dT2 = 1, D =.3, W =-.25, WF 1 213 5. 20. Threshold functions for Case IV; d 2 = 1, D =. 3, T p W =-.5, W = 2 214 A Fr 5.21. Threshold functions for Case IV; dT2 = 1, D =.3, WX = -25, WF 2 215 5. 22. Threshold functions for Case V; d2 = 1, D =.3, T p W = -1, W =1 218 X F 5. 23. Block diagram for m = 2 maximum detection time decision device 219 5. 24. Threshold functions for Case I; dT = 1, D =.5, p W 1, W 1 221 X F 5. 25. Threshold functions for Case IV; d2, D =.5, T p WX =- 1, WF = 1 223 5.26. Threshold functions for Case; dT 9, D 3, T P xiv

LIST OF ILLUSTRATIONS (CONT.) Figure Title Page 5. 27. Threshold functions for Case IV; d2 = 9, D =.3, T p WX =-.5, WF = 1 224 5. 28. The MF decision device (a) respond-and-hold case, (b) respond-once case 228 5. 29. The SKE paradigm for signal-to-noise ratio dT2 230 5. 30. The SKE paradigm for signal-to-noise ratio dT2/ 2 231 5.31. The region containing the projection of X y(2) 233 5. 32. The region containing the projection of AX (2) 234 5. 33. The projection of XAy(2) into the RX = 0 plane for d = 1,3 236 5. 34. The ROC curve X, (1) 238 5. 35. Performance curves for the ROMF decision devices 239 5. 36. The performance of the R-O decision devices 241 5. 37. The projection of Ap (2) into the RD = 0 plane (a) dT =1 243 5. 37. The projection of Xy(2) into the RD = 0 plane (b) dT = 3 244 5. 38. The projection of XA(2) into the DD = 0 plane (a) dT = 1 245 xv

LIST OF ILLUSTRATIONS (CONT.) Figure Title Page 5.38. The projection of X) (2) into the D = 0 plane (b) dT = 3 246 5. 39. The projections of the performance curves for the RHMF into the RD = 0 plane 247 5. 40. The projections of the performance curves for the RHMF into the D = 0 plane 248 A. The loss function for the classical detection problem 260 B.1 The temporal relation between 0k 0k-'.. 01 264 B.2 The case a e tk - Tp, tkl) 266 B.3 The case Ua e [tkl- Tp, tk- Tp) 267 B.4 The case q e E[-(T - A), tk- 1- T ) 269 C. 1 The density P(OkIJkl) for ok-l [E A) U {T (a) k [0, A) (b) 0k1 =T 278 D.1 The region in which p(0 201), 01 [OA), is positive 287 1.1 The function s(t, 0k, Okl) (a) 0k-1 (0, A), =Tp (b) 0kl=Tp 0 k [T- A, Tp) (c) 0k- [, 0 k (Tp )] oke[T- AT ) 331 s kk-ik-1 p k k- 1 I.2 S(t, Bk' k-l); ok-lE [0, Tp ) ok k-l 332 L.I The recursive procedure 363 M.1 The generic form for the decision devices of Chapter V 369 xvi

LIST OF ILLUSTRATIONS (CONT.) Figure Title Page M. 2 The relation of the events 3 to the decisions ak1 and ak and the state 0k 370 M.3 State transition diagram for the random variables 0k 371 xvii

LIST OF SYMBOLS Symbol Definition Ak(.,.) response set at time tk C' action space a parameter of the discrete prior probability law ak decision at time tk a total decision C correct rejection outcome D detected pulse outcome M(m) the set of decision devices for a fixed m Dp,DDDF pulse duty, detection duty, false alarm duty F false alarm outcome F(.) commutative distribution function IB( ) indicator function for the set B K(-) threshold function for the m=1 Bayes decision device 0 ^ K (. ),(K ) threshold function for the m=2 Bayes decision device at time tk given that ak_ =0, the limit of the sequence K0k() xviii

LIST OF SYMBOLS (Cont.) Symbol Definition Kt ), K () threshold function for the m=2 Bayes decision device at time t given that ak 1 = 1, the limit o} the sequence Kk(') L(', *) total loss function Lk(*, *,*) loss at time tk LD' LF' LM' LX the loss per detect pulse, false alarm, missed pulse and extra detection, respectively T T T T L LT LT LT the loss per unit correct rejection C' D LF LM time, per unit detection time, per unit false alarm time and per unit miss time respectively Pk(x k[Xk-S 0k' On k) the Raydon-Nikodym derivative of Pk(dx 0 k' 0 k-1 ) with respect to P(dx I _l) fk(xk O 0 k-l1 So), the carry-over pulse likelihood fk(Xk co k1S~)ratio, the current pulse likelihood k(xk I 0k Sc)' ratio, the combined pulse likelihood ratio fk(xk | k k 1 Sco S) ratio 2k(xk) the likelihood ratio for white noise and translation invariant pulses M missed pulse outcome MPD maximum pulse detection device MDT maximum detection time device m the number of decision opportunities per pulse xix

LIST OF SYMBOLS (Cont.) Symbol Definition n the stochastic process representing noise N the total number of decision times NC, ND, NF, NM, NR, the total number of correct rejection outcomes, detected pulse outNX comes, false alarm outcomes, missed pulse outcomes, rest outcomes, and extra outcomes, respectively NN, NS the total number of decision times where noise alone is present and where signal is present, respectively NR N NR the total number of decision times N S at which responses are made and noise is present, and signal is present, respectively N noise power per unit bandwidth N the total number of pulses p,0( *' *' *) odds ratio density at time tk Pk carry over odds ratio p(t) the basic pulse shape p (t) the basic pulse shape with arrival time t - T K p P(' I *) conditional probability measures on the observation space P( kI 0k 1) prior conditional density p(O k) prior unconditional density PD, PF probability of detection and the probability of false alarm xx

LIST OF SYMBOLS (Cont.) Symbol Definition Pr[ B] probability of the event B q number of allowable pulse arrival times per T seconds p R rest outcome R(xkk, ak, Tk) posterior risk at time tk RC, RD, RF, RM, RR, the normalized rates for a correct rejection, a detection, a false alarm, RX a miss, a rest, and an extra detection, respectively EI~t~R ~the set of real numbers IRo, T] the set of real valued functions defined on [0, T] R-O respond- once decision device R- H respond- and- hold decision device rC, rD, rF, rM, rR, rX the unnormalized rates for the six basic outcomes listed above rN, r'S the expected value per unit time of NN and NS r the pulse rate P s(t, ) total signal s the number of allowable pulse arrival times per A seconds A3, SP respond-once performance set and respond-and-hold performance set xxi

LIST OF SYMBOLS (Cont.) Symbol Definition T length of the total observation interval Tp, TDV TF TM total amount of correct rejection time, detection time, false alarm time, and miss time, respectively TN, TS total amount of noise alone time and signal time, respectively T minimum time between pulse arrivals Tt(, (*, * ) test function at time t' e-y set of allowable decision times U('*, *) odds ratio density updating function WF, WX normalized false alarm loss, normalized extra detection loss W (WF) the function defining the inhibit rule x, X total observation, total observation space Xk, Xk current observation, current observation space Xk, Xk total past observation, total past observation space X extra detection outcome ca parameter of prior probability law 6 (*) decision rule 5k(') component decision rule xxii

LIST OF SYMBOLS (Cont.) Symbol Definition A decision time separation Ec}() the measure that assigns unit {IC}I~ ~mass to the point c 77i inter-arrival time between i- st and ith pulse 0, O total parameter, total parameter space 0k' Ok parameter for the interval [tk-Tp tk), [tk- T tk] Xdz, A~ ~respond-once ROC surface, XO~/^, ^ ~ respond-and- hold ROC surface ~~~~v ~separation between the allowable pulse arrival time II( ) prior and posterior probability distribution functions on the space O i. arrival time of the ith pulse t most recent response function xxiii

LIST OF APPENDICES Appendix Title Page A THE CLASSICAL DETECTION PROBLEM AS A GENERAL DECISION PROBLEM 258 B DERIVATION OF THE PRIOR PROBABILITY LAWS - PART I 264 C DERIVATION OF THE GENERALIZED DENSITIES FOR THE PRIOR PROBABILITY LAWS 27 5 D DERIVATION OF THE PRIOR PROBABILITY LAWS - PART II 281 E EXPRESSIONS FOR THE RATES r rN AND rS 296 F THE PROOF OF THE MOST RECENT RESPONSE RULE 299 G THE DERIVATION OF THE RECURSION RELATION FOR THE BAYES DECISION RULE 304 H DERIVATION OF THE UPDATING EQUATION FOR THE ODDS RATIO DENSITY 315 I EXPRESSIONS FOR THE ODDS RATIO DENSITY 328 J THE PROOF OF THEOREM 4.10 342 K THE RECURSIVE RELATIONS FOR THE m = 2 BAYES DECISION DEVICE 345 L THE COMPUTER ALGORITHM FOR OBTAINING THE m - 2 BAYES DECISION DEVICE 361 M THE SIMULATION PROCEDURE 367 N THE RATES RX, R AND RD FOR THE SKE DECISION DEVIE RD 373 xxiv

CHAPTER I THE FREE RUNNING DETECTION PROBLEM The free running detection problem (FRD) is a decision problem that is concerned with the detection of randomly occurring pulses as they occur. In this study we will develop the basic decision theoretic model for this problem and use this model to obtain the optimum (Bayes) decision rules. This is done in Chapters II through V. In this chapter, we introduce the FRD problem as the decision problem associated with two different pulse-monitoring systems. We then examine certain properties of the decision rules for the FRD problem from an intuitive point of view. 1.1 The Continuous Action and the Discrete Action Pulse Monitoring Systems By a pulse monitoring system, we mean a system which takes as its input randomly occurring pulses (unknown arrival times) and produces at its output an action for each pulse that occurs. Such a system is shown in Fig. 1.1. In this section two different pulse monitoring systems are examined. The first system of interest will be referred to as the continuous action pulse monitoring system. This system is characterized by the fact that it takes an action at each instant of time that -1

-2Randomly Pulse Monitoring Actions for Each Pulse Occurring System That Occurs Pulses Fig. 1.1. General pulse monitoring system a pulse is present. A simple example of such a system is an alarm system. Here the appropriate action is to sound an alarm precisely at those times at which a pulse is present. In contrast to the continuous action system, we also consider the discrete action pulse monitoring system. This system is characterized by the fact that it takes a single action each time a pulse occurs. Perhaps the simplest example of a discrete action system is a system that provides a running count of the number of pulses that have occurred up to the current time. In this system, the appropriate action is to increment a counter each time a pulse occurs. Additional insight into the nature of these systems is obtained by decomposing the general system model into two components as shown in Fig. 1.2. The components in this system configuration have the following interpretation. The decision device takes as its input the randomly occurring pulses and extracts from these pulses the information that is needed by the action device in order to take the appropriate

-3Decision Action Device Device Fig. 1.2. The general pulse monitoring system as a decision device and an action device action. The action device then takes the necessary action. The continuous action system can be interpreted in terms of this model by noting that the information required by the action device is simply the knowledge of those times when signal pulses are present. Thus, we may take the decision device to be a device which produces the function 1 if some pulse is present at time t a(t) - (1. 1) 0 O otherwise We may then take the action device to be the device which produces the appropriate action at those times when a(t) = 1 Now consider the discrete action system. In this system, the information needed by the action device is the knowledge of a single time when each pulse is present. This information is provided by a decision device that produces the function

-4(1 if some pulse arrives at time t a(t) = o (1.2) 0 otherwise As above we may take the action device to be a device that is on for a(t) = 1 and off otherwise. In the work to follow the output functions of the decision devices, a(t), are referred to as response functions and the values a(t) = 1 as responses. A typical input sequence of pulses and the corresponding response functions for both decision devices are shown in Fig. 3 a, b, c. Input Pulse Sequence (a) a(t) (b) a(t) "It (c) Fig. 1.3. The relation between the input pulse sequence and the response functions for the continuous action and the discrete action decision devices. (a) input pulse sequence, (b) continuous action response function, (c) discrete action response function

-5The point to be noted in connection with the above system model is that both the continuous action and the discrete action systems can be characterized by a decision device that produces a binary valued response function a(t). In both of these systems the responses (a(t) = 1) occur at times when the presence of a pulse calls for an action to be taken. Thus, in both of these systems, we may view the decision device as a device that detects randomly occurring pulses as they occur. The decision devices for the two systems differ, however, in the fact that the continuous action decision device seeks to respond as long as a pulse is present, whereas the discrete action decision device seeks to respond once and only once to each pulse that occurs. To emphasize this distinction we will hereafter refer to the continuous action decision device as a respond-and-hold (R-H) decision device and we will refer to the discrete action system decision device as a respondonce (R-O) decision device. 1.2 The Performance Degradation Due to Noisy Inputs When the input to the decision devices of the preceding section consists only of the randomly occurring pulses, the ideal performance illustrated in Fig. 1. 3 can be realized. But, if noise is also present at the input to the decision device, this performance will be degraded. In this section the nature of this performance degradation is examined.

- 6We consider first the effect of noise on the performance of the R- H decision device. Figure 1.4 shows a typical pulse sequence, a corresponding noisy input and a typical respond-and-hold response function. Pulse Sequence tI t1t4 t7 t10 T Input t a(t) 1 t2 34 t5 t6 t7t8 t t10 T Fig. 1.4. Typical inputs and outputs for a respond-and-hold decision device By comparing the pulse sequence and the response function in the above figure it is clear that the time axis can be divided into four sets, detection times [a(t) = 1 and pulse present], false alarm times [a(t) = 1 and no pulse present], miss times [a(t) = 0 and

-7pulse present] and correct rejection times [a(t) = 0 and no pulse present]. For example, if we let TD, TF, TM and TC denote the total amount of detection time, false alarm time, miss time and correct rejection time in the interval [0, T] respectively, then from Fig. 1.4 above, TD = (t3-t2)+ (t-t8) TF = (t6-t5) F 6- M =(t2 t)+ (t4-t3)+ (t8-t7)+ (t1-t9) TC = t1 + (t5-t4) + (t7-t6)+ (T-t10) A numerical expression for the performance degradation of the R-H decision device can be obtained in terms of a loss function. A natural definition for the loss function for this system is L = L T + LT L T T (1.3) D D M M F F C C T T T Here L, L, L and Lc represent the loss per unit D M' F C detection time, per unit miss time, per unit false alarm time and per unit correct rejection time, respectively. It should be noted that, in all physical situations, detection time is preferred to miss time and correct rejection time is preferred to false alarm time. T LMT and LT T LF Thus we may assume that L < L and L < L D - M C - F

-8A more general definition of the respond-and-hold loss function is needed for systems that must show special concern for the possibility of missing a pulse altogether. For example, in Fig. 1.5 we have, shown a typical pulse sequence and two different response functions. Both of these response functions have been constructed to have the same values of TD, TM, TF, TC so that by Eq. 1.3, both response functions have the same loss. But, the response function a1(t) detects all three pulses (responds at least once to each pulse), whereas the response function a (t) detects only one of the pulses. This fact may be incorporated into the loss function by adding in the term Pulse Sequence (a) al(t) t. i (b) a2(t) (c) Fig. 1.5. A pulse sequence and two different response functions

-9LD ND 4 LM NM where ND and NM represent the number of pulses that are detected and the number of pulses that are missed, respectively, and LD and LM represent the loss for a detected pulse and the loss for a missed pulse, respectively. The complete expression for the respondand-hold loss function is then L LDD + LMNM T T T T + L T+ L+L T T L TC (1.4) D D M M F F C C As above, physical considerations allow us to assume that LD < LM Before turning to the analysis of the respond-once decision device, it should be noted that the six quantities involved in the respond-and-hold loss function are not independent. This fact is a consequence of the following observations. First, note that each pulse that occurs must either be detected or missed. Thus, if we denote the number of pulses to occur by Np, then we may write N ND + NM (1.5a) Next, note that, if a pulse is present at time t', then t' must either be a detection time or a miss time. On the other hand, if no pulse is present at t', then t' must be either a false alarm time or a correct rejection time. Thus, if we denote the total amount of time

-10occupied by pulses by TS and the total amount of time occupied by noise alone by TN, then we may write TS = TD+ TM (1.5b) and N = TF+ TC (1.5c) We may then substitute for NM, TM, and TC from Eq. 1.5 into Eq. 1. 4 to obtain the equivalent expression for the respond-and-hold loss function. T T T T L = - L ND + (LD LM) TD + (L LC) TF] T T + [LMN + L T + LCTN] (1.6) Now the point to be noted here is that the second bracketed term on the right hand side of Eq. 1. 6 depends only on the pulse content of the interval of concern, [0, T], and not on the particular decision device. Thus, for the purposes of comparing the performance of different decision devices, it is sufficient to consider the loss function T T T T L (LD- LM) ND + (L - LM) T + (L LC) TF (1.7) Next we consider the performance degradation of the respondonce decision device. First, recall from the preceding section that

-11the objective of the discrete action system is to take a single action each time a pulse occurs. This is accomplished in the noiseless case by requiring that the decision device respond at the instant that each pulse arrives. (See Fig. 1. 3c.) But, when noise is present at the input to the system, this is no longer a reasonable requirement. Instead, it is usually sufficient to require only that the decision device respond within some prespecified time of the arrival of a pulse in order to detect that pulse and fulfill the objective of the system. Thus, we shall say that a pulse is detected if some response occurs within T seconds of its arrival. A detected pulse is shown in Fig. 1.6. a(t) +T Fig. 1.6. A detected pulse In order to insure that one response cannot detect more than one pulse, we will also assume that the arrival times of the pulses are separated by at least T seconds. Now consider a typical pulse sequence, a corresponding noisy input and a typical respond-once response function as shown in Fig. 1.7.

-12Pulse Sequence al (1 +Tp or2 c2+T (T 3 o3+T Input a(t) * * I I I t.,,,I I Ua- o+Tp 2 a2+T a3 o3+T 1 1 p 2p 3 3p Fig. 1.7. Typical inputs and outputs for a respond once decision device Here, it is seen that the effect of noise on the decision device performance is to cause some pulses to be missed, some pulses to be detected more than once, and some false alarms to be made. Let NF be the number of false alarms that occur and let NX be the number of extra detections that occur. Then the performance degradation can be expressed numerically by defining the loss function for the respond-once decision device to be

-13L D ND M NM X NX F F (1.8) where LX and LF are the loss per extra detection and per false alarm, respectively. As above the respond-once loss function may be separated into a part that depends only on the pulse content of the interval of concern. This is done by substituting Eq. 1.5a into Eq. 1.8. The result is L - L+LMNp (1.9) where L = (LD- LM)ND N+ LNX F NF N(1.10) In the above paragraphs, loss functions have been introduced as a means of measuring the performance degradation of the respondand-hold and the respond-once decision devices. The basic assumption here is that, in the noisy case, the overall objectives of the system can be characterized by assigning specific values to the losses that appear in these functions. Thus, this section concludes with several important examples that illustrate the role these losses play in characterizing the system objective. First, consider the respond-and-hold loss function of Eq. 1.7. The most important special case here occurs when LD = LM. In this

-14case, the loss function depends only on the total amount of detection time TD and the total amount of false alarm time TF but not on the number of detected pulses N. Thus, this case corresponds to a decision device that seeks only to respond as long as possible to each pulse that occurs with no extra penalty for missing a pulse. T T T T It might be noted here that if we also set L - L L = L then D M F C this loss function provides a measure of the mean square error between the ideal response function for the noiseless case and the actual response function. In the following, this loss function is referred to as the maximum detection time (MTD) loss function. As a contrast to the MTD loss function suppose that T T LD - LM << L - LM < D M D M In this case, the system incurs a large gain, -(LD- LM), for each T T detected pulse, but only a small gain, -(LD - L ), for the total amount of detection time. Thus, this decision device seeks to respond at least for a short period of time to each pulse that occurs in order to detect that pulse, but the actual amount of detection time is not that important. Next consider the respond-once loss function of Eq. 1.10. Here, LX is the loss for an extra detection and LF is the loss for a false alarm. Since both of these outcomes are undesirable, we may assume that LX, LF > 0. The case LX > LF corresponds XA F- X F

-15to a system that prefers false alarms to extra detections and, if LF < LX, the converse is true. The case LX = LF corresponds to a system that views extra detections and false alarms as equivalent as is the case in the example of the running counter of Section 1.1. An important special case of the respond-once loss function is defined by the condition LX = 0. In this case, the loss function depends only on the number of pulses that are detected and the number of false alarms. Thus, this decision device simply seeks to respond "at least" once to each pulse that occurs rather than "once and only once" as is the case when LX > 0. In the following, we will refer to this case as the maximum pulse detection (MPD) case. 1. 3 The Free Running Detection Problem In the preceding section we considered the effect of noise on the performance of two decision devices that arise in certain pulse monitoring systems. As a measure of the resulting performance degradation, a loss function for each device was introduced. In this study we shall be concerned with the problem of obtaining the decision devices that are optimum in the sense that they minimize the average loss. This problem shall be referred to as the free running detection problem (FRD). In this section, a formal statement of the FRD problem is presented. We begin by specifying the decision device input. As discussed

-16in the preceding sections, this input consists of a sequence of randomly occurring pulses together with noise. Consider first the pulse sequence itself. In this study it is assumed that each pulse is identical in shape and completely known except for its arrival time. These pulses may then be characterized in terms of a signal s(t, at, a2'...) defined by s(t, al, 2''" ) = 2 P(- (j) (1.11) J where a. is the arrival time of the jth pulse and where p(t) is the shape of the pulse with arrival time t = 0. In addition, it is assumed that the pulses are non-overlapping in the sense that there exists a number T with the property that aj+1 > a' + T for all j (1.12) and such that p(t) is non-zero only over the interval [0, T ). The number T is referred to as the pulse duration. A typical pulse p(t) and signal s(t, o1, a2,...) are shown in Fig. 1.8. It should be mentioned in connection with the definition of the signal s(t, al, a2) that we allow p(t) to be zero over part of the interval [0, T). For example, the waveform shown in Fig. 1.9 is an acceptable pulse waveform. To complete the definition of the decision device input it is assumed that the noise is additive. That is, if the input is denoted

-17p(t) I...... It T p s(t, al, a2) A (a) l a2 a3 U1 U2 U3 (b) Fig. 1.8. A typical pulse p(t) and signal s(t, ol, a2,...) p(t) P I - T Fig. 1.9. A pulse p(t) that is zero over part of the interval [0, T ) P

-18by x(t), then x(t) = n(t) + s(t, al, (2'..) t [O, T] where n(t) is a realization of some known stochastic process which represents the noise in the system. In the following we will also refer to x(t) as the observation and we will refer to the interval [0, T] as the total observation interval. Next, consider the output of the decision device. As discussed in the previous sections, this output is a binary valued function a(t), a(t) {0, 1 }, called the response function. For the respondonce decision device, we will impose the condition that the number of responses in the interval [0, T] is finite. That is, it is assumed that the set of response times, {t;a(t) = 1} consists of a finite number of isolated points. For the respond-andhold decision device, we will stipulate that responses may occur only over half-open intervals. More specifically, we will assume that the set of response times can be written as a finite union of disjoint intervals of the form [a, b). In the following, the value of the response function at a particular time t', a(t'), will often be referred to as the decision at time t'. Let us now turn to the description of the decision device itself.

-19This device operates in real timne to produce the response function a(t) from the observation x(t). Thus, we may formally consider the decision device as a function 6 which maps observations x(t) into response functions a(t). We may incorporate the fact that the decision device must operate in real time by requiring that 6 have the property that a(t') can depend only on x(t") for t" < t'. In the following we will also refer to 6 as a decision rule. The basic definition of the FRD problem is completed by defining the respond- and- hold and the respond- once loss functions. From Eqs. 1.4 and 1.8, it is seen that the respond-and-hold loss function is a weighted sum of the six quantities TD, T, TC, TF, ND and NM and the respond-once loss function is a weighted sum of the four quantities, NF, NX, ND and NM. Thus, in order to define these loss functions, it is only necessary to specify how these quantities are calculated from each response function a(t) and signal s(t, a l a2'...). To this end, we will classify each possible decision a(t') into one of six different outcomes, a false alarm (F), a current rejection (C), a detected pulse (D), an extra detection (X), a missed pulse (M) and a rest (R). It will then be shown that the quantities that appear in the loss functions can be defined in terms of these outcomes. First, define a false alarm and a correct rejection. Now both of these outcomes occur when no pulse is present. Thus, if we define

-20the condition that no pulse is present at time t' by the condition that t' [a, a. + Tp) for any a., then: J J A false alarm (F) occurs at time t' iff a(t') = 1 and no pulse is present at time t' and A correct rejection (C) occurs at time t' iff a(t') = 0 and no pulse is present at time t' Next, we define the detected pulse outcome and the extra detection outcome. Both of these outcomes occur when a pulse is present at time t'; that is, when t' e [a., a. + T ) for some a. 3 ] P J Now, the idea here is to consider the first response to a pulse as a detected pulse outcome and any other response to that pulse as an extra detection. Formally, A detected pulse (D) occurs at time t' iff a(t') =1 and t'e[a., a.+T ) for some a. J J P P and a(t") = 0 for t"e (a., t') alnd

-21An extra detection (X) occurs at time t' iff a(t') = 1 and t'e [a., aj+T ) for some a. and a(t") = 1 for some t" e (a, t') Finally, we define the rest outcome and the missed pulse outcome. These outcomes also occur when a pulse is present. Here the idea is to define a miss outcome as the decision not to respond to an undetected pulse at the last remaining opportunity to detect that pulse. Any other decision not to respond when a pulse is present will be a rest outcome. Formally, A missed pulse (M) occurs at time t' iff a(t') = and t' = a.+T for some a. J P ] and a(t") = 0 for all t" (aj, a.+Tp) and A rest (R) occurs at time t' iff a(t') = and t' (a.j, a+T ) for some j or a(t') =0 t' =a.+T and a(t") = 1 for i p some t" E(a., r.j+T ) The above definitions are illustrated in Fig. 1.10. The outcomes associated with the decision times tl,..., t7 are shown beneath these times.

-22s(t, a(' a2) al (l+Tp a2 a2+T a(t) ~ * ~ t1 t2 t3 t4=a1+T t4 t5 t6 t7=2+T D X R R F C R M Fig. 1.10. The different possible decision outcomes It remains to show that the quantities ND, NM, NX and NF for the respond-once loss function and the quantities ND, NM, TD, TF, TC and TM for the respond-and-hold loss functions can be expressed in terms of the above six outcomes. This is done as follows. First, set the quantities ND and N equal to the number of detected pulse outcomes and the number of missed pulse outcomes respectively. These quantities are always well-defined, since there can only be one detected pulse outcome or one missed pulse outcome for each pulse that occurs, and there can only be a finite number of pulses that occur in the interval [0, T]. Next, set NF and NX, for the respond-once device, equal to the total number of false alarm outcomes and extra detection outcomes, respectively. These quantities

-23are well-defined, since a false alarm or an extra detection can occur only when a(t) = 1 and, by definition, the respond-once response function has only a finite number of responses. Finally, define TD, TMTC and TF for the respond-and-hold decision device by: T = the total extra detection time T = the total rest time M TC = the total correct rejection time TF = the total false alarm time F It should be emphasized that TD, TM, TC and TF as defined here are numerically the same as the TD, TM, TC and TF of the preceding section. In the above paragraphs, we have formally defined the basic model for the FRD problem. In Chapter II, we will interpret this model in terms of the general decision theory model and in the remaining chapters we will use this theory to obtain optimum (Bayes) decision rules and to analyze their performance. 1.4 An Intuitive Look at the FRD Decision Device Since, for the most part, the analysis in this study is of a formal nature, we will pause at this point to analyze the structure of the decision device from an intuitive point of view. We begin by considering the quantities that determine a particular decision a(t'). First of all, note that in general this

- 24decision depends on the total past observation x(t"), t" < t'. Thus, it can be expected that the decision device contains a memory for storing this past observation. In Chapter IV, however, we will show that, if the noise satisfies a certain independence condition, then it is sufficient to store a certain density function which can be updated as time progresses. Next, consider the dependence of the decision a(t') on the past decisions a(t"), t" < t'. This dependence is due to the fact that, if a pulse is present at time t', then the outcome of the decision a(t') depends on whether or not that pulse has been previously detected. For example, a response at time t' to the pulse shown in Fig. 1.11 results in a detected pulse, if a(t") = 0 for t" (a, t'), but it results in an extra detection if a(t") = 1 for some t" e(, t'). On the other hand, a decision not to respond at time a+T results P in a missed pulse if a(t") = 0 for t" e (a, a+T ), but it results in a rest if a(t") = 1 for some t" (a, a+T ) Since the outcome of the decision a(t') may depend on the previous decision a(t"), t" < t', it also seems necessary to expect that the general decision device must contain a memory for storing these past decisions. But note from the above arguments that this dependence occurs only if a pulse is present at time t' and then only through the condition that thse has been previously detected. Now we may also express this condition if we know only the time

-25a t' (+T P Fig. 1.11. A pulse present at time t' at which the last response was made. To see that this is so, denote by T, the time of the most recent response prior to time t'. Then, from Fig. 1.12, it is seen that, if Tt, > a, then the pulse present at time t' has already been detected, but, if 7tr < a, then that pulse has not been detected. Thus, it seems reasonable to conclude that the general decision device need remember only the time of the most recent response and not the total past decision history. In the following, this conjecture is referred to as the most recent response rule. In Chapter III it will be shown that essentially all decision devices obey the most recent response rule. We now consider the form of the decision device. In the above paragraphs, it has been argued that the decision device must contain a memory for storing the total past observation and a memory for storing the value of the most recent response function. Now the

-26(x t' (a) O ~ It r Tt t' (b) r r?- t' TtT (c) Fig. 1.12. An illustration of the most recent response time (a) a pulse present at time t' (b) the case rt > a (c) the case Tr, < a decision at time t', a(t'), is based on some function of these two quantities. Let us denote this function by Tt,(x(t"), t" < t'; r, ). Then without loss of generality, we may assume that the decision a(t') is determined according to a(t) i 1 if Tt,(x(t"), t" t; t ) > 0 0 otherwise It is then possible to characterize the form of the general decision device as shown in Fig. 1.13.

-27Past Observation Most Recent Memory Memory x(t')_ H Calculate |t 1, > 0 T t,(x(t"),t" <t'; Tt,) = at ) 0 aV ( ~tt~t? < t rt?).... - __ - 0 T, < 0') Fig. 1.13. The form of the decision device The intuitive idea behind Fig. 1.13 is that the decision device first filters or smooths the observation by a time varying filter which also depends on the time of the most recent response. The filter output, Tt,, is then compared with a threshold of zero to determine the current value of the response function.. The operation of this configuration is illustrated in Fig. 1.14 in terms of a typical signal s(t, a, a...), a corresponding input x(t), the filter output Tt, and the resulting response function a(t) for a respond-and-hold decision device.

-281s(t,' 2) ~x~~~~~t) W~~~~~~~~~~~~~~~~~~ a(t) Fig. 1.14. The signals present in a respond-and-hold decision device with the configuration of Fig. 1.13

-29In the above paragraphs it has been argued that the generic form of the optimum decision device is as illustrated in Fig. 1.13. This argument is based solely on a consideration of the quantities that determine a particular decision a(t'). In the next paragraphs we will draw several conjectures on the character of specific decision devices based on a consideration of the loss functions. Consider first the respond-once device. This device seeks to respond to each pulse that occurs while avoiding false alarms and extra detections. One way to preclude extra detections and yet still have the opportunity to detect each pulse is to avoid making more than one response in any interval of length T seconds. (Recall that pulses are separated by at least T seconds.) Stated another P way, the decision device can refuse to respond at time t' regardless of the observation if the time of the most recent response Tr, is greater than t'- T. In the following, this principle is referred to as the inhibit rule, since any decision device that obeys the inhibit rule is inhibited from responding for T seconds after each response. The reasoning behind the inhibit rule and the fact that large values of the loss LX reflect a strong desire to avoid extra detections suggest the following conjecture. Conjecture I. If the loss LX is sufficiently large in relation to LF, then the optimum respond-once device satisfies the inhibit rule.

-3 0Next, note that if, in fact, a decision device satisfies the inhibit rule, then the number of extraneous detections NX is equal to zero, so that in this case the loss does not depend on LX. (See Eq. 1.10. ) Thus, it is also reasonable to conjecture that the structure of those decision devices that satisfy the inhibit rule does not depend on LX. Stated formally, Conjecture II. For each LF, there exists a Lx(LF) such that for L > L (L) the optimum decision device satisfies the inhibit rule X- LX F and does not otherwise depend on the particular value of LX. Finally, consider the respond-once decision devices associated with small values of L. These values reflect only a mild desire to avoid extra detections. Thus, it is reasonable to suppose that the optimum decision device for small LX will not satisfy the inhibit rule if in so doing it can increase the number of pulse detections without increasing the number of false alarms. This reasoning suggests Conjecture III. For the optimum respond-once decision devices that do not satisfy the inhibit rule there is a trade-off between the average number of detected pulses ND and the average number of extra detections NX. Moreover, the maximum average ND occurs for the MPD decision device (WX = 0). We now turn to the respond-and-hold decision devices. First,

- 31recall that the purpose of the decision feedback loop in the system diagram of Fig. 1.13 is to help the device distinguish between an initial response to a pulse and a later response to the same pulse. This was seen to be necessary, since the initial response resulted in a detection, whereas the later response results in an extra detection. But, note that the loss function for the MDT decision device does not distinguish between initial detections and extra detections, since this function does not depend on the number of detected pulses ND. Thus, it can be asserted that, in this case, the decision feedback loop is not necessary. Hence, Conjecture IV. The optimum MDT decision device does not contain a decision feedback loop. Finally, consider the respond-and-hold loss function for which (LD- LM) << (LD - LM) < 0. As mentioned above, this function places a high premium on the number of detected pulses but a low premium on the total amount of detection time. A reasonable procedure for the optimum decision device in this case is to respond only for short periods of time during the center of each pulse as illustrated by the response function a1(t) in Fig. 1.5. This would increase the likelihood that each pulse is detected but it would reduce the amount of false alarm time that might otherwise result from responding for a longer period of time to each pulse. On the other hand, this would also reduce the total amount of detection but this is of little

-32T T consequence since the gain for detection time - (L - L ) is small. D M Thus, it is reasonable to assume that the effect of increasing the number of detected pulses is to decrease the amount of detection time. That is, Conjective V. For the respond-and-hold decision device there is a trade-off between the average number of detected pulses and the average detection time with the maximum average detection time occurring for the MDT decision device. We conclude this section with an example of an R-H decision device and an R-O decision device. It will be demonstrated later that both of these decision devices are sub-optimal by comparing their performances with the performance of the optimum decision device. Nevertheless, these decision devices are simple and practical and, in certain situations, they perform almost as well as the optimum decision devices. To define these decision devices, it is sufficient to specify their test functions Tt. For the R-H decision device, we define Tt by Tt,(x(t"), t" t'; t) = Ct'(x(t"), t"< t')- where 3 is some prespecified threshold and t' Ct,(x(t"), t" < t') = x(t) p(t-(t'-T ))dt t'-T p

-33For the R-O decision device we have essentially the same definition except now we enforce the inhibit rule. This is done by setting Tt, equal to - Il (e an arbitrary non-zero number) whenever r, > t' - T. The resulting definition is Ttt X t" < t'; Tr Tj(x(t"), t"< t'; <t,) Ct(xt"), t" < t')- < t'- T c ptt p -le t, >t'-Tp The engineering significance of these decision devices lies in the fact that the function Ct, can be realized as the output of a filter that is matched to the basic pulse shape p(t). That is, Ct, can be obtained by filtering the observation x(t) with a linear timeinvariant filter with impulse response h(t) = p(T - t) Thus, we may realize these systems as shown in Fig. 1.15. In the following discussion, these decision devices are referred to as matched filter (MF) decision devices.

-34x(t) Matched 1 if Ct> a xi ivi~~~aicet)_ - _ [Ia(t) Filter a(t) = 0 if C t< (a) Y (t) Matched a(t) - 1 iff C > a(t) - Filter r ~ t- T and T < t - T It P (b) Fig. 1.15. The matched filter decision device (a) respond-and-hold case, (b) respond-once case 1. 5 Historical Remark To this author's knowledge there is no existing theory for the free running detection problem. This lack of theoretical results is probably due to certain mathematical and conceptual difficulties that arise in carrying out a theoretical analysis. For example, the fact that different decisions are interdependent makes it impossible to determine the optimum decision device by considering a single decision independently of the other decisions. Nevertheless, in the psycho-physics literature there has been some attention given to the problem that we have characterized here by the respond-once decision device. We shall mention only three

-35of these references. A more complete bibliography is found in Ref. 3. The first reference is a paper by Egan, Greenberg and Schulman (Ref. 1). This paper serves mainly to define the basic respondonce paradigm and to present techniques for experimentally evaluating the performance of the human decision-maker. The second reference is a paper by Lucas (Ref. 2). In this paper, Lucas introduces essentially the same decision device as the respond-once MF of the preceding section and he calculates certain bounds on the performance of this device. The third reference is a paper by Luce (Ref. 3). This paper assumes that responses occur according to some renewal process and on the basis of this assumption certain probabilities are calculated. 1. 6 Organization of This Study In Chapter I we introduced the free running detection problem and formulated several conjectures on the character of the respondonce and the respond-and-hold decision devices. In this study, we will develop the theory for the FRD problem in three parts: (1) the definition of the basic decision model, (2) the description of the performance for the two classes of decision devices, and (3) the derivation of the optimum (Bayes) decision devices. As a result, we will be able to verify the conjectures of this chapter. The organization of this study is as follows. Chapter I provides

-36the basic definition of the FRD problem. In Chapter II we proceed formally to obtain the basic decision model for this problem as a special case of the general decision problem. In Chapter III, we develop a procedure for evaluating the performance of different decision devices in terms of the average rates of detections, detections and false alarms. This performance characterization is then used to verify some of the conjectures in Section 1. 4. In Chapter IV we analyze the model of Chapter II to obtain a general recursive description of the optimum decision rules. Then, some sharper results are derived under the assumption that the observations are conditionally independent. We also verify Conjecture IV of Section 1. 4 in this chapter. In Chapter V, we turn to the computer to obtain a description of certain optimum decision rules and to evaluate numerically the performance of these decision rules as well as the performance of the matched filter decision devices. Chapter VI summarizes the results of this study.

CHAPTER II THE MATHEMATICAL MODEL FOR THE FREE RUNNING DETECTION PROBLEM The analysis of any problem involving the making of decisions requires precise definitions of many things. First of all, the nature of those entities for which decisions are desired must be known. For example, in the detection of randomly occurring pulses, it must be possible to characterize all of the pulse sequences that may occur. Also, it is necessary to be able to express any prior knowledge regarding these pulses such as the rate at which they are occurring. Secondly, one must know precisely what kind of decisions are to be made. That is, is a typical decision a statement of the form "a pulse is present at the current time" or is it a statement of the form "the pulse present at the current time arrived at time to? Thirdly, one must know precisely what information is available on which to base these decisions and how this information is statistically related to the unknown quantity. Lastly, one must have a criteria by which to judge the "quality" of different decision procedures. In most decision problems the above quantities can be specified in a fairly straightforward manner without resorting to a formal mathematical structure. But, in contrast to most decision problems, the FRD problem is quite complex. Moreover, it involves the use of -37

-38certain concepts not usually found in most of the classical decision problems. For this reason, we shall call on the formal theory of decision-making to provide a theoretical basis for the mathematical model. The first section of this chapter reviews the basic elements of decision theory. The remainder of the chapter is devoted to interpreting the FRD problem in terms of this theory. 2. 1 Basic Elements of Decision Theory In this section we introduce the basic elements of decision theory. The definitions introduced here closely parallel the development found in Chapters I and 11 of Ref. 4. To begin, consider the triple (Ie 0, L(', )). The first element of this triple,cI, is known as the action space. It is the set of all possible actions or decisions available to the decision-maker. The second element, 0, is the state space. It is the set of all possible states of nature that are of interest to the decision-maker. The third element, L(-,.) is the loss function. This is a realvalued function defined on the product a x O. These elements have the following interpretation. A state, 0e 0, unknown to the decisionmaker, is in effect. The decision-maker chooses an action a e and then incurs a loss given by L(a, 0). The objective of decision theory is to provide the decisionmaker with some rational rule for choosing an action, a e, so as

-39to incur a small loss. To assist in this choice, the decision-maker is allowed to observe some quantity that is statistically related to 0. Let X denote the space of all of these possible observations. The decision-maker then observes an observation x e X and on the basis of that observation he chooses an a eI. The rule governing the choice of a e jd for each x e X is known as the decision rule employed by the decision-maker. Formally, the decision rule, 6, is a mapping from X to /. The statistical dependence of x e X on 0 is provided by the family of probability measures {P(dxl 0); o e O) which are assumed to be known. We are now in the position to present the formal definition of a decision model. A decision model is a triple (J, 0, L(, * )) coupled with a random experiment which consists of an observation space X and observation probabilities {P(dx 1 0); 0 e }). A solution to a decision problem is a decision rule 6 e where!1 is the set of all functions from X to V that are measurable with respect to P(dx I 0) for each 0 e. It should be mentioned that, in the above definition of a decision model, the action space i must be a measurable space so that the definition of 6 as a measurable function makes sense. In a given decision problem, it is desired to obtain a decision rule which satisfies some specific criterion of optimality. One

-40such criterion of optimality is the Bayes criterion. The procedure used to obtain a decision rule that is optimum in the Bayes sense is known as the Bayes procedure and the resulting decision rule is known as the Bayes decision rule. In the bulk of the work which follows, we shall be concerned with Bayes decision rules. To investigate further the notion of Bayes optimality, it is necessary to introduce two additional entities. The first of these is the prior probability law. The prior probability law, II( ), is a probability measure defined on the state space 0 (0 must now be considered as a measurable space), with the interpretation that I(dO) represents the decision-maker's "subjective feelings", prior to the observation, as to which state of nature 0 is in effect. The second concept needed for the Bayes procedure is the Bayes risk. This is defined as follows. First, recall from above that the loss function L(, ) is a real-valued mapping onWx 0. The space Ix O can be interpreted as a measurable space since both a/ and O are measurable spaces. If we now require that L be a measurable mapping into the real line, then, for each 6 e, the mapping L(6(x), 0) is a measurable mapping defined on Xx O. With this provision we define the Bayes risk r(6) of the decision rule 6 e! with respect to the prior probability law II( ) as r(6) = E{L(6(x), 09)

-41where the expected value is taken with respect to the joint measure defined by P(dx, do) = P(dxl 0) Il(d0) We are now in a position to define the Bayes decision rule 6. Specifically, a decision rule 6 is said to be Bayes with respect to I( ) iff r(6~) = inf r(6) The number r(60) is referred to as the minimum Bayes risk. It should be mentioned in passing that, although, in a given decision problem, a Bayes decision rule may not exist, the minimum Bayes risk always exists. Before proceeding, it is desirable to introduce some additional notation. In applying the Bayes procedure it is often necessary to take the expected value with respect to different probability laws. Which law is intended can be conveniently denoted by writing underneath the expected value operator the random variables involved. As examples, E denotes the expected value with respect to x, 0 P(dx, d0), E denotes the expected value with respect to P(dxl 0) xl 9 and E denotes the expected value with respect to II(dO). We then have the well known identities

-42E {*} =E {E {* } x, 9 0 xl | and E { } = E fE {I } x, 9 x x O where of course P(dx) = P(dx,d0) and P(dO Ix) - P(dxO) )(dO) P(dx) To provide an example of the above definition and to introduce some techniques and results which are useful later, we have cast the well known classical detection problem in the notation of the general decision model. This is done in Appendix A. The remainder of this chapter is devoted to interpreting the FRD problem in terms of the general decision model. 2. 2 The Action Space The action space is the set of all possible actions available to the decision-maker. In Chapter I it was seen that we can identify these actions with response functions a(t). Thus, in general, we should take the action space for the FRD problem to be the set of all

-43 - binary valued functions satisfying the definitions of Section 1.3. To deal directly with such an action space, however, introduces certain mathematical difficulties in determining the Bayes decision rule. To circumvent these difficulties we will follow a strategy that is common in problems that deal with the continuum. Specifically, the action space will be further restricted to response functions that allow decisions to be made at only a finite number of equally spaced decision times. It is then reasonable to expect that, as the separation between the allowable decision times becomes small, the resulting Bayes solution will closely approximate the solution obtained from an unrestricted action space. We proceed with the definition of the restricted action space. To begin, define the set of allowable decision times, by = {titi iA, i=l,..., N} (2. 1) where the decision time separation A and the number of decision times N are related by A = T/N (2. 2) Now denote the value of the response function at time t. (the decision at time t.) by a.. That is a) a(ti) = a.

-44Then we define the action space for the respond-once decision device to be those functions a(t) such that o tal a(t) = (2.3) ~a. te6 and we define the action space for the respond-and-hold decision device to be those functions a(t) such that a(t) = ai te [ti_, ti) (2.4) An example of a typical respond- once response function and a typical respond-and-hold response function appears in Fig. 2.1. The technical advantage of the above definitions lies in the fact that a response function for either the R-O case or the R-H case may be represented by the binary N-tuple a = (a1,a2,..., aN), ai {0,1} Thus, we may characterize the action space for either case as the set of all such N-tuples, Ad = {a = (a1,..., aN); ai {,1}' Henceforth, an element a e 3d will be referred to as a response vectol'. TEli es1 Spflns( flunclti)onl (t) Uass(ci'ated withlI tlie( r1csl))ns vec(tor

-45a(t) 1 * *.;; I,I.... I. I. I I * * I I:0 tl t2 t3 t-1 ti t1 ti+2 tN-2 tN tNT (a) a(t) I,, ~. I...I *.' I I; ~ tl t2 t3 ti-1 ti ti+1 ti+2 tN-2 tN-l tN=T (b) Fig. 2.1. Typical response functions (a) respond-once response function (b) respond-and-hold loss function a is obtained from either Eq. 2.3 or Eq. 2.4 depending on the case at hand. We conclude this section with a discussion of the significance

-46of the decision time separation A. First, it should be noted that for a fixed pulse duration, T, A determines the number of decision opportunities available to detect each pulse. For example, suppose T and T are fixed and A = Tp/2. Then, any pulse that occurs covers two allowable decision times, so that the decision-device has two opportunities to detect each pulse. On the other hand, if A is decreased to T /4, then each pulse covers four decision times so that now there are four opportunities to detect each pulse. These cases are illustrated in Fig. 2.2. I I, ti i+l i+2 ti+3 (a) T p I I I. 1 t i ti+l i+2 ti+3 ti+4 ti+5 ti+6 (b) Fig. 2.2. The number of decision opportunities per pulse; (a) A = T /2 (b) A = T /4 P p

- 47In the light of the above comments, it is reasonable to expect that the performance of the decision-device should improve as the number of decision opportunities per pulse increases. Indeed, this is shown to be the case in Chapter IV. Thus, it is not the absolute durations of A and T that are important but rather the ratio of p these quantities since this ratio gives the number of decision opportunities per pulse. To simplify the following analysis we shall assume that A is only allowed to take on values such that the number of decision opportunities Tp/A is integer valued. We will denote this number by m = T /A. 2.3 Parameter Space and Prior Probabilities As discussed in Chapter I, the signal present at the input to the decision-device consists of a train of non-overlapping, randomly occurring pulses. In this section we shall make this notion precise. This is done by means of a probabilistic description of the arrival times of the individual pulses. This description is then used as a basis for defining the parameter set 0 and the a priori probability law on nI(d ). To begin we recall from Section 1.3 that a signal is represented as a function Np s(t,,1.'," aN) =; p(t- i) p i=l 1

- 48This representation can be interpreted as a mapping which identifies signals with sequences of arrival times al, a2,..., oNp p Therefore, it is possible to answer probabilistic questions concerning the presence of pulses, if we have a probabilistic description of the arrival times. In the discussion which follows we will obtain such a description. The theoretical basis for the probabilistic description of the pulse arrival times is found in the theory of renewal processes. We shall pause briefly to review those aspects of renewal theory that are of interest here. Formally, any sequence of positive, independent, identically distributed random variables, {Vi }, can be used to define a renewal process. The random variables 7i are known as inter-arrival times. Physically, 7i represents the elapsed time between the (i-1)st event and the ith event. The arrival times, a., are related to the inter-arrival times, 7i, by a. = j 7i i i=l An important concept in renewal theory is the concept of the arrival rate. Let N (al,.. ) be the number of arrivals in the interval [0, T]. Then the arrival rate is defined as rE[N ] rp(T) = T (2.5)

- 49In general r (T) depends on T. In fact, the fundamental result of renewal theory, the renewal theorem, states that under suitable conditions, r (T) - (E[Ti]) as T --. However, in some situations, it is desirable to deal with processes for which r (T) does not depend on T. A renewal process with this property can be obtained by allowing the arrival time to the first event, r77, to have a distribution different than ri, i > 1. The resulting process is then known as a "delayed" renewal process. Specifically, if the cumulative distribution function for 77, F (77(), is redefined by (771) - - [1 - F (u)] du (2.6) h1 o 2E[72] 0 2 then the resulting "delayed" renewal process has the rate r (T) = 1 (2.7) P E[r2] A more comprehensive discussion of renewal processes along with precise statements and proofs of the above facts can be found in Chapter XI of Ref. 5. We shall now apply these concepts to obtain a probabilistic model for the pulse arrivals. The simplest renewal process is the Poisson process or the "No Memory Waiting Time" process. This process is characterized by the fact that the inter-arrival times are exponentially distributed.

- 50It is the Poisson process which is usually used to describe the notion of "randomly occurring events". The Poisson process shall be modified in two ways to obtain the model for the pulse arrival times. The first and simplest modification will be to incorporate the provision that the pulses be nonoverlapping in the basic Poisson process. This is accomplished simply by translating the density of the arrival time T units in time. Thus, the density of the inter-arrival times f (17) shall be defined as r0 r < Tp f7i (7) (2.8) -ol(nl-T ) ae 7 > T - P This density is illustrated in Fig. 2.3. f (n)'I i T Fig. 2.3. The density of the ith inter-arrival time The second modification to the Poisson process will be to redefine the density of the first arrival time, f (r7), so as to obtain r17

- 51a delayed process with a constant rate. This modification is made for two reasons. First, it greatly simplifies the mathematical model to deal with signals whose pulses are arriving at a rate independent of T. The second reason is more subtle. In all ordinary renewal processes, with the single exception of the (unmodified) Poisson process, the time origin takes on a special significance. Physically, this significance can be interpreted in the following way. If the pulses can be considered as being generated by some device, then the ordinary renewal process describes the pulse arrivals as if the generator were turned on at time t = 0. When this is the case, the process appears to "evolve" beginning from the time t = 0. The evolution of the ordinary renewal process can be seen in terms of the behavior of the random variable vt, known as the residual waiting time. Specifically, vt is the time elapsed from the current time, t, to the next pulse arrival. Now, for an ordinary renewal process, the distribution of vt depends on the specific time t. It is well known, however (Ref. 5), that as t - x the distribution of vt approaches a limiting distribution, thus indicating that the process has reached a "steady state". Moreover, it can be shown that the limiting distribution of vt is precisely the distribution of the random variable 771 in the delayed renewal process. This fact allows one to interpret the delayed renewal process as a renewal

- 52process which began at time t = -x, or, stated in other terms, as a renewal process which has reached its "steady state". Thus, the inclusion of the second modification assumes that the decisionmaker knows only that the pulse generator was turned on in the far distant past and that the pulses are occurring at some constant rate r p In summary, the pulse arrival times will be assumed to be defined by the delayed renewal process with inter-arrival times i }. The density of,i' i> 1, is given by Eq. 2.8. The density of 7I can readily be determined from Eq. 2.6. The result is 0 7 <0 f (7) = (T +(a-1)) 0 < r < T 77! P -1 -1 (Tp+(a )) exp [-a(7-Tp)] Tp < (2.9) p pThe pulse rate r is determined from Eq. 2.7. Specifically, -1 1 r= (T + o ) (2.10) P P Before proceeding to the description of the state space, we shall present a "discretized" version of the above process. Physically, this amounts to restricting the pulse arrivals to occur only at regularly spaced points on the time axis. To be more specific, we

-53divide the time axis into equal intervals of length v = Tp/q for some fixed integer q. Then it is assumed that pulses can arrive only at the times t. = iv. To obtain the probability law in the discrete setting, we appeal to the fact that randomly occurring events are described by geometrically distributed inter-arrival times. Thus, in analogy to the continuous case, the probabilities of the inter-arrival times 7i' i > 1 will be defined as O j j<q Pr[i = j] = (2.11) a l- a) j> q This probability law is illustrated in Fig. 2.4. Pr[i =jiv] _1,.I. I I -ll.l,,.. l v 2v ~ ~ c ^14. 1 I + Fig. 2.4. Discrete inter-arrival time probability law

- 54The probability law for the first arrival 71 will be defined as 0 j< Pr[1 = jv] = [q + a/(l-a)]1 1 < j < q (2.12) [q + a/(1-a)]1-aj-q+l q<j It is shown in Appendix E that with {7i } as defined above the rate r is constant and equal to P r = [[a/(1-a)]v T] (2.1 3) We turn now to the definition of the state space 0 and the prior probability law II(dO). In the beginning of this section it was shown that the signals s(t) can be identified with the pulse arrival times ao, a2,... Therefore, we might take the set of all possible sequences of arrival times as the state space O. Then, the prior probability law, nI(dO), would be defined directly in terms of the delay renewal process. However, this would introduce the following difficulty. At each allowable decision time t. E A, the decision-maker is concerned with whether or not an undetected pulse is present at time t.. The difficulty lies in the fact that the events of this kind are difficult to express directly in terms of the arrival times or inter-arrival times. It would be far more advantageous to have a description of O in terms of Ntuples, W = (HO,..., ON), where Qi is defined so as to express

-55the prior information relevant to the decision-maker at the time t.e oj. In the next paragraphs, the state space 0 is constructed in terms of random variables 0. that have this property. The random variables 0i, i =1,..., N are defined in terms of the pulse arrival times {a.j }. Specifically, for i = 1,..., N, we define - (t- T) if some a [ti - Tp, ti) 0. = (2.14) T otherwise P These random variables are well-defined since we have assumed that aj+l > a. + T so that it is not possible for more than one pulse to arrive in the interval [t.- Tp, ti). We might also point out that this definition assumes that the signal s(t; a;,..., aN ) is defined over P the interval [-(T - A), T] The above definition of 0i is interpreted with the help of Fig. 2. 5. Here it is seen that, if a pulse arrives in the interval [ti- T, t), 0i 1 p 1 Fig. 2.5. The relation of 0. to a pulse arriving in the interval [t. - T,t.) 1 p 1

-56then 0. gives the time at which that pulse arrives as measured from time t. - T. Note that when a pulse is present at time t., 1 p i E [0, Tp). On the other hand, if no pulse is present at time ti, then. = T. Thus, the range of the random variable 0. is i = 1 p 1 1 [0, T]. p With the above definition at hand we may define the state space O as the set, O=~) {= ( N): iOi -[O T p] These definitions are illustrated in Fig. 2.6, which shows a typical pulse sequence and the resulting values of 0 = (0.'..' 0N) For the above definition of 0 to be valid, we must be able to represent the signal s(t,al, a2'... aN ) in terms of. To p show that this is possible and to establish a result for later use we will prove that N s(t,W) = 0 I (O p (t (i +ti - T) (2.15) i= l 1 P {0e [0, A)} We proceed as follows. Suppose that ol a2,.... aN denote the p arrival times of the pulses in s(t, al, a2,..., aN ). Consider the p jth pulse, arriving at time ao. This pulse is contained in the interval [aj, a. +T ) and no other pulse may also arrive in this interval. Let i* be the largest integer such that the decision time t.- T is less P

- 57s(t, 1', a2...) 1 U2 o3 tN (a) /i 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~.T 0 ~~~~~0. 6 ~ ~ ~ ~ ~ ~ ~0* * 0 t t iN (b) Fig. 2.6. An illustration of 0 (a) a typical pulse sequence (b) the corresponding value of = (01..., N) than or equal to r. as shown in Fig. 2.7. From this figure it is clear that the interval [aj, aj+T ) contains only the m decision times ti*+m- ti* The random variables associated with these decision times are

-58-i*+m- 1 t/ t -TT t<-T C. t t/ t i*+m-l i*-lT p i* p j i*- i* t i* - T t -T =t -p i*+l p i*+m-1 Fig. 2.7. The definition of i* Also from Fig. 2.7 it is clear that the only one of these random variables that takes on a value in the interval [0, A) is 0i*. Thus, the only contribution from the jth pulse to the sum in Eq. 2.15 is the i*th term, I (*) P(t - i* ti* Tp)) {0 i* e [0A) But this term is equal to p(t- a.) since by definition a. =.i* + ti* T. Since this reasoning applies for all of the pulse arrivals al, a2,... ON' Eq. 2.15 is verified. P The remaining task in this section is to specify the prior probability law II(d0). This law is obtained as that induced from the delayed renewal process by the transformation of Eq. 2.14. The

- 59computations involved in obtaining II(dO) are for the most part tedious and uninteresting and for that reason they are left to Appendices B and D. The main result of these computations is that for both the discrete and continuous pulse arrival time cases, the induced probability law, I(dO), is a stationary Markov probability law on the random variables 01,'...' N In Appendix B, II(dO) is shown to be Markov. In the continuous case, the conditional cumulative distribution function F(ck C _l ) = Pr[Ok < cl kl = Ck- ] is given by [ C < Ck-_1+ (T -A) k~ 1 P- 4^k-(c- (Ck l+T -A))< T<c P- k F(clckl) =C<Ck-l- A 1 Ck-1'[^'Tp) f0 c < T -A ll-eCP T -A< c< T k-=T I-6 p p - p I k-1 p l.cT < c P - (2.16) This function is illustrated in Fig. 2.8.

-60F (cl ck-l) (a) T - Ck- T p k-I p F( cl ck 1) F(cl ck 1) l I I I,. i c T T -A T k-1 P P p (b) (c) Fig. 2.8. Conditional cumulative distribution function F(cl ck ) (a) Ck- [O, A), (b) Ckl e [A, Tp) (c) ck = T For the discrete case it must be assumed that the decision separation time, A, can be divided into an integral number of intervals of length v. If this number is denoted by s = A/v, then in the discrete case the conditional probability law is given by

- 61Pr[O=jk = lkl =1 ] = 0 Q < j+q-s (1-a) a j+q-s < Q < q je {0,..., s-1 } aS-& =q O Q j-s je {s,..., q-1} j-s 0 f < q-s (l-a) a-(q-s) q-s < Q < q j = q a Q=q (2.17) This function is illustrated in Fig. 2.9. In Appendix D the Markov law for II(dO) is shown to be stationary. In the continuous case, the unconditional cumulative distribution for Ok, k > 1 is given by -1 c(Tp+ (a)-1) ce[0,T ) Pr[k < c] = (2.18) i1 T < c P - and in the discrete case

- 62Pr[k =l v Ok1 =jv] 1—........ - 11111 (l,v (q- s+j)v qv (a) Pr[6k=f v lOk6 I=jv] Pr[Ok f=v 0k-1 =V ] 11 _ _..._- 1 L -..__ _ _ _ _ _ _ I l I i i (j-s)v qv (q-s)v jv (b) (c) Fig. 2.9. Conditional probability law P(0k = v 0k-1 = jv) (a) jE {0,..., s-1}, (b) je {s,..., q-1}, (c) j =q F[q+a/(1-a)]-1 e {0,..., q-1} Pr[ k=tV] (a/(1-a)) [q + a/(1-a)]1 =q (2.19) Both of these probability laws are seen to be uniform for 0k [0, T ) p with a jump at 0k = T. It has been shown above that the prior probability law II(d0) is expressed in terms of discrete probability laws and cumulative distribution functions. In the later chapters it will be of great value

-63to be able to express both of these probability laws in terms of densities. For this reason we will introduce a concept which is common in mathematical statistics, specifically, the generalized density. The theoretical basis for the generalized density is the RaydonNikodym theorem. Theorem (Radon-Nikodym). If w, j, are two measures defined on the measurable space (Y, eo) such that co is absolutely continuous with respect to A. (written o << /), and ti is a-finite, then there exists a function p:Y - R such that for any measurable set A ecm, w(A) = S p(y) /i(dy) A The application of this theorem that is of interest here is due to the fact that if co is a probability measure, then Pr[A] = w(A) - p(y) g(dy) A so that the function p(-) has the usual property of a probability density. In this setting, the function p(-) is commonly referred to as the generalized density of the probability law w with respect to the measure t, and is often written as p(y) = wo(dy)/p(dy). A more comprehensive discussion of the generalized density can be found in Ref. 6.

- 64In Appendix C we apply the above ideas to obtain generalized densities for the prior probability law II(d ). Definitions of absolute continuity and a-finite measures are also found there. 2.4 The Observation Space and Real Time Decision Rules In this section we specify the observation space and the structure of the decision rules for the FRD problem. The observation space, X, is considered first. This space is the set of all possible observations that may occur together with a family of probability measures {P(dxl 0); 0 e 0) }. Now from Chapter I it is recalled that the FRD decision device observes a waveform that consists of a train of pulses immersed in additive noise. Stated more formally, it has been assumed that some noise process, n = {n(t); t [0, T] }, defined on some appropriate probability space (Q,, P), is specified and that the observation x consists of a realization of that process plus some signal s(t;0), 0 e O. That is, x(t) = n(t)+ s(t, 0) 0 6 t [0, T] (2.20) This equation defines for each 0 e 0 a stochastic process x {x(t); t e [0, T] }, on (,t,, P). Now the viewpoint to be adopted here is as follows. For each w e iQ, x(t, o) is a real-valued function defined on [0, T], so that the stochastic process x can be viewed as a mapping from the space (S1,,*, P) to the set of realvalued functions defined on [0, T], IR[0'T. Moreover, it is possible

-65to define a a-algebra, B, of subsets of IR[0' T] so that the mapping x can be considered as a random variable taking values in IR[']. Thus, for each e 0, we may speak of the probability measure P (dx) as that measure induced by the transformation of Eq. 2.20. We may then take as the observation space X, the space (ER[0 T],2) and we may define the family of conditional probability measures {P(dxl0); 0 e O } by P(dxl ) = P0 (dx). Further details on this point of view can be found in Chapter 12 of Ref. 6. The observation space X defined in the above paragraph is the set of all the observations x(t), t e [0, T]. This space shall be referred to as the total observation space and an x e X shall be referred to as a total observation. Now in the FRD model,decisions are made at each of the decision times t. e,J. Since each of these decisions is based only on the observation up to time ti, we must also be able to speak of the observation x(t), t e [0, t). This can be done in the following straightforward way. For a given set of allowable decision times'= {ti; ti = iA, i = 1,..., N), consider the intervals [til, ti). These intervals partition the interval [0, T). For each i = 1,..., N, we can consider the stochastic process X. = {x(t); t E [ti, ti). An observation space X. can be defined 1 1 for x. in precisely the same manner as above. This observation space shall be referred to as the space of current observations and an x; e X, as a current observation. In addition, we can define the

-66observation space i Xi = X x..Xi = n Xi j =1 This observation space will be referred to as the space of observations up to time t. and an observation X. = (x1,..., xi) e Xi will be referred to as the observation up to time t. i The probability distributions for x. and x. have the obvious definitions in terms of the noise process n and the definitions of i = {x(t); te[ti.,ti)} and x = {x(t); te[O,ti)}. Afewproperties of these distributions which will be of use later might be mentioned at this point. First, it is easily verified from Eq. 2.15 that for te [0, ti), s(t, 0) can be written as i-i s(t,0) I ( t. - T J j js l (0.) p(t-(o j ~t-T)) j { [0, A)} + I (i) p (t- ( i + t. - T )) {ie [0, Tp)} t e [0, ti) so that for te [0, ti), s(t, 0) and thus x(t) depend only on'. = (01'.''., i). Therefore, P(dx il) depends only on i. By the

- 67same reasoning it can be concluded that for t c [til, ti), s(t, 0) can be written as s(t, 0) = I (O) Pt - i + t1 { i-1 [0, Tp)} + I ( i) p(t (i+ ti-T)) {0e[0, A)} ti e [ti_1' ti) (2. 21) Thus, P(dxil0) = P(dxil0i, 0i-). The remainder of this section is concerned with the decision rule 6. The decision rule 6 is a mapping from the observation space X to the action space g. In the general decision problem, nothing more need be said about the structure of the decision rule. In the FRD model, however, it is assumed that the decisions are made in real time. This assumption imposes a special structure on the decision rule 6. To investigate this structure, consider a decision time ti e. At this time the decision-maker either responds, a. = 1, or he doesn't respond, a. = 0. But since the decisions are to be made in real time, the decision- maker can base this decision only on the information which has become available up to time t.. This information is of two kinds, the observation up to time t., x., and the knowledge of what responses have already been made, a. = (a1,..., a;).

- 68Therefore, the decision at time t. can be a function only of x. and 1 1 a.. This function will be denoted by 6. and referred to as the com1 1 ponent of the decision function 6 at time t. We then have a. = 5i(x., i). The total decision rule 6 is defined as the N-tuple 6 = (1'..., N). It should be emphasized in connection with this definition that in order to specify a. at some time t. by a. = 6(, ) 1 i 1 11 the past decisions ai.1 must have already been specified. 2.5 The Loss Function It is recalled from Section 2.1 that the loss function is a mapping of pairs (a, 0) c/x 0 into the real numbers. The interpretation here is that if the decision-maker chooses the response vector aec e and if the parameter vector 0 e O is in effect, then a loss L(a, 0 ) is incurred. Now, in Section 1. 3, two loss functions, a respond-and-hold loss function and a respond-once loss function, were defined. In this section, these loss functions will be expressed in the notation of this chapter. It will be seen that both of these loss functions can be described in terms of a single canonical form. We begin by expressing the six basic outcomes defined in Section 1. 3 in terms of a and 0. First it is recalled that these outcomes depend on the following conditions: whether or not a response is made at time ti, whether or not a pulse is present at time ti, and whether or not a pulse, present at time ti, is detected prior to time t. Now, by definition, a response is made at time ti if a. = 1,

- 69and no response is made at t. if a = 0. Furthermore, a pulse is present at time ti if 0 e [0, T ), and no pulse is present at time t. if 0. = T. Thus it remains to find an expression for the condi1 1 p tion that a pulse present at time t. is detected prior to time t.. 1 1 To accomplish this, define the most recent response function.; i = 1,..., N as follows. Suppose that at time ti. c responses have been made at the times t < t <... < t < t.. Then r! r2 r 1 define the value of the most recent response function at time t. by t (t Tp) if tr e (ti p i) 0 if t E (ti- Tp, ti) The interpretation of this function is as follows. If no responses occur in the interval (t.- Tp, t.) then T. =. On the other hand, if at least one response occurs in this interval, then T. specifies the time of the most recent response as measured from t.- T. This 1 p case is shown in Fig. 2.10. The past responses are denoted by crosses. To apply the above definition, note that if a pulse present at time t. is detected prior to time ti, then a response must have occurred after that pulse has arrived (see Fig. 1.11). Thus, we must have

-70*^ ------- T7'. ---— T. 1 X —-- X —H+I —**- **-... I- - t.-T t t t. i p rl-1 Fig. 2.10. The definition of the most recent response function i —----— i t.-T a t t. i p r Fig. 2.11. A previously detected pulse t > a r or tr (ti-Tp > a- (ti- ) or finally

-71T. > 0i Conversely, the condition that a pulse present at time t. is not detected prior to time t. is equivalent to the condition, T. < 0. 1 - 1 With these observations at hand we may immediately write down the definitions of a false alarm, a correct rejection, a detected pulse and an extra detection in terms of a and 0 from the corresponding definitions in Section 1.3. A false alarm (F) occurs at time t. iff a = 1 and = T 1 1 p A correct rejection (C) occurs at time t. iff a.= 0 and. = T 1 1 p A detected pulse (D) occurs at time t. iff a. = 1, i0e[O,T ) and 7. < 0 A e d t i 1- i An extra detection (X) occurs at time t. iff a. = 1 0 i[0,Tp) and Ti> 0i i p 1 i~~~~~~~

-72To carry over the definitions of a missed pulse and a rest, note that the condition that time t. is the last decision opportunity to detect a pulse present at time t. is equivalent to the condition i e [0, A). (See Fig. 2.12.) Thus we must replace the statement t. = a + T 1 p in the definitions of Section 1. 3 by the statement i e [0, ). The resulting definitions for ti < t are as follows. A missed pulse (M) occurs at time t. iff a.=0,. =0 and ie [0, A) A rest (R) occurs at time t. iff a. =0 and 0i [A, T or a. = 0 and T > 0 and 0 [[0, A) 1 [, 1 t.-T t. 1p 1 Fig. 2.12. The condition that time ti is the last decision opportunity to detect a pulse present at time t. 1

-73The terminal time tN = T requires special consideration since if aN = 0, any pulse present at time tN that has not been previously detected will be missed. A missed pulse occurs at time tN iff aN=- TN< 0N 0Ne[,OTp) A correct rejection occurs at time tN iff aN= 0 N N ONE [0, T ) The above definitions of the six basic outcomes are illustrated in Fig. 2.13 below. The outcome associated with a particular decision time t. is shown beneath that time. Next it is shown that both the respond-once and the respondand-hold loss functions defined in Chapter I can be written in terms of a single canonical form. First, consider the respond-once loss function. According to Eq. 1.10, this function is given by L - (L - LM) ND + LX NX + LF NF where it is assumed that LD< LM, LX> and LF> 0 In the remainder of this report it will be convenient to normalize

-74s(t) J1 a2 3 (a) t0 (b) a CCCDRRCCCCFCCCDRXCCCCRRMCC (c) Fig. 2.13. An illustration of the six basic events (a) a typical pulse sequence, (b) the corresponding values of 0., and (c) a typical response vector and tlhe corresponding decision outcomes

-75this loss function by dividing through by LM- LD The resulting loss function can then be written as L WX NX + WF NF - ND (2.22a) where W = L ( - L >0 (2.22b) Wx = LX/(LMLD) (2.22b) and WF = LF/(LM- LD > 0 (2.22c) Next, consider the respond-and-hold loss function. From Eq. 1.7, T T T T L = (LD-LM) ND + (LD L) TD + (LF-LC) TF where T T T T LD < L <L and L < L D-M mL D M F C First, it is necessary to relate TD and TF to ND, NX and N. TD F D X F To do this, note from Fig. 2.14 that, if it is assumed that the pulses can arrive only at discrete times and that these times are also decision times, then T = ANF (2.23a) and

-76TD = (NX + ND) (2.23b) s(t, 0)'':" I I Ip a(t) ~ II C C R D X C F F C R D detection time false alarm detection = 2A time = 2A time = A Fig. 2.14. The relation between TD, TF and NX, ND and NF It might also be pointed out that if the pulses are allowed to arrive continuously, then Eq. 2.23 still holds approximately, providing A is small. In either case, we will substitute Eq. 2.23 into Eq. 2. 22a to write the loss function as

-77L [(LD- LM)+ A(L - LM)] ND + (LD LM) N T T + (LF - LC) NF Finally, this function is normalized by dividing through by LM- LD T T + a(LM LD). The resulting respond-and-hold loss function is L = WX" NX + WF" NF - ND (2.24a) where X = (L' LM)(LM LD) + A(L - L(2 24b) X D M M'/UM D M D>(2.24b) and WF = (LF- LC/LM- LD) + T (2. 24c) In the above paragraphs we have shown that both the respondonce and the respond-and-hold loss functions can be written in terms of the single canonical form L = WXNX +WFNF- ND (2.25) Thus, for the purpose of analysis we can work with the loss function of Eq. 2.25. When it is necessary to interpret this function in terms of a specific decision device, set WX and WF equal to WX' and WF' or to WX and WF" depending on whether the decision device r ~~ F

-78is a respond-once or a respond-and-hold decision device. It might also be pointed out here that we may distinguish between the two cases by whether or not WX > 0. That is, it is easily concluded from Eqs. 2.22 and 2.24 that the respond-once decision devices are associated with values of WX > 0 and the respond-and-hold decision devices are associated with values of WX satisfying -1 < W < 0 The special case WX = 0 is the maximum pulse detection (MPD) device and the special case W = -1 is the maximum detection time (MDT) device. To complete the definition of the loss function it is necessary to find expressions for NX, ND and NF in terms of a and 0 To this end, we appeal to the definitions of the extra detection outcome, the detected pulse outcome and the false alarm outcome. From these definitions it is clear that N NX(a,') = I (i' a, i) (2.26a) X i=l 1 1 1 {ai=1 i, i<Ti', i[0,Tp)} N ND(a,) = I (Ti, ai, i) (2.26b) {ail, 0i> i' 0 ei[0e Tp)}

- 79and N NF(,) -= I (ai, 0i) (2.26c) i=1 ^1 {ai=l, 0 i Tp where IB(0) denotes the indicator function for the set B. We may then substitute these expressions into Eq. 2.25 to obtain N L(a,O) = 3 Li(7',ai 0) (2.27a) i=1 where L(i, ai, ) = WF I (ai, 0i) + W I (T7, ai, i) {ai=l, 0i=Tp} {ai=1, 0i< Ti, 0i[O,Tp)} - I (T, ai, i) (2.27b) {a=l, 0. > T. 0 e[0 Tp)} This completes the derivation of the loss function for the FRD problem.

CHAPTER III THE PERFORMANCE OF FRD DECISION DEVICES In Chapter II we introduced the mathematical model for the FRD problem. In Section 2. 1 of that chapter the Bayes procedure was introduced as the means of obtaining optimum decision rules. This procedure is useful primarily because it has the advantage of being applicable in any decision problem where it is possible to characterize the losses and the prior knowledge. The Bayes risk r(5) can also be used to compare the quality of suboptimum rules. Specifically, if 6' and 6" are two decision rules, then 6' is said to be "better than" 6" if the Bayes risk r(6') is less than the Bayes risk r(6"). This means of comparing performance, however, has one major drawback. Namely, it loses meaning if the losses and the prior probabilities are not held constant. For example, in classical detection theory it is possible to find two decision rules 6' and 6", where 6' is the optimum Bayes rule for a particular set of losses and prior probabilities, 6" is suboptimum for a different set of losses and prior probabilities, but yet r(6") < r(6'). What is needed is a means of comparing the performance of different decision rules which is independent of the specific losses and prior knowledge. In classical detection theory, this is done in -80

-81terms of performance sets and ROC curves. In this chapter we will develop an analogous characterization for the FRD problem. 3.1 A Review of Performance in Classical Detection Theory In classical detection theory, two numbers PD and PF, the probability of detection and the probability of false alarm, are calculated for each decision rule. These numbers are plotted as a point (PF, PD) in the unit square. The locus of all points corresponding to all possible decision rules is known as the performance set. Through the notion of the randomized decision rule it can be shown that the performance set is convex. Moreover, it contains the points (0, 0) and (1,1), the former corresponding to the receiver that never responds and the latter corresponding to the receiver that always responds. Since this set is convex, it also contains the line segment connecting these points. This line segment is usually referred to as the chance diagonal. An additional property of the performance set is that it is symmetric about the point (1/2, 1/2). That is, if (PF, PD) is a point in the set corresponding to a particular decision rule, then there is a decision rule, known as the antipodal decision rule, whose performance plots as the point (1 - PF' 1 - PD). A typical performance sst is shown in Fig. 3.1. A criterion of performance can be defined in terms of the performance set as follows. Specifically, a decision rule corresponding

-82PD P ^ P3. F Fig. 3. 1. A typical performance set for the classical detection problem to the point (P, P' ) is said to be "better than" a decision rule coroteponFP' D responding to the point (P', Pj) if the point (Pb, PD) lies to the left and above the point (Pp, PP). That is, if Pp <_P and PD > PI It is quickly recognized that the'best" decision rules are those corresponding to points on the upper boundary of the performance set. This upper boundary is referred to as the optimum receiver operating characteristic (ROC). One of the main results of classical detection theory is that the ROC is actually contained in the performance set, so that, in fact, the best decision rules do exist. A more comprehensive discussion of these concepts can be found in Ref. 8. In classical detection theory, the Bayes procedure can be interpreted in terms of the performance set in the following manner.

-83Recall from Appendix A that the Bayes risk of a decision rule' can be written as r(6) = Pr[SN] (LD- LM) PD + Pr[N] (LF - LC) PF + (Pr[SN] LM + Pr[N] LC) or W D = P F r(6) - (WM WC/L)] (3.1) where WF (LF-LC)/(LM- LD) WM =LM/(LM- LD), WC = Lc/(LM- LD and L = Pr[SN]/Pr[N] In the unit square this equation plots as a line with slope WF/L0 and intercept r(5)- (LM + L/L c(6) r- - L c The Bayes decision rule 60 is then characterized by the point

-84(P0F PD) that supports the Bayes risk line of Eq. 3.1. This is illustrated in Fig. 3.2. PD Bayes Risk Line c( ~ ). PF 3. 2 Performance Sets for the FRD Decision Rules In the preceding section, it was seen that, in classical detection theory, the performance set is defined in terms of two probabilities, PD and PF. Since FRD decision devices also make detections, false:. PDF alarms and in addition extra detections, it might seem reasonable to define the performance set in terms of three probabilities, PD, PF and P. The ddifficulty here, however, is that these probabilities ano nd PX n he difficulty here, howeve is easily seen to be the now depend on the decision times t.. This is e asily seen to be the case by considering the decision device that does not respond at the

-85decision times ti, for i even. For this device, PD, PF and PX must be zero at these times, since a detection, a false alarm or an extra detection can occur only if the device responds. Thus, in general, we must associate sequences of probabilities to the FRD decision rules. The difficulty in dealing directly with sequences of probabilities can be avoided by dealing instead with the rates at which the pertinent events occur. This approach will be followed here. In this section, we first define the rates of the various events. Next, we investigate the relations that exist between these rates. Finally, we define two performance sets, one for R-O decision devices and one for R-H decision devices, in terms of certain "normalized rates. " To begin, we define the rate of the event 3, where 3 denotes one of the six events: false alarm, detection, extra detection, rest, correct rest, or miss. Recall from Section 2. 5 that N/(6(x), 0) denotes the number of times the event 3 occurs when the observation is x, the decision rule 6 is used, and the pulses are described by 0. Now, the rate of the event /, denoted by r((6), is the average number of times the event 3 occurs per unit time. Thus we have the definition E [N3(6(x),0)] r (6) = xO (3.2) /3 T

-86An alternate form for this definition can be obtained by substituting the expressions for N, (6(x), 0) given in Eq. 2. 26 in Section 2. 5 into Eq. 3. 2. Then, N E 2 C I (aj, 0j) x,J =1J rx = j {a.=1, 0 =T rr(6) T N r E I (aj 0I j=1 x, 0 [ {a l, 0.=T T or N Pr[a.=1, 0=T ] r,(6) E= -p (3.3a) jF =l T Similarly, N Pr[a =1, <; j e[0,T )] r (6) = i -- ] - ]- - — p- (3.3b) j=l T and N Pr[a.=1,T >0j; e[O,T )] rX(6) = V - (3.3c) j=1 T Thus, in general, ra(6) can be expressed as

-87P.f ~r:~(56~) = i^~ T(3.4) where P = Pr[event 3 occurs at time t.] j J It should be emphasized that P. is not the same as the corresponding P in classical detection theory. For example, PD is the probability that a detection occurs at time t. whereas PD is the probability that a detection occurs given that signal is present. Next, we investigate the relation that exists between the different rates. To this end, note from Fig. 2. 13 that at any decision time that is not covered by a pulse either a false alarm or a correct rejection occurs. Thus, if NN(0) is the number of decision times in the interval [0, T] that are not covered by a pulse, then NF( (x), ) + NC (6(x), ) = NN(O) (3.5) Furthermore, at any decision time that is covered by a pulse, either a detection, an extraneous detection, a rest or a miss occurs. Thus, if NS(0) is the number of decision times in the interval [0, T] that are covered by a pulse, then ND(6(x), ) + NX(6(x), ) + NR(5(x), ) + NM(6(x), ) = NS(0) (3.6)

-88Now, recall from Eq. 1. 5 that ND(6(x), ) + NM(6(x), ) = N (6) (3.7) If Eq. 3. 7 is substituted into Eq. 3. 6 the result is NX(6(x), ) + NR(6(x), ) = NS( ) - N (6) (3.8) Finally, divide Eqs. 3. 5 and 3. 6, and 3. 8 by T and take expected values to obtain rF + rC = rN (3.9a) + rR = rS - r (3.9b) rD + rM = r (3. 9c) The significance of Eq. 3. 9 is as follows. Each decision device 6 is completely characterized by the six rates rF(6 ), rc(6), rD(6), rM(6), rX(6), rR(6). Now if rp, rN and rS are fixed, then the right hand sides of the above equations are constant. Thus, the performance of each receiver can be completely specified by the three rates rD(6), rF(6) and rx(6). The dependency of Eq. 3.9 on the numbers r, rs, and rN can be removed in the following way. First, note from Appendix E that

-89Prl iO o, Tp)] r T (3. lOa) p Pr[ i = T ] rN m -— T- (3. 10b) rN =m T Pr[oi [0 T )] p rS = m = m r (3. lOc) Here Pr[Oi,e [O0T )] and Pr[Oi = T ] are the prior probabilities of the events "a pulse arrives in an interval of length T " and "no pulse arrives in an interval of length T ", respectively. Now make the following definitions: RD rD/rp RX = /rp R = r /r R = rR/r RF = rF/(rN/m) = rF/[Pr[T ]/Tp ] RC= rC/(rN/m) = rc/[Pr[0=Tp]/Tp (3.11) Finally, substitute Eq. 3. 11 into Eq. 3. 9. The result is RD + RM = 1 (3 12a) RX+ RR = m-1 (3.12b)

-90RF + RC = m (3.12c) These equations depend only on the number of decision opportunities per pulse, m. The quantities Ri in Eq. 3.11 will be referred to hereafter as the normalized rates of the events 3 or, if it is clear from the context, as simply the rate of the event f3. These normalized rates can be interpreted as follows. The rates rD, rx, rM and rR are rates of events that occur when a signal pulse is present. Thus, RD, R, RM and RR are the results of normalizing these rates by dividing through by r = Pr[O e [0,Tp)]/T. On the other hand, the rates rF and rC are rates of events that occur when no pulse is present. Thus, RF and RC are the results of normalizing rF and rC by dividing through by rN/m = Pr[i = T ]/T. N P lim r = lim-1 N N-oo N-oo i We now define the performance set for the respond-once decision devices. As in classical detection theory, we should like the performance set to characterize the performance of all of the possible R - O decision devices. From Eq. 3. 12 it is seen that this can be done by specifying for each device 6 the three rates, RX(6), RF(6 ), and RD(6 ). Thus, we shall make the following formal definition.

The performance set 5(m) is the subset of R3 defined by J'(m) = { (Y1, Y2,) e R3; there exists a decision rule 6 e5(m) with RX(6)=Y1, RF(6)=Y2 and RD(6) =Y3} We shall sometimes refer to 97(m) as the image of the set of decision rules 95(m) and to a point {R(6) = (RX(6 ), RF(6), RD(6)) as the image of the decision rule 6. The performance set Y(m) is most aptly suited to the respondonce decision devices, since, for these devices, the quantities of interest are RX, RF and RD. In dealing with the respond-and-hold decision devices, however, we should like to relate these quantities to the average detection time and the average false alarm time. This can be done as follows. Define the normalized detection duty DD and the normalized false alarm duty DF by E{TD/ T} D - E{T/ T} and

-92E{T /T} F = E{TN/T} Now recall from Eq. 2. 23 of Section 2. 5 that TF = NF and TD = (ND + NX) Also, note that by employing the same reasoning used to obtain Eq. 2.23, we may write T =AN S S and TN = A N N N Now, substitute for TD and TS into the equation for DD to obtain EA(ND + Nx)/T} D = D E{A NS/T} rD + rX =

-93rD + rX mr p or finally D = - (RD + R D m DRX) with the last equality following from Eq. 3. 9. Next, substitute for TF and TN into the equation for DF to obtain E{A NF/T} rF rN rF 1 or finally or finally RF D = F m Now the point to be made here is that if we have the performance point,

-94R(6) = (Rx(6), RF(6), RD(6)) then we can associate with this point another performance point D(6) = (DD(6), DF(6), RD(6)) by the transformation DD(6) = (RX(6) + RD(6) )/m (3.13) DF(6) = RF(6)/m RD(6) = %(6) Thus we can also define the performance set for the respond-andhold decision devices, J(m), either by the definition 9'(m) = {ye R3;there exists a 6 e!(m) with DD(6) = Y' DF(6) = Y2 and RD(6) = y3} or as the image of 9(m) under the transformation of Eq. 3. 13. In either case, we will hereafter refer to the performance sets 9(m) and (T(m) as the respond-once performance set and the respondand-hold performance set, respectively.

-953.3 Some General Properties of the Performance Sets In this section, some general properties of the performance sets 9(m) and c9m) are developed. First some simple bounds on these sets are derived. Next, a notion of randomized decision rules is introduced to insure that'(m) and (m) are convex. This convexity is then used to extend the notion of a chance diagonal to the FRD problem. Next the relationship between the performance of a given decision device 6 and the performance of the "inverse" antipodal decision device is determined. Finally, we verify the "most recent response" rule introduced in Section 1. 3. We first show that the set J(m) is contained in a triangular shaped wedge in the positive quadrant of JR3. That ~(m) lies in the positive quadrant is immediately evident from the fact that, by definition, r (6) is non-negative and, hence, RF(6) is also nonnegative. Furthermore, from Eq. 3.12 it follows that 5'(m) must lie in the rectangular volume. 0 RX < m-1 0 RF < m o0 RD < 1 The final result appears in the following theorem and its corollary.

-96Theorem 3. 1 The performance set 9m) is contained in the region defined by 0 < RF < m (3.14) 0 < RX (m-1) and Rx/(m-1) < RD < 1 Proof. From the above remarks, it is only necessary to show that RX m-1 RD We reason as follows. An extra detection cannot be covered by a pulse that is not detected. Furthermore, any pulse that is detected cannot cover more than m-1 extra detections. Thus, NX < (m-1) ND. Dividing by T, taking the expected value, and then normalizing yields the desired result. Corollary 3. 2 The performance set 7(m) is contained in the region. 0 < DD < 1 (3.15) 0 < D < 1 DD D RD I min[1, DD/m]

-97Proof. The result follows immediately by applying the linear transformation of Eq. 3.13 to the triangular wedge of Theorem 3. 1. The regions of Theorem 3.1 and its corollary are illustrated in Fig. 3.3. RD mII _____ —----- 1'1 =, >RF m-1 RX RD DF 1 _ DD Fig. 3.3. The region containing 92(m) and 9(m) Next we show that the performance sets are convex. In classical detection theory, this is accomplished by introducing the notion of the randomized decision rule. The same idea can be extended to the FRD problem. The physical interpretation is as follows. Suppose that 6' and 5" are two decision devices. Further, suppose that, before a decision device is to be put into operation, we

-98perform a random experiment with two outcomes, say "A" and "B"' with probabilities p and 1-p respectively. If the outcome "A" is observed, we use 6', and, if the outcome "B" is observed, we use 6". Then, the 3-rate for the randomized receiver 6 is E[N (6(x),) 6=6'] p + E[NP(6(x),O) 6=6"] (1-p) r3(6) = T = r(6') p + r(6") (l-p) Dividing through by the proper normalizing factor, yields R8(6) = R((6') p + R(6") (l-p) (3.16) From Eq. 3.16 we obtain the desired theorem and its corollary. Theorem 3.3 The performance set 5(m) is convex. Proof. It is recalled that a set S is convex, if, given any two points a, b e S, the line segment ap + b(1-p),p e [0,1] is in S. Now, we have shown that, given any two decision devices 6' and 6" with images R(6') = (Rx(6'), RF,'D), R()) and

-99R(8") -(R6"), RF(6"), RD(6") ) in Y(m), there exists a decision device 6 with image R(6') p + R(6") (l-p) in 9(m). Since this holds for all p e [0,1], it follows that 92(m) is convex. Corollary 3. 4 The performance set 9(m) is convex. Proof. The set.P(m) is obtained as the image of 9P(m) under the linear transformation of Eq. 3.13. Thus, the conclusion follows, since the image of a convex set under a linear transformation is convex. The convexity properly can be used to extend the notion of a chance diagonal to the FRD problem. The procedure is basically the same as that followed in classical detection theory. First, consider the decision device that never responds. Since a detection, a false alarm or an extraneous detection can occur only if the decision devices respond, we have rD = rF = rX = 0, and thus R = RF R =0. Hence, the point (0, 0, 0) (m). D F X Next consider the decision device that always responds. For this device, rC = rR = rM = 0, since correct rejections, rests and misses occur only when the device does not respond. Thus,

-100RC RR = RM = 0 and by Eq. 3.12 RX = m-l, RF = m and RD = 1. Thus, the point (m-1,m, 1) e Y(m). Finally, consider the decision device that responds every mth time. Its performance is determined as follows. First, note that since each pulse covers m successive decision times, each pulse must cover one and only one decision time at which a response is made. Thus, N = N and N = 0. It follows that RD= 1 D p X' and RX = 0. Now, rF is given by N Pr[a. = 1, 0. = T ] r = T 1 l p F. T i=l But, if i f m, a. = 0 so that Pr[ai = 1, 6. = T ] = 0. On the 1 1. 1 p other hand, if i = m, then a. = 1 so that Pr[a. = 1,. = T ]! 1 p= Pr[i. = T ]. It then follows that, (N/m) Pr[O = T ] rF NA Pr[8. = T ] 1 P T p =rN/m so that by Eq. 3.11 R =1 F

-101Thus, the receiver that responds every mth time has the performance point (0,1,1). The convexity theorem can now be applied to the performance points (0,0,0), (0,1,1) and (m-l,m,1) to obtain additional points in Y7(m). By Theorem 3.3 every point on the line segments connecting these points must be in (m). Moreover, by convexity, every point in the segment of the plane bounded by these lines and segments is in,T(m). (See Fig. 3.4.) T 1 i (0,1,1) / (m-, m,l) m RF m-1 / A/ RX Fig. 3.4 The randomization between the performance points (0,0,0), (0,1,1) and (m-l,m,1) The segment of the plane shown in Fig. 3. 4 can be interpreted as follows. First, note that every point on the line connecting the points (0,0,0) and (m-l, m, 1) can be obtained by

-102randomizing between a decision device that always responds and a decision device that never responds. Thus, this line has the usual property of a chance diagonal. Next from the remarks preceeding the statement of Theorem 3. 8 in Section 3. 4, it is clear that, among all decision devices that have images in the RX = 0 plane, the decision device with image (0, 1,1) responds as often as possible. Thus, the line connecting the points (0,0,0) and (0, 1,1) has the usual property of a chance diagonal. Finally, it is noted that any other point in the segment of the plane illustrated in Fig. 3. 4 can be obtained by randomizing between points on the lines. (0,0,0) p + (0,1,1) (l-p) pe [0,1] and (0,0,0) p + (m-l,m,1) () p) E [0,1] Thus, the segment of the plane shown in Fig. 3. 4 can be interpreted as a "chance plane" for the performance set 9(m). The equation of this plane is easily seen to be RX+ R - RF = 0 (3.17) A chance plane for the performance set Y(m) can be obtained by applying the transformation of Eq. 3. 13 to Eq. 3. 17. The resulting chance plane is then given by DD - DF = 0 (3.18) F

-103The segment of this plane that is contained in 92(m) is illustrated in Fig. 3. 5. RD (OOO~ I. _....DF /xA, / - -- DD Fig. 3. 5 The chance plane for (m) Next we investigate a certain relationship that exists between the points in 9(m) and 9G(m). To begin, it is recalled that, in classical detection theory, the antipodal decision rule 6a of a decision rule 6 is the decision rule that responds only when 6 does not respond, and conversely. This idea can be extended to the FRD problem as follows. For a given decision rule 6 = (6= 1,' )

-104define the antipodal decision rule, 6 a, by 6a (6a 6 "'. 6N) where 1 if 6 (xi, ai-) = a 1 6 i (i, ai-) = 0 if 6i (xi, ai1) =1 i 1 The interpretation here is that 6 a responds at those times that 6 does not respond and fails to respond at those times that 6 responds. The performance of the antipodal decision rule 6 ais related to the performance of 6 by the following theorem and its corollary. Theorem 3. 5 If 6 is the antipodal decision rule for 6, then R() = m - RF(6) (3.19a) R a) + RX(6a) = m - [R(6) + RX(6) ] (3.19b) Proof. Fix the decision device 6, the observation X and the pulse sequence 0, and let N (6(x), 0) be the number of reN sponses that occur at the decision times that are not covered by a pulse. Similarly, let N (6(x), 0) be the number of responses rS occurring at decision times that are covered by a pulse. Then it follows immediately that,

-105NT (6 (x),) N (6(x),) N and N (6(x), ) = ND(6(x), ) + NX(6(x), ) Now the antipodal device 6 responds at those times that 6 does not respond at and vice versa. Thus, N ( a(x),) NN() - N (6(x), ) N N and N (6 (x)) = NS() - N (6(x),0) rS s S Combining the above equations results in NF(6a(x),) = NN(O ) - NF(6(x), ) and ND(a(x), 0) + NX(6a(x),0) = NS(0) - (ND(6(x),0) + NX(6(x),0)) Dividing by T, taking expected values and normalizing yields Eq. 3.19. This completes the proof. Corollary 3. 6 If 6 is the antipodal decision rule for 6, then D,(6a) = 1 - DF(6)

-106DD() = 1 - DD(6) (3.20b) Proof. Eq. 3. 20 follows immediately from Eq. 3.19 by the transformation of Eq. 3.13. The above theorem and its corollary have an interesting geometrical interpretation. To see this, we first combine Eqs. 3. 19a and b to obtain RX( ) + R(6 ) - RF(6a) = - [RX(6) + RD(6) - RF(6)] (3.21) Next, it is noted that the plane defined by RX + R -RF =c is a plane parallel to the chance plane and located at a normal distance of c away from the chance plane. Thus it is evident from Eq. 3. 21 that the two points R(6 ) and R(6) are located on opposite sides of the chance plane and at equal distances from the chance plane. Precisely the same reasoning can be used to conclude that the two points D(6 ) and D(6) are on opposite sides of the chance plane for g(m) and at equal distances from that plane. This section is concluded by strengthening the most recent response rule. Recall that it was conjectured in Chapter I that it

-107is sufficient to consider only decision devices that depend upon the previous decisions only through the time of the most recent response in the interval (ti-Tp,t). This conjecture is now shown to be true in the following sense. Given any point in the performance set, 9(m), there is a decision device that obeys the most recent response rule and that has that point as its image. Formally, we have the following theorem. Theorem 3. 7 If (Y1, Y2 Y3) 9e (T(m), then there exists a 6 = (6..., 6N) e-i(m) such that Y1 = Rx(6), Y2 = Rx(6) and y3 = RD() and 6(xi' a. ) = 6i(X ri) where T. is the most recent response function of Section 2. 3. The proof of Theorem 3.7 is found in Appendix F. 3. 4 The Inhibit Rule and the RX = 0 Plane In Section 1. 4 we introduced the ROMF decision device. This device responds if the output of a matched filter exceeds a fixed threshold 3 and if no responses have been made at any of the previous m-1 decision opportunities. This last condition is imposed to insure that no extra detections result. We have called such a condition the "inhibit" rule. In this section, we investigate decision devices which obey the inhibit rule.

-108Before proceeding, the notion of the inhibit rule must be made more precise. In Appendix F it is shown that the component 6. of any decision rule 6 can be expressed in terms of a response set A.(xi-1, T)' Specifically, i 1 1 if x. e A.(xi- r) 6i(, (i ) = 0 if xi Ai(x, Ti) The interpretation of the response set A(xi_, Ti) is that for Xi- and Ti fixed, Ai(xi_, Ti) is the set of current observations, x. e X., for which the decision device responds at the decision time t.. Now, the condition T. > 0 is equivalent to the condition that at least one response has been made within the last m-1 decision times. Thus, if the probability of the set A.(xi, Ti) for x.i1 fixed and any pulse configuration 0i is greater than zero for T. > 0, then the decision device is allowed to violate the in1 hibit rule. Therefore, for a decision device to satisfy the inhibit rule,we must have T.)r 0.^-r ei] =0 ~ for T.> 0 Pr[A.i(x. X-1' i-1 ] for 71 > 0 This probability depends on all the past observations xi 1 and all the possible current pulse configurations 0. Mathematically speaking, it is only reasonable to lrequire that this probability be

-109zero except perhaps on subsets of X. x 0 of probability zero with respect to P(dxil, dO i). It is common practice in measure theory to indicate that a certain proposition holds, except perhaps on subsets of measure zero with respect to a measure p, by appending to the statement of that proposition the symbol [1L], which is read "almost everywhere with respect to A. " If we follow this convention here, then the formal definition of the inhibit rule is as follows. The decision rule 6 satisfies the inhibit rule iff for i = 1, 2,..., N Pr[A i( Ti) |Xi- 1.i] =0 for Ti > 0 [P(dx 1' dit)] (3. 22) The significance of this definition is easily seen from Eq. Fl. b in Appendix F. Specifically, if 6 satisfies the above definition then the probability of making an extra detection at time ti,Pi, is px m-1 =0 Z f f P(dxix _l, ) p(dx' it) P(di) 0 e[OT ) jjA>O Ie P i >X 1 Ai(Xi-1 IXTi=j^A) = 0

-110since f P(dxi ji- i ) = 0 [P(dxi_, doi)] 1 i-l' i ijA) Furthermore, from Eq. 3.4 it follows that RX = 0. Thus, the performance point for a decision device that satisfies the inhibit rule lies completely in the RX = 0 plane. The main result of this section is that, under relatively mild conditions on the family {P(dxl0); 0 e 0}, the converse statement holds. Namely, the only decision devices with images in the RX = 0 plane are those that obey the inhibit rule. Theorem 3. 8 Suppose for all i P(dx i i-'0) > P(dx Xixi-'0 =T.) [P(dxi.l'd.i] i- p i-1 Then, given any performance point in the RX = 0 plane of,(m), there exists a decision device that satisfies the inhibit rule and that has that point as its image. Proof. It is desired to show that given any performance point in J9(m) with RX(6) = 0, then the response sets for the decision device 6 satisfy

-111f P(dxixi1,&) = 0 for j = 1,..., mi i-1',i=jA)... (3.23) except perhaps on a set A CXi_1 x Oi with zero probability. Now suppose it can be shown that Eq. 3. 23 holds except perhaps on a set A CXi1 x[0,T ) with zero probability. Then, since, by p hypothesis, P(dx xi_l, i) >> P(dxi xl, iT ) [P(dx-i di )], we have P(dxi xi10i=Tp) = 0 for j = 1,..., m-1 Pp Ai(xi- 1' Ti=iA) Thus, it is sufficient to verify Eq. 3. 23 up to subsets of probability zero contained in X ix[O, T ). The proof is by contradiction. Suppose there exists some j e {1,..., m-l} and some set B CXi_x [0,T ) with S P(dxi,dOi) > 0 B such that f P(dx xi-1,'0) > 0 for (Xi-,6.) e B

-112Then it follows from Eq. F 1 b in Appendix F that RX(6) > P > J / S P(dxii X_,0i)P(dxi l0i )II(dei ) [O,T ) Xi- A(Xi-Ti=jA) Moreover, by the definition of A.(x_, = jA) in Appendix F, B C Xi x [O,Tp) so that f- j f P(dxil xl,i)] ( i) I (di [0o,T) T X A.iX i ti= A) S / / P(dXi|Xi-1^0i) P(dxi.,d i) Xi- x[O,Tp) Ai( i-1 j) > f f P(dx i Xli- 0 i) P(dxi-,d i) B Aii- (X T=Ja) > 0 Thus, RX > 0 which contradicts the assumption that 6 has its image in the RX = 0 plane. This completes the proof. As a corollary to the above theorem, we may strengthen Theorem 3. 5 for those decision devices that satisfy the inhibit rule.

-113Theorem 3. 9 Under the hypothesis of Theorem 3. 8,if 6 satisfies the inhibit rule and if m> 1, then the antipodal decision device, 6, has the image RF ) = m - RF(6) RF(6 ) (6a) = m-1 - RD(6) R(6) = 1 Proof. Since 6 satisfies the inhibit rule, the responses must occur no more frequently than every mth decision time. Therefore, the antipodal decision device cannot make no-response decisions any more frequently than every mth decision time. Hence, every pulse that occurs must be detected. Thus ND(6(x),) = Np(e) p so that RD(Sa) 1 Substituting RD(6) = 1 and R 6) = 0 into Eq. 3. 19 gives the desired result.

-1143. 5 Optimality and ROC Surfaces In this section we borrow the idea of the ROC curve from classical detection theory and apply this idea to describe the optimum performance of the FRD decision devices. The result is two ROC surfaces, one for respond-once decision devices and one for respondand-hold decision devices. The procedure used to define the ROC surfaces is in complete analogy with classical detection theory. We first define a concept of "better than" in terms of the performance sets and then we use this concept to define the ROC surface as representing the "best" possible performance. Consider first the respond-once case. The basic idea here is to say that one decision device is "better than" another if on the average it detects more pulses without making more false alarms or extra detections. More precisely, we will say that 6' is better than 6" iff RX(6 )< Rx (6,) RF(6) < RF(6") RD(6') > RD(6") To obtain a geometrical interpretation of this definition we define the

-115"inner quadrant at the point z" as the subset QZ C IR3 by z Qz = {y IR3; yl < z, Y2 < z2, Y3 > z3} Then, it follows from the above definition that the set of performance points that are better than the point R(6") = (RX(6"), RF(6"), RD(6")) is the set QR(6") n s Such a set is illustrated in Fig. 3. 6. RD R() D QRi n,(m) F 3 T s o e m e n b RX Fig. 3.6 The set of performance points better than R(6")

-116We will now define the respond-once ROC surface to consist of those points that represent the best possible performance. From the above figure it is clear that R(6") represents the best possible performance if QR(6") is just touching that portion of the upper surface that is closest to the RX = 0 plane. But, this occurs when the point R(6") is the only point in QR(6 n ". Thus we have the definition: The respond-once ROC surface X 9.(m) is the set, = {y e IR3; Qy n 9(m) = {y} } A typical respond-once ROC surface is shown in Fig. 3. 7. R R Fig. 3. 7. A typical respond-once ROC surface We treat the respond-and-hold case in the same way except now we deal with the performance set $r(m). The underlying idea here is to say that one decision device is "better than" another if it

-117detects more pulses and has more detection time but less false alarm time. That is, we will say that 6' is better than 6" if DD(6') > DD(6") DF(6') < DF(6" ) RD(6') > RD() The geometrical interpretation of this definition is now expressed in terms of the "outer quadrant at the point z"i. This is the set Q defined by QO = {y IR3; Y1 > Z1' Y2 < z2' Y3 > z3} It then follows that the set of performance points that are better than the point D(6") = (DD(6"), DF(6t), RD(6")) is the set QD(6") n An illustration appears in Fig. 3. 8.

-118RD Fig. 3. 8. The set of performance points better than D(6") D(6") D F DD Fig. 3. & The set of performance points better than D(6 ") The respond-and-hold ROC surface is the set of the best performance points in 19(m). In the same manner as before, this set is seen to consist of that portion of the upper surface of 29 that is farthest from the DD = 0 plane. The formal definition of this surface is as follows. The respond-and-hold ROC surface X is the set 3 Q0 X - ={ye IR3; Q n 9={y} } A typical respond-and-hold ROC surface is shown in Fig. 3. 9.

-119RD DF DD Fig. 3. 9 A typical respond-and-hold ROC surface As an application of the ideas of this section we will obtain a result on the effect of increasing the number of decision opportunities per pulse, m, on the performance. Specifically, it is shown that the performance gets better, or at least gets no worse, if m is doubled. This result follows from the fact that the ROC surfaces are contained in the upper boundaries of the performance sets and from the following theorem. Theorem 3. 10 J(m) C 9C(2m) (3.23a) 97(m) c 99(2m) (3.23b) Proof. Since the performance set I(m) can be obtained from

-120the performance set 9,(m) by the transformation of Eq. 3. 13 it is sufficient to prove only Eq. 3.23a. This is done as follows. It is desired to show that given any decision rule 6 e M(m) with performance point R(6) e 9(m), there exists a rule 6' e!(2m) with the same performance point. To show this, define 6' = (61 62', 6N' ) by 6 (,') i even s',,') = 0 i odd Now, note that if J is the set of decision times for decision rule in 5(2m), then F can be written as i= {ti = iA; i even} U {t. = iA; i odd} But, the set {ti = iA; i even} is precisely the set of decision times for decision rules in 2(m). (See Fig. 3. 10.) Since 6' responds iff 6 does for t. e {ti = iA; i even} and never responds for t. e{t.i = i; i odd}, it follows that the rates RX, RF, RD must be the same decision rules. This completes the proof.

-121II 1 I ---- 0 (a) I I I I I I I I t1 t2 t3 t4 t5 t6 t7 t8 (b) Fig. 3. 10. The relation between the decision time for decision devices in 9(2m) and!2(m) (a) the decision times associated with!Y(m) (b) the decision times associated with )(2m) 3. 6 Optimality and the Bayes Decision Rules In classical detection theory the performance of the Bayes decision rules is geometrically related to the ROC curve by the fact that the Bayes risk line intersects the ROC curve at the point of optimum performance. (See Fig. 3. 2.) This relation can be used both to show that the concepts of Bayes optimality and ROC optimality are equivalent and to relate the losses in the Bayes risk to the probability of detection and probability of false alarm. In this section, these ideas are extended to the FRD problem. We begin by establishing a relationship between the Bayes risk and the respond-once performance set J(m). To this end, it is first necessary to obtain an expression for the Bayes risk of a decision rule 6 in terms of the Ry(6), RF(6) and R%(6). This

-122is done as follows. In Section 2.1 we defined the Bayes risk as the average loss. In this section only, however, it will be more convenient to normalize this risk by dividing by the length of the total observation interval T. We then have r(6) E[L(6(x), 0)] T Substituting for the loss function L( *,) from Eq. 2.25 yields r(6) = WxNx(6(x), 0) + WFNF(6(x), 0) - ND(6(x), 0) T E { Nx(6(x), 0)} E{NF(6(x), 0)} E{ND(6(x), 0)} WX T F T T WXr (6) + WFrF(6) - rD(6) Next, substitute for rx(6), rF(6) and rD(6) from Eq. 3.11. The result is r = WXRRX() + WL RF(6) - R(6) (3.24) p where L = rp/(rN/m) = Pr[ie[OTp)]/Pr[0.iTp]

-123Now, note that Eq. 3. 24 defines the plane in (RX, RF, RD) space that is normal to the vector (Wx, WF/Lo, -1) and which has intercepts RX = r(6)/rp W RF = r(6)/r WFLo0 RD = -r(6)/rp The significance of this plane lies in the fact that it contains all performance points R(6) that result in the same Bayes risk. Moreover, if we denote the Bayes rule by 6, then as r(6) is allowed to approach its minimum value of r(6~), (the minimum Bayes risk), then this plane approaches the plane that intersects the upper surface of J'(m) at the point R(6~). Thus, in complete analogy to classical detection theory, the performance points for the Bayes decision rules may be characterized as those points for which the Bayes risk plane intersects the upper surface of _,(m). The geometrical relation here is shown in Fig. 3.11. We are now in a position to relate the respond-once Bayes decision rules (the case WY > 0, WeF, > 0) to the respond-once ROC

-124RD R /TRF Bayes Risk Plane / \ 0 Upper Surface of,9(m) / i-~ (6 Rx Fig. 3. 11. The Bayes risk plane for the respond-once case surface Ay. To this end, fix R = RF and consider a section of the upper surface of,(m). Since 27(m) is convex, this section appears as a continuous convex function fR (RX) = RD in the R *=R^~~J F RF = RF plane as shown in Fig. 3.12.? Bayes Risk Line with Slope WX RD i F RF [ RX R* F Fig. 3. 12. A section of the upper surface of 9(m) at RF = RF F F

-125Now, let C R be the smallest value of Rx for which F fR* (RX) takes on its maximum value. Specifically, C = inf; fR;(R) = sup fR* (RX) RF L F RX F Then the section of the respond-once ROC surface at RF = RF is given by R f * (R) for RX [OCR] D RF ~ F Next, note that the Bayes risk plane for the Bayes decision rule 6~ with RF(6o) = RF must intersect the section f* (RX) = RD as a line RD = WRx + [WFLO ) RF r(60)/rp] as shown in Fig. 3.12. Since the slope of this line is W, it is immediately clear that this line intersects the respond-once ROC curve iff WX > 0. Thus, all Bayes rules associated with WX > 0 have performance points lying on the respond-once ROC surface. Moreover, for any point on the respond-once ROC surface, there is a Bayes rule for some W > 0 with that performance point. Before turning to the respond-and-hold case, conjectures I, II and III of Section 1.4 are verified. This is done in the following two

-126theorems. The first of these theorems says that if the loss WX is increased past a certain value then there is only one Bayes rule and that rule is an inhibit rule. This is the content of conjectures I and II. Theorem 3.11. For each WF > 0 there exists a WF(WF) > 0 * such that, if 6* is a Bayes rule for the losses WX and WF, then (1) 6 is an inhibit rule and (2) if WX > WX(WF), then 6* is also a Bayes rule for the losses W and WF. Proof. Fix WF = W and define W (WF) to be that value of WX such that the Bayes risk plane is tangent to the upper surface of 9S at some point in the RX = 0 plane. Denote the Bayes rule associated with the losses WX(WF) and WF by 6. This construction is shown in Fig. 3.13 in which the Bayes risk plane appears as the line tangent to the section fRF(6*)(RX) at the point RX =. Now, by construction, RX(6*) = 0 so that, from Theorem 3. 8, 6 * must be an inhibit rule. It remains to show that 6* is also a Bayes rule for the losses ** WX > Wx(WF) and WF. To accomplish this, denote the Bayes risk of an arbitrary rule 6 for the losses WX and WF by r(6) * * * and the Bayes risk of 6 for the losses Wx(WF) and WF by r(6). Then,

-127Bayes Risk Line with Slope W*(W*) R D x F RF(6*)(RX) = RD Rx Fig. 3.13. The Bayes risk line tangent to fR (6*)(WX) at the point RX = r(6) = WXRX(6) + WF L1 RF(6) - RD(6) > W(WF) R(6) + WF L RF(6) - R( 6) But, by the definition of a Bayes rule, r(6*) < r(6) Thus r(6*) < r(5) The proof is then completed by noting that, since Rx(56*) = 0,

-128so that 6* satisfies r(6*) = inf r(6) 5 The next theorem says that the smaller the value of WX the larger the value of RD for a fixed value of RF. This is the content of conjecture III. Theorem 3.12. If 6' and 6" are Bayes rules for the losses WX and W' respectively and if RF(6') = RF(6") = RF, then < < W < W X X implies RD(6') > RD(6") In lieu of a formal proof of this theorem we will refer to Fig. 3. 14. Here it is seen that, since the slope W' of the Bayes risk line for 6' is less than the slope WX of the Bayes risk line for 6", RD(6") must be greater than or equal to RD(6"). Moreover, the maximum value of RD for RF = RF will occur for the Bayes rule with WX =. The only comment that might be made about the above geometrical argument is that it is possible that the section fR*(RX) is a F

-129Bayes Risk Line (Slope WX') RA / Bayes Risk Line (Slope = WX') RD(6') fR* = RD RD(6") F C RX R* F Fig. 3.14. The intersection of the two Bayes risk lines with the section fR * (RX) F decreasing function of RX for RX > 0. In this case, Theorems 3. 11 and 3. 12 together say that there is only one Bayes rule with * RF = RF and that this rule is an inhibit rule corresponding to the loss WX = WX(WF) = 0. We might point out here, however, that at least in the actual computations described in Chapter 5 this was not the case. We now turn to the respond-and-hold case. First, we establish the geometrical relation between the Bayes risk and the respondand-hold performance set 99(m). To do this, substitute for RX(6) and RF(6) from Eq. 3.13 into Eq. 3. 24 to get an alternate expression for the Bayes risk r(6). Namely,

-130r() (WXm) D(6) + (WLO m) DF(6) - (1 + W) RD(6) (3.25) P As above, this equation also defines a plane except now in terms of the variables DD, DF and RF. The normal vector is (WXm, WF L m, - (1 +W )) and the intercepts are DD = r(6)/rp Wx m DF= r(6)/r W L m RD = -r(6)/r (1 + W) The geometrical interpretation of the Bayes risk plane and the Bayes decision rule 6 is the same as in the respond-once case. This relation is illustrated in Fig. 3.15. RD i DF ~_ Upper Surface of S9 D(a0 Bayes Risk Plane DD Fig. 3. 15. The Bayes risk plane for the respond-and-hold case

-131To relate the respond-and-hold Bayes decision rules (the case -1 WX < 0, WF > 0) to the respond-and-hold ROC surface A, we proceed as before. First, denote the section of the upper surface of, at DF DF by gF*(DD)= R. Then, define C * to be DF DF the largest value of DD for which gD* (DD) = RD takes on its maxiF mum value. Specifically C sup (D; gD (Di) sup gD(DD) F F DD F Finally, note that the Bayes risk plane intersects gD* (DD) as the line -1 RD -1 DD [ x D* r(6) "D 1 + W ~1 +W DF r (1+WX ) These ideas are illustrated in Fig. 3.16. Then proceeding as before, note that the portion of the section g * (DD) that is contained in the F respond-and-hold ROC surface is given by R = g *(D ) for DD > C D F F Moreover, as WX ranges from -1 to 0, the slope of the Bayes risk line ranges from -cc to 0. Thus, as before, it follows from Fig. 3. 16 that all respond-and-hold Bayes rules have performance

-132RD mWx Bayes Risk Line with Slope 1 WX 1 + Wy gD*(DD)...... [..~ DD CD* Fig. 3. 16. The relation of the Bayes risk line with slope WXm/(l+WX ) to the section gD*(DD) F points on the respond-and-hold ROC surface and, for any point on A,' there is a respond-and-hold Bayes rule with that performance point. Next, we verify conjecture V of Section 1.4. This conjecture says that as Wx increases, the number of detections increases but the detection time decreases. This is stated formally by the following theorem. Theorem 3.13. If 6' and 6" are two Bayes rules corresponding to the losses W~ and W? respectively and such that DF(6') * DF(6") = DF, then

-133WX > W implies RD(6') > RD(6") and DD(6') < D(6") As with Theorem 3.12, we will reason from the geometrical relation between the section gD* (DD) and the Bayes risk line. F This relation is shown in Fig. 3.17 below. R ^Wx m RDl Slope = 1"+ Wx RD(6') RD(6) Slope = i~xm _ —I I —.. —DD DD(6t) DD(6") D Fig. 3. 17. The intersection of the Bayes risk lines with the section g *(DD) F

-134Here, it is seen that, since 0 > W I> WX > -1 implies that the slopes of the Bayes risk lines satisfy WIm Wtm 0 > X > W - c o + I +w 1- -WI? x X then the intersection point corresponding to W' must be to the right of the intersection point corresponding to W. Thus, it is clear from Fig. 3.17 that RD(6') > RD(6") and DD(6 ) < DD(6") Moreover, it is to be noted that the maximum value of DD and the minimum value of RD for DF = DF will occur for W = -1 since then the slope is -cc. This section is concluded by summarizing the results of the above theorems. The main ideas of these theorems are contained in Fig. 3.18. In the diagram of Fig. 3. 18 the region Wx > -1 WF > has been divided into five special cases. The first three cases

-135Wx RespondOnce (I) Inhibit Rule Holds Case W (W) (II) Inhibit Rule Does Not Hold..'~ o -- (III) MPD Respond- (IV) and-HoldV) Case iL -1 I (V) MDT Fig. 3.18. The relation of the losses WX, WF to the behavior of the Bayes decision rules correspond to the general respond-once decision device and the last two correspond to the general respond-and-hold decision devices. The important characteristics of these special cases are as follows. Case I WX > WX(WF) WF > 0 The Bayes decision devices associated with losses in this region satisfy the inhibit rule and, as a consequence, they depend only on the value of W,. Their performance is characterized by

-136the fact that they attain the lowest values of RD and RX for a fixed value of RF. Case II 0 < W < Wx(W ) W > 0 The Bayes decision devices associated with these losses do not satisfy the inhibit rule and therefore they depend on both the losses WX and WF. Their performance is characterized by the fact that an increase in RD can be obtained at the expense of an increase in RX for constant RF. This takes place for decreasing values of W Case III WX =0 WF > 0 This set of losses defines the MPD decision devices. These decision devices also do not satisfy the inhibit rule and their structure will depend on both WX and WF. The singular importance of these devices lies in the fact that they achieve the highest possible RD for a fixed RF. It might also be noted that on the basis of the numerical calculations of Chapter 5, the structure of these decision devices is unique among the different cases. Case IV -1 < WX < 0 WF > 0 X F The Bayes decision devices associated with these losses are respondand-hold devices. Their performance is characterized by the fact

-137that increases in RD are obtained at the expense of decreases in DD for fixed values of RF. These increases occur with increasing values of W. Case V W = -1 W > 0 X F The Bayes decision devices defined by these losses are the MDT devices. Their special significance lies in the fact that these devices achieve the maximum DD for a fixed DF. In the next chapter, it will be shown that these devices also have a unique structure.

CHAPTER IV THE BAYES DECISION RULES FOR THE FRD PROBLEM In Chapter II the FRD problem is cast in terms of the general decision model. In this chapter, we use this model to develop a recursive procedure for obtaining the Bayes decision rules. Then, the resulting equations are used to establish a general model for the Bayes decision device. Next, we investigate two special cases for which the recursive procedure is easily implemented. Finally, the additional assumption of conditionally independent observations is applied in order to simplify the general decision device model. 4.1 The Bayes Decision Rule In Section 2.1, a Bayes decision rule was defined as any decision rule 6~ satisfying r(6~) = inf r(6) where r(6) = E [L(6(x), 0)] x, 0 The philosophy for minimizing r(6) to obtain 6~ = (6 1 6. 1' 2'' bN~) is well known. Specifically, r(6) is minimized over the components 6k one at a time starting with 6N and proceeding through -138

-139decreasing k. The result is a recursive scheme for obtaining 6k~ in terms of 6k+. The formal derivation of this procedure, however, is mechanical in nature and as such it sheds little light on the character of the Bayes rule itself. For this reason, we will content ourselves with a heuristic argument here, leaving the formal derivation to Appendix E. As a preliminary, we will investigate in further detail the most recent response function Tk of Section 2.5. It will be shown that k can be written as a function of Tk-1 and the preceding decision ak_. Specifically,!T -A ak-1 ~P ~ak-1 = k = (4.1) nax [ 0, k-1-A] ak- = We proceed as follows. Recall from Section 2.5 that the most recent responses function Tk is given by t - (t- T ) if the most recent response occurs at time tr e(tk-Tp, tk) Tk = 0 if no response occurs in t- Tp, Suppose first that ak1 = 1, so that a response occurred at time tk-l Then, Tk =tkl - (tk-T) = T -A. This verifies Eq. 4.1 P- 1kkp

-140when a k- = 1. Next, suppose that akL = 0. There are two cases,'k < A and'- > A. If rk-1 < A, then it follows that no rek-I - k-i k-I - ) sponses occurred in the interval (tk-Tp, tk ). Thus, in this case, Tk = 0. On the other hand, if Tk > A, then from Fig. 4.1 it is seen that the most recent response occurred at time t e (tk-Tp, tk_ ) and that k k- k-1 T tk -T L-T tr tk-I tk k- p p r k- 1k Fig. 4.1. The relation of Ti. to T. when a. = 0 1 1 and T. > A Thus, for either of the two cases associated with the condition ak- = 0, Tk can be written as Tk = max[0, k 1-A] This completes the verification of Eq. 4.1.

-141The computational procedure for determining the Bayes decision rule will now be developed. To begin, we fix the decision time tk e' and consider the dependency of the total loss L(a, ) on the decision ak. First, recall from Eq. 2.27 that the total loss is given by N L(a, 0) = Lk(Tk ak, k) k=l where Lk(Tk, ak, Ok) represents the loss arising from the current decision ak when the most recent response is given by Tk and the pulse configuration in the interval [tk-Tp, tk) is given by 0k. Clearly, the total loss L(a, 8) depends on the decision ak through the "immediate" loss Lk(Tk, ak, k). But now consider the situation at time tk+1. At this time, a decision ak+i is made and the loss Lk+l (k+l, ak+l, 0k+1) is incurred. But, at time tk+, the value of the most recent response function Tk+l depends on whether or not a response was made at time tk according to T -PA if ak =1 1k+ T(a k)Ok) = max[O, Tk-A] if ak 0 Thus, the loss at time tk+, Lk+(Tk+l, a+l, k+l) depends on ak The above reasoning can be extended to any future decision time tk+ j=l,..., N-k, by noting that Tk+j depends on all of the k"'J k'*j

-142preceding decisions and hence on the decision ak. Thus, we may conclude that the decision ak affects the total loss L(a, 0) through each of the functions Lk+j(k+j, ak+j 0k+j) j =0,1,..., N-k (4.2) As a concrete example of this dependency, consider the situation shown in Fig. 4.2. +0k Tp T-p tk tk tk-Tp tk+l-Tp tk tk+l1 Fig. 4.2. An example of the effect of the decision at time tk on the decision at time tk+1 Here we have a pulse that is present at both the times tk and tk+1. As shown, Tk = 2A which is less than 0k so that at time tk the pulse has not been detected. First, suppose that the decision device responds at tk. Then a detection results, so that

-143Lk(k = 2, ak =, 0k> k) = LD Then at time tk+, Tk+ = T -A which is greater than k so that the loss at time tk+ is LX ak+l =1 Lk+(k+l k T' - ak+' k+l< k+l lo ak+1 - 0 On the other hand, if the decision device does not respond at time tk, then ak = 0, and Lk( k 2A, ak Ok > zk) = 0 But, Tk+1 = max[0, Zk-A] = A which is less than 0k+1 so that the loss at time tk is given by D if ak+l1 Lk+l(Tk+1 = ak+1, k+ > Tk+1) = L0 if ak+I = 0 The next step in the derivation of the Bayes rule 6~ is to determine an expression for the posterior risk associated with the decision ak. This risk, to be denoted hereafter by Rk(xk, Tk' ak), is the posterior average of the loss due to the decision ak. We can immediately write down the contribution to Rk(xk, Tk, ak) due to the immediate loss L(-,',,). Specifically,

-144E Lk(Tk, ak,'k) Now, at time tk+l, the posterior risk associated with the loss Lk+l(.,,) is E Lk+1 (rk+l, ak+1, k+l) k+1 lxk+1 But, at time tk only the observation up to time tk is known. Thus, we must average this term with respect to the "unobserved" observation Xk+l, conditioned on the observed value xk. The resulting contribution to Rk(Tk, ak) is E 0 E - {Lk+ (k+l, ak+l, 0 k+l) Ek+ 1 xkkk+llxk+l Similarly, the contribution to the posterior risk of the decision ak due to the loss at time tk+, j = 1,..., N-k, is E E {L- k+i(Tk+j, ak+j', k+j)} xk+l''*'xk+jlxk {kk+jlXk+j Adding these contributions gives, for the total posterior risk of the decision ak,

-145Rk(, TkT ak) (4.3) {Lk(k, ak' k)} OkIxk N-k = ~ E {L k+j(Tk+j ak+j 0 k+j). k< N 6 = 0(61'. 0..' N ) kN we apply the well known principle that 6 is the function which minimizes the posterior risk of the decision ak. In the problem at hand, ak takes on one of two values, ak = 1, denoting a response at time tk, and ak = 0, denoting a no-response decision at time tk. Thus, the posterior risk of the decision ak is minimized by the decision device that responds at time tk iff the posterior risk of responding, Rk(xk, Tk, ak=l), is less than or equal to the posterior risk of not responding, Rk(k, Tk, ak=0) Stated formally, k= if Rk(xk, Tk, ak- ) < Rk(xk, k' k' 0) 6k(k, o'Xk) = (4 4) otherwise

-146It is to be noted in connection with Eqs. 4.3 and 4.4 that since Rk(Xk' Tk ak) depends on the decisions akj= 6j( k+ xk) j =1... N-k the complete Bayes rule 6 = (61, 6N ) must be obtained recursively by solving Eq. 4.4 for 6N then 6N and so on An alternative expression for the Bayes decision rule can be obtained in the following way. For each 7ke {., m-1 and each Xk 1 e Xk 1 define the set of current observation Ak(xV 1' Tk) CXk by Ak(Xk-1' k) = {Xk;Tk(k' k) - } (4. 5) where Tk(x Tk) = R(xk,' k' ak=O) Rk(xkk' ak ) (4.6) The set Ak(xk1_, Tk) is the response set for k(x ) 0(see Section Tk(tck, Tk) 3. 3), and the function Tk(xk, Tk) is the net posterior risk of not responding at time t. In terms of Ak(', ) and Tk(*, ), we can write 60 either in the form 1 Xk Ak(Xk_l, 7k) 6k(Xk', k) = (4.7a) X,~ x,Ak(Xk 1, Tk)

-147or, in the form o-0r Tk(xk' Tk) > O k(xk, Tk) = (4.7b) o Tk(x, Tk) < o It is to be noted from Eq. 4.7b that the function Tk(xk, Tk) plays the same role as the test function of Section 1.4. Our next task is to obtain a block diagram for the Bayes decision rule. To this end, we appeal to Eqs. 4.7b, 4.6, 4.3 and 4.1 and conclude that the configuration of the general Bayes decision device is as illustrated in Fig. 4.3. The operation of this device is as follows. At time tk_, the decision device has observed the observation xk1 and made some decision ak-1. The observation Xk_1 is then stored in memory so that when the current observation Xk is observed, the decision device will have available the total past observation x = (,xk)- Simultaneously, the decision ak-i is used to calculate Tk according to T =maX[O, Tk-l-A] ak1 =0 The value of Tk is then stored in a memory so that it too is available at time tk. Finally, at time tk, the decision device calculates Tk,(xk, k) according to Eqs. 4.6 and 4.3 and responds iff

Most Recent Response Memory and Updater k+1 T -AT ] akl =1 P k- I max[0, T^-A] a^ =0 Delay Past Observation Memory Delay ak xk Tk (x, Tk) a 1 T >0n k Rk(xk TkCO)- kk k, 71) 0 T <0 k Fig. 4. 3. The general Bayes decision device

-149Tk(X' k) > 0 Theoretically, the general Bayes decision device is completely determined by the block diagram of Fig. 4.3. In practice, however, this device is difficult to implement for at least two reasons. First, since the past observation memory must store the total past observation from time t = 0 up to the current time t = tk, its capacity must grow linearly in time. Thus, it is not practical to think of allowing this decision device to operate indefinitely. The second difficulty in implementing the general Bayes decision device arises in the calculation of Tk(k, k). To see where this problem lies, replace the expected value operators in Eq. 4.3 by integrals with respect to the appropriate probability measures and note that k+~1 k+l'ak+' k+( k' ak)) r+l(xk+l' 1, Tk+1l(k'a)) for xk+l Ak+1(, Tk+ l. Xk+lk+l', k+(<ak)) for xk+le OAk(x, 6 where X At ) \ <t = 1 tX kl\Ak+l((k' k+l) tAkl (x-& % +1) AklX (XkT) t=-O

-150The resulting expression for Rk+1(.,', ) is Rk(k, 7k' ak) (4.8) f Lk( k, akk) II(d kl k < + ft Rkl+l1 ak+l=t k+1(ak Tk)) P(dxk+ lxk) k+l (Xk, Tk+l) k< N L(TN,aN,N) II(dNOXN) k= N N) NN ON Now, from Eq. 4.8 it is seen that in order to calculate Rk6(, 7k, ak) and hence Tk(Xk, 7k, ak), it is necessary to have available II(dOklk) and P(dxk+1 I.). But, in the general case, these probability measures can be obtained from the known probability measures {P(dxk| 0); 0 e E0 and II(d ) only by integrating over the whole space RxO. It is certainly not practical to carry out these integrations at each decision time tk. Fortunately, both of the above difficulties can be eliminated if we place certain restrictions on the observation probability laws. The result is that the probability measures relevant to the calculation of Tk(.k, Tk) can be calculated from the probability measures relevant to the calculation of Tk_ (k-' Tk 1) along with the current observation xk. When this is the case, the computational procedure

-151is often referred to as an updating procedure. In Sections 4.3 and 4.4 we will investigate such a procedure. In the next section, two special cases of the general Bayes decision device are examined. 4.2 Two Special Cases: The Maximum Detection Time Decision Device and the m = 1 Decision Device In Section 4.1 it was seen that at time tk the general Bayes decision device must calculate Tk(xk, Tk) according to Eqs. 4.6 and 4.8. But, these equations involve the optimum component decision rules at time tk+l through the response set Ak+1(''). Thus, to get an explicit expression for Tk(xk, 7k), one must first solve for TN(XN, TN) and then proceed recursively to solve for TN_1(xN_1 TN_1),.., Tk+l(xk+l, 7k+1), Tk(X, k). There are two special cases, however, where the solution of this recursive procedure can be obtained easily. In this section, we treat these cases. Case 1. The Maximum Detection Time (MDT) decision device. The MDT decision device is a respond-and-hold decision device that is characterized by a loss function that depends only on the amount of detection time and false alarm time. In Chapter III it was seen that among all R-H decision devices, this device maximizes the detection duty DD for a fixed false alarm duty DF. Thus, the structure of the MDT decision device is of particular interest. This structure is given explicitly in the following theorem.

-152Theorem 4.1. The test function for the MDT decision device is given by Tk(k, k) = Tk() (4.9) = n(d0lxk)- WF S n(dOkk) 0 kE[OTp) 0k=T k k P'pk p Proof. We will show that Eq. 4.9 holds by also showing that Rk(j, k, ak) E Lk(k' ak' k)+fk(x) (4.10) klxk where fk(xk) does not depend on Tk. The proof is by induction on the index k. To begin, substitute for the loss function from Eq. 2.27 of Section 2. 5 to conclude that E Lk(k'ak, k) ~ klxk 0 k 0 akO < WxS II(dOkl|) - f n(dOk|k) {Ok< Tk, Oke[O,Tp)} {0k> Tk' Oke[O,Tp)} +WFf n(dOkk) ak=1 {Ok=Tp } Next, combine the first two integrals on the right hand side of the

-153above expression into a single integral by using the facts that W = -1 for the MDT loss function and that {Ok> rk' Oke[OTp)} U {k< Tk Oke[, Tp)} {0k [0,Tp)} The result is E Lk(Tk, ak, k) (4.11) klxk 0 "k'0 0 ak=0 - a nI(dkXk ) - WF S I(dk ak 0ke[0, T) {0k =T } Now, for k = N we note from Eq. 4.3 that Eq. 4.10 is satisfied with fN(N) = 0. Moreover, TN(XN' TN) = RN(X TN aN=O) - RN(N' N' aN=) = f l(doN x N) - WF f I(doN IN) {0NE [0, T)} 0 =T so that Eq. 4.9 holds for k = N. Now, suppose that Eq. 4.9 and Eq. 4.10 are satisfied for indices k+1,..., N. Then, Tk+ ( k+T ) does not depend on

-154Tk+l Thus, from Eq. 4.5, Ak+l(xk' Tk+) = {Xk+l; Tk+(i,' Tk+l) > 0) {Xk+l;Tk+l(Xk) > O} or Ak+l(xk, k+l) Ak+l(k) (4.12) Therefore, the response set Ak+l does not depend on Tk+l. Next, substitute Eq. 4.11 into Eq. 4.10 to obtain Rk+l +l' k+' ak+l +l) pfk+ l('r+1 ak+)l = - I(dok+l Xk+l)- wF S II(dklxk) + fk+l (k+l) ak+l1 ok+1 e [ Tp) k+1 =Tp (4.13) Then, substitute for Rk+1 from Eq. 4.13 and for Ak+l from Eq. 4.12 into Eq. 4.8. The result is

-155kk' k' ak) k = S Lk(Tk, ak, ~k) II(dekx) 0k+1 (Xk) k k + f I H(dok+ Xk+l)- WF S II(dOk+1lXk+l ++l(k+ 1+!Ak+ (k) a k+lc[0,1T ) k+-T k+xk) k+1 p k+1 p1k X P(dxk+llxk) or Rk(xk Tk, ak) E Lk(k, ak, 0k) + fk(k) Oklxk where we have set the bracketed term equal to fk(_k), since it depends only on xk. Thus we have established Eq. 4.10 for the index k. The proof is completed by substituting for E Lk( k, ak,' k) from Eq. 4.11 into Eq. 4.10 and then substituting into Eq. 4.6. The result is Eq. 4.9 as was to be shown.

-156On the basis of the above theorem we can simplify the block diagram for the MDT decision device to that shown in Fig. 4.4. Delay Xk-I ak xk I() I > ak II(dO kl) - WF fn(dOktxk) k [O, Tp) =T 0 p k p k Fig. 4.4. The MDT decision device Several properties of the MDT decision device might be pointed out here. First, it is no longer necessary to provide a decision feedback loop to calculate Tk, since 7k does not appear in the test function Tk. Thus, we have verified Conjecture IV of Section 1.4. Secondly, it is noted that the test function Tk no longer depends on the distribution P(dxk+l X). Thus the only distribution that is involved in the decision at time tk is the posterior distribution I(dO k k). The single most important property of the MDT decision device, however, is contained in the following corollary to Theorem 4.1.

-157Corollary 4.2. The MDT decision device may be considered as the optimum detector, in the sense of classical detection theory, for detecting the arrival of a pulse within the last T seconds. Proof. In Appendix A it is seen that, in general, the optimum detector responds iff f n(dO Ix) > WF S n(dO Ix) OSN 3N where OSN are those parameters to be associated with a response decision and ON are those parameters to be associated with a noresponse decision. Now, from Section 2.3 it is clear that if we wish the optimum detector to respond at time tk iff a pulse has arrived in the last T seconds, then we must take = x=(..., k) x [0, T ) 1 X Xk-1 X[O, Tp) and oN = { =(o,..., k);OkT } N { (O1'' k); k = T} = x... Ox x{T } Also, in the notation of Section 2.4, X = Xk

-158Thus, we can write the response condition for the optimum detector as /n(dk,.,d0xk) WF nI(dk,...,dolxk) Elx...xo k_ [0, T) 8x...xek- {Tp} or, equivalently, as f n(dok >) > WF S n(dOkI|) [O,Tp) {Tp} But, this is precisely the same response condition as the response condition Tk(xk) > 0 where Tk(k) is given in Eq. 4.2. Thus, the optimum detector from classical detection theory is equivalent to the MDT decision device. This completes the proof. In the above paragraphs we have seen that, because of the special nature of the MDT loss function, the structure of the decision device is considerably simplified. This also turns out to be the case for the Bayes decision device that makes decisions at intervals of T seconds. P

-159m = 1 Case The quantity m is the number of decision opportunities per pulse. In Section 3. 5 it was seen that, in general, the performance of FRD decision devices increases with increasing m. Thus, the case m = 1 represents a lower bound on the performance of the Bayes decision devices. The structure of the m = 1 Bayes decision device is given in the following theorem. Theorem 4.3. The test function for the m = 1 Bayes decision device is given by Eq. 4.9. Proof. For m = 1, A = T, so that the value of the most recent response function Tk is given by T -A = T-T = k- =1 P P Pkl1 max[O, Tk_-A] = max[O, k_-T] = 0 ak- =0 Now, substitute Tk = 0 into Eq. 2.27 for the loss function to conclude that

-160E Lk(k, ak''k) d0k ak-O - II(dOk X - WF f I(dOk ) ak= [0k [0, Tp) k=Tp But, this is precisely the same expression as Eq. 4.11 in the proof to Theorem 4.1. Thus the conclusion to Theorem 4. 3 follows by the same inductive argument. The remarks following Theorem 4.1 also apply to the m = 1 case. The block diagram of Fig. 4.4 also describes the structure of the m = 1 decision device and the interpretation of this device in terms of classical detection theory is also appropriate. It is emphasized, however, that Theorem 4.1 applies to a specific loss function (the MDT loss function) and for any m > 1, whereas Theorem 4.3 applies to any FRD loss function so long as m = 1. 4.3 The Updating Equation for Conditionally Independent Observations In Section 4.1 we discussed the desirability of obtaining an updating procedure for the general Bayes decision device. In this section the basic updating equation is derived. In the next section we will use this equation to simplify the general Bayes decision device.

-161Generally speaking, updating procedures do not exist without certain restrictions on the prior probability measure II(dO) and the observation measures {P(dxl 0); 0 e }. One set of restrictions that yields tractable results requires that II(d ) be Markov and that {P(dx 0); 0 e O } be conditionally independent. Now, in Chapter II it was seen that II(d ) satisfies the Markov property, II(dok ok_1 ***. 01) = I(dOk 0k-1) Thus, we only need the assumption of conditionally independent observations to get a tractable updating procedure. In the signal plus noise setting of Section 2.4, the assumption of conditionally independent observations is really an assumption on the independence of the noise in different observation intervals. This is seen as follows. Recall from Section 2.4 that the observation xk depends only on the noise in the interval [tkl, tk) and on the signal parameters 0k and 0k_. Now, suppose that the noise in the interval [tkl, tk) can be considered as being independent of the noise in the interval [tk 2, tk- ). Then, it follows that for a fixed signal parameter 0, the observations xk and xk_l are independent. That is, P(dxk l'dXk0) = P(dxkl0)P(dxk[0) P (dx-i/ (k-l'B- 2)P(dxk18 k, Bk

-162If this assumption holds for all of the intervals [tk_, tk) then the observations are said to be conditionally independent. Formally, we have the following definition. The observations are conditionally independent iff for each OeO, and k =1,..., N, k P(dx,...,dxkO) n P(dxj O, ) (4.14) j =1 We now state three theorems which together provide the theoretical basis for the updating equation. Theorem 4.4. If II(dO) is Markov and the observations are conditionally independent, then P(dxk(0ke l) (dOk) f P(d n(dOk k-1) (dk-l xk-l) P(dxk xk-l) k-I (4.15) The proof of this theorem is interesting in its own right. The philosophy of the proof, however, is already established in the existing literature, most notably Ref. 9, and although there are some differences here, it is felt that this proof, along with the proofs of the remaining theorems in this section, is best left to Appendix H. In the next theorem we replace the ratio

-163P(dxk ok' k- l)/P(dxk xkl) in Eq. 4.15 by a Raydon-Nikodym derivative. Theorem 4. 5. If the hypothesis of Theorem 4.4 holds, then the Raydon-Nikodym derivative P((xkxk IOk' Sk-l) = P(dxkl(kO k-l)/P(dxkxk 1) (4.16) exists and n(dAkk) = f (xk xkk, k, k )k-1) (dokl k-1) n(d k-1 xk k (4.17) Next we determine a generalized density for II(dOklxk). To do this, recall from Section 2.3 that we must first find a a-finite measure uL that dominates II(dOklxk). (A measure J dominates n(dk0Xk) if n(d OkXk) << i(dOk).) Now, for practical reasons, it is desired that this measure / work for all xk. Thus, we seek a measure g that dominates each member of the family {I(dOklk); k e Xk }. Such a measure is not always possible to find. As an example, it can be shown that there is no a-finite measure tp that dominates the family {II(dO k 0k-1) 0k- 1 [0, T ] }, when II(dOk 0k-1 is the continuous prior distribution of Section 2.3. Nevertheless, consider the physical situation at time tk. The prior probability of 0k is given by the density k

-164(Tp+o 10 k[O,Tp) P k p P(0k) [ -1 -1 o- (Tpc + ) Ok=T p k p with respect to the measure Ii(dOk) d0k + e {T }(dOk). Heurisp tically speaking, it seems reasonable to expect that the effect of any observation xk on the distribution of 0k should be to no more than "continuously" redistribute the probabilities. That is, it might be expected that {I(dkj 0); xk- Xk} is dominated by A(dOk) = dOk + e {T }(dk). A statement to this effect is provided by Theorem 4.6. P Theorem 4.6. If the hypothesis of Theorem 4.4 holds, then the family {II(dOklIk); xkc Xk} is dominated by the measure pd(dOk) [O, T )(dOk) + i {T }(dOk) where dOk for continuously arriving pulses [OT )(dO k) =(4.18) P q-1 for pulses with l e. (dO k) discrete arrival j=jl {J} time and there exists a generalized density p(0klxk) with respect to ji(dOk) such that II(d0,_ = P(|_) = P(0 x) (dO) (4.19)

-165An expression for p(0klxk) in terms of P(0k_l k_l) and ((XkXk-l' 1 0 k-1 ) is found in Eq. H.8 of Appendix H. This expression provides the means for updating the posterior densities p(Oklk). The difficulty, however, is that this expression involves the function (xkxkl, 0k' 0k-1) which depends on Xkl. Thus, it is still necessary to provide a memory for storing the past observation xk1. In the next section, however, it will be shown that it is sufficient for the decision device to have available the function P(6k!). --- -k - 0o [0,T ) >1 k0 =T k p k(0klk - P(OkI ) 0 k[O Tp) P(0k=TpI) O 0 0k T (4.20) The advantage of this fact is that the function 0k(Oklxk) can be updated solely in terms of 0k(0k xk) and xk. Thus, the decision device need only provide a memory for 0k(Ok|xk) This result is a consequence of the following theorem. Theorem 4.7. If the hypothesis of Theorem 4.4 holds and if in addition

-166P(dxk0k, 0k-) << P(dxkl0k=Tp 0k-_ =T all 0k- 0k then Ok( k tk) (Ok) f (xkl k0 0 k-1) P(Ok 0k-i) 0k-1(0k- 1lXk-l) (dOk-l) {[Tp-, Tp) k-1 E [,0 k-(Tp-A)] U {T I +I(k) f(xk kl(Sk-l =k, k ) k-1()k-l1 0k+Alxkl) {[O, Tp-A)} fS (%lOk T p' Sk-1) P( k k-l k- 1(k-l 1i l) k-(dokl) k_1 E [O, a) u {Tp (4.21) where P(dxk,1 0k-_l1) f(Xkl0k', k-1P) P(dxkl T ) (4.22) P(dx0k= P k- lp) In the proof to Theorem 4.7 above we use only the hypothesis that the observations are conditionally independent, that 0 is Markov with conditional density given in Appendix C, and that the family {P(dxk0k',k-1)} is dominated by P(dxkl0k= kl=TTp ). But, we have also assumed in this study that the observation x is statistically related to the state 0 by the "signal plus noise" relationships of Section 2.4. This added assumption makes it possible to obtain

-167more explicit expressions for the updating equation. These expressions and their derivations are found in Appendix I. We now apply Theorem 4.7 to deduce the updating procedure for the function Ok(0klj). To this end, note from Eq. 4.21 that Ok(0kXk) depends on the past observation k_1 only through the function Okl(0 k- 1 X k ) More precisely, suppose xk_' and Xk 1" are any two past observations such that k-l(' kXk-l') - Ok-l xkl" ) Then, for any xk, if xk' = (',xk) and " k-1'' then 0k(- IXk') - OXk' thx) Thus, without loss of generality we can write Ok(Ok xk) = Ok(Ok' Xk' Ok-l) We can then consider Eq. 4.21 as defining a function U that maps the current observation xk and the function Ok_ (.,xk, Ok- 2) into the function Ok(. Xk, Okl) Specifically, Ok(" Xk k- 1) = U(Ok_ l('k-l Ok2)' k) (4.23)

-168Then, in terms of this function, the updating procedure can be illustrated as in Fig. 4. 5. Ok(' xk' Ok-1) k(,'k' Ok-1) xk = U(Okl(. k-l Ok-2)'Xk) Ok- (I Xk-1' Ok- 2) Delay Fig. 4.5. The updating procedure for Ok(,xk, Ok 1) We conclude this section with an interpretation of the function 0k,(0 Xk' Ok-1) To do this, consider the integral f Ok(O k' Xk Ok-1) P.(d k) b( Pk(Okjk) - I(dok) /Pk(Ok=T Xk) pn he enointorof h, eprsi w a [In the denominator of this expression we have In the denominator of this expression we have

-169Pk(kxk) - Pr[0kT pTj] = Pr[no pulse arrives in [tk-Tp, tk))Xk] In the numerator, we have fPk(klxk) (dok) = Pr[ k [0,Tp) k] e' = Pr[a pulse arrives in [0, T )I] Thus, the integral of Ok(0k k' Ok-1) over the set {k e [0, T)} gives the ratio of the posterior probability that signal is present to the posterior probability that no signal is present. Now, in the literature it is customary to refer to this ratio as the odds ratio. Thus, in this context k(0, Xk' Ok- 1) can be viewed as an "odds ratio density, " since its integral gives the corresponding odds ratio. 4.4 The Bayes Decision Device for Conditionally Independent Observations In Section 4.1 it was pointed out that there are two basic difficulties in implementing the general Bayes decision device: the fact that the past observation memory must grow linearly in time and the fact that the distributions H(d0k|xk) and P(dxkl| I1)

-170must be calculated at each decision time tk. In this section we will apply the updating equation of the preceding section to replace the observation memory by a memory of fixed size and to replace the distribution P(dxkl |Xk) by the known distributions P(dxk+l I k+l' k) and II(d k+l 0 k) This is done in the following three theorems. Theorem 4.8. Under the assumptions of Theorem 4.4, Tk(, 0 k) Tk(Xk Tk' k) P(klk) (k (4.24) k where Tk( 7k, ) = Rk(, Tk' ak', k) - Rk(k, ak, 1 0 k) (4.25) and'](Xk,' 7k ak' k) (4.26) Lk(Tk' ak' k) |t=0|ff Rk+ l+ l' Tk+ l' + tok+l) P(dxk+l k+1' k) x n(dok+lk) k <N LN(TN aN, N) k = N \,~ ~ k1xW'+

-171Proof. We begin by showing that Rk(k, 7k, ak) can be written as Rk(x T' a) = (,k' ak' 0 ) II(dokx (4.27) 0 where R(xk, Tk, ak, 0k) satisfies Eq. 4.26. The proof is by induction on the index k. For k = N, it follows from Eq. 4.8 that RN(XN, TN, aN) S LN(TN aN, N) II(d ONIXN) ON = f VRN(N N aN.,0N) l(dOI ) where RN( xN' N' aN N)= LN(N, aN 0 N) Thus, the result holds for k = N. Now, assume Eq. 4. 26 and Eq. 4.27 are satisfied for indices k+l,..., N. From Eq. 4.8 we have Rk('k, Tk' ak) (4.28) = Lk(k, ak, k) II(d klx k 1th s. 4 2 by B (xk ) t=O A+1 ( Xk' 7k+l) Denote the second term in Eq. 4.28 by Bk(Xk, 7k+l) and apply the

-172induction hypothesis of Eq. 4.27 with k replaced by k+l to conclude that Bk(Xk, k+l) (4.29) 1 = O t fT Rk+l('k+1, k+ ak+l=t, k+l) = tAk+l(Xk, kk+l) k+l II(dOk+Xk+l) P(dXk+l Next, substitute for II(d k+l |xk+l) P(dxk+, tXk) from Theorem 4.4 into Eq. 4.29 to obtain Bk(xk,) k+l)'k[/ f f(k+ 1' k+1, ak+lt' ok+l) P(dXk+ll k+l'0 k) k+1 tAk+(X k' k+l) x H(dOk+l Ok) n(doklk) (4.30) Finally, substitute Eq. 4.30 for the second term of Eq. 4.28 and combine the two integrals into a single integral. The result is

-173Xk('Yl'k' "k) f L(Tk, ak, Sk) k 1 f f Rk1l( xk+,l' k+1' ak+t1 Ok+l) P(dxk+l 0k+l' Ok) t=O0 - tA 0 t= Ok+ Ak+l(xk',Tk+l) x H(dOek+l k) (dOklk) S^ Rk(x, 7k, k, O k)(dO xik) k as was to be shown. It remains to verify Eqs. 4.24 and 4.25. To do this, substitute for Rk(x, Tk, ak) from Eq. 4.27 into Eq. 4.6 and combine the two integrals into a single integral. The result is Tk(Xk' Tk)= Rk(Xk Tk' k) n(dOklxk) k where Tk(Xk, Tk' Ok) is given by Eq. 4.25. The proof is then completed by using Theorem 4.6 to write I(dOklxk) as P(0klxk) gL(d0k). In the next theorem the test function Tk(xk, 7k) is written in terms of the odds ratio density Ok(0 k Xk' Ok- 1) Theorem 4.9. Under the hypothesis of Theorem 4. 6, the component decision rule 6k(x.,.k) is given by

-1741 if Tk( k) > 0 6k(Xk, ) = (4.30a) 0 otherwise where Tk(k Tk) = Tk( k', k) 0k(0k' Xk k-) (d0k) (4 30b) Ok Proof. Expand the integral in Eq. 4.24 to obtain Tk(Xk, T'k) = Tk(Xk Tk' Ok) P(0k Xk) L[0, T )k 0ke [0,Tp) + Tk(Xk k' 0k=Tp) P(k= Tpk) First, suppose P(0k=Tpl k) f 0. Then, we may write kp + Tk(I'Tk, IkTp) Ok where the last equality follows by first substitutning the odds ratio density from Eq. 4.20 and then applying the definition of i (do ) from

-175Theorem 4.6 to combine the two integrals into a single integral. On the other hand, if P(Ok=T p k)=0, then [0, Tp) = f Tk(' k' ok) Ok(Ok'Xk' Ok-l) (dok) ok In either case, the condition Tk(x, Tk) > is equivalent to the condition Tk(xk' Tk) T(xk' Tk' k) Ok(0k, xk Ok-i )(de) > 0 Ok as was to be shown. As the final step, we show that the decision rule at time tk depends only on xk, k1 and 7k. Theorem 4.10. The functions Tk(xk, 7k) and Rk(xk, Tk, ak, 0k) and the response set A(xk -' Tk) depend on the observation x only through xk and 0 k. That is, Tkk, 7k) = Tk(Ok_-1,xk, Tk-1) (4.32a) Rk(', Tk, ak', k) = Rk(O _, xk, Tk, ak, 0k) (4. 32b)

-176and Ak(xk_,'k) = Ak(Ok-' k) (4. 32c) The proof to this theorem is a straightforward proof by induction. The details are found in Appendix J. On the basis of the above theorems, we may replace the past observation memory in the block diagram of Fig. 4.3 by a memory for the previous odds ratio density Ok01 and an updating device for obtaining Ok from xk and Ok1. The resulting block diagram is shown in Fig. 4.6.

kk+ _ -A ak \ 1 ~~~^ 0^ = U(0^^,x^j (k) max[0, 7,-a] a=0'-i~~k-i k- }o I 10i | Ok-1 ~ —DeaDelay 7 3 Delay Xk ak~~~~~~~~~~~~~~~~~~~ k, xk I k(Ok- I xk k) a=I I..~~~ ~~~~ Tk> I 0 Tk < 0 Fig. 4.6. The Bayes decision device for conditionally independent observations

CHAPTER V A NUMERICAL SOLUTION FOR THE OPTIMUM BAYES DECISION DEVICE AND ITS ROC SURFACE In the preceding chapter we developed the structure of the optimum Bayes decision device up to the solution of the recursive relation for the test function Tk (Xk, Tk). In this chapter we will carry out this recursive computation numerically for the m = 2 Bayes decision device. For the purposes of comparison we will also discuss the m = 1 Bayes decision device and the matched filter decision devices of Chapter I. Finally, we will obtain a numerical description of the ROC surfaces for these decision devices. 5. 1 The Basic Setting In order to simplify the computations in this chapter, certain additional restrictions will be made on the probability laws involved in the general model. Specifically, it is assumed that the pulse arrivals are discrete, that the noise is white Gaussian noise and that the pulse waveforms satisfy a certain translation property. In this section these assumptions are stated precisely and their effect on the basic mathematical model is determined. To begin, we review the basic temporal structure for the m = 2 case. According to the definitions of Chapter II, the decision time separation A is -178

-179A T /m = T /2 P P the set of allowable decision times Y is s' = {tk; tk = (Tp/2)k;k = 1, 2,..., N and the current observation xk is given by Xk = x(t) t e [tk - Tp/2, tk) The temporal relation between these quantities is shown in Fig. 5.1. -l__ -L TT =2A t- 1=t,-Tp/2 tk t 1 =tk+ T/2 t k-1 k p k k+1k p k+2 Xk Fig. 5.1. The temporal relation between A, Tp, and xk We now state the assumptions for this chapter. Our first assumption is that the pulse arrival times are discrete and that there are two allowable pulse arrival times per T P seconds. Thus, in terms of the notation of Section 2.3, we assume that q is equal to 2 and that the pulse arrival time separation v = Tp/q = Tp/2. It is to be noted that the pulse separation time v is Pp

-180equal to the decision separation time A As a consequence of this assumption, the random variable 0k takes on only the three values. 0k = 0, T /2, are associated with K p pulse arrivals in the interval tk - T, tk] and the value 0k = T P p denotes "noise alone" in the interval [tk - Tp, tk]. The pulse configurations associated with each of the possible values of 0k are illustrated in Fig. 5.2 along with the temporal relation between v and T p To obtain the specific form of the prior probability law, we appeal to the relevant equations of Section 2.3. First, from Eq. 2.19 with q = 2 we have 1 2 + a/(1-a)] 01 Pr[ = ] = (5. 1) [a/(1-a)] [2 + a/(1-a)] = 2 Next, the conditional prior probabilities Pr[ 0 = 6 1k =j6] are obtained from Eq. 2. 17 by setting q = 2 and s = A/v = 1. The results are most conveniently expressed in terms of the transition matrix {pi }, where p, j = Pr[0k = I k-1 = jv]' This matrix is 0 (1-a) a {P, } = 1 0 0 (5.2) 0 (1-a) a

T h --— P ——. —i 0k =0 ok tk-Tp tkT2 tk (a) " 0 =T /2 _~/ k p tk-Tp tk-Tp/2 t (b) -h- — 1 0, =T k p I I * tk-T tk-T/2 tk (c) Fig. 5.2. The pulse configurations for the different values of 0, (a),=O, (b)k =T/2, (c) k =Tp

-182Finally, we determine the pulse rate r from Eq. 2. 13 with q = 2 and v = T /2. The result is ~1 2 r r T [ 2 + a/ (1-a)] P or r = D (5. 3) p T p P where D = 2 (5.4) p [2 + a/(1-a)] From Eq. 5. 3 and the definition of the pulse rate r, we have E {T N } D - p p T But, since the pulses are non-overlapping, the quantity T N is P P equal to the total amount of time occupied by signal TS, so that E {TS } D = p T Thus, we may interpret D as the average pulse duty. p Our next assumption is that the noise is white Gaussian noise. From this assumption we may obtain the update equation for the odds ratio density by substituting Eq. I. 21 into Eq. I. 15 in App. I, and then setting q = 2, s = 1 and v = T /2. The result is P

-183Ok(O k Xk' Ok 1) (5. 5a) Pk(xkl0kkOSc) a [kl k l k O k- k1 ( ok-l= 1, Xk 1- O k- 2) -&k(xk Ok=Tp/2 Sc) a- (1-a) k =Tp k k k= p c k p 1 0 k T where, from Eq. 1.19, tk Ck(XkOk,Sc ) = exp x(t)p (t- Ok) dt10kS) tk-Tp/2 k1: p tk 1N [P (t- 0k)]2 dt (5.5b) o tk-T /2 The quantity N is the noise power per unit bandwidth. The above equation for 0k(k, Xk, Ok- ) can be expressed in a more compact form by denoting the quantity that appears in brackets in Eq. 5. 5 by Pk-1. Thus we shall define Pk- by p Qk-^k'V2 k-i Ok 1 (O k= Tp/2, Xk, Ok-2) k-1 1 + Okl(ok=O, x'kOk-2) First, we show that Pk 1 also satisfies an update equation. To do k- 1i this, replace k-1 by k in the above equation and substitute for Ok(, *, ) from Eq. 5. 5. The result is

-184kXk } k OkTp/2, S )a (l-a) Pk = T 12 0 -1 [ ok ( k-= p/'2; xk- r k-2) 1+ k( (Xk Ik =,0 Sc) a k-l - (' ( k- ~k-2 - 0k-1 0k-i'Xki1 k-2 -1X-ki Pr"k_1 0 k Xk 1, T Next, multiply the numerator and denominator by ra" and note that the bracketed quantity in the denominator is Pk-is The resulting up= Pr[Xk i = Tp/2, S ) (k-a) P (5. 7) k a + k xk I k =, S) Pr[k1 = T o obtain an interpretation for Pk substitute Ok-1 (k-' Xk-' Ok-2 = 2 Pr[ 0k-1 = Tp I k-1] into Eq. 5. 6 and multiply numerator and denominator by Pr[ 9 k_1 = Tp xk- ]. Theresultis Pr[k l-= Tp/2 2ik-l] p k-1 Pr[gk-1 = Tk k-1] + Pr[[k-T 1: Otk-1] or, since Prtk^k-1 p vk-1 k-k-i 0k-1 = 1 - ^k-1k2 -l we may also write Pr[ akl Tp/2 tk-l] I I-Pr[O k-1 = Tp/ 2 K;k_-l

-185Froni the above equation it is evident that the quantity Pk-1 can be interpreted as the posterior odds based on the observation Xki that 0k - = T /2. The event k = T /2 is shown in Fig. 5.3. k-i p - p tk-2 k-1 tk Fig. 5.3. The event 0k_ = Tp/2 Now, the significance of this event at the decision time tk is that Ok 1 = T /2 represents a pulse that is present at time tk and that also has energy in the interval [tk, tk). In Appendix I we referred to such a pulse as a carry-over pulse. Thus, we will refer to the odds ratio Pk-1 as the carry-over odds ratio. Finally, we assume that the first half of the pulse p(t) "looks the same" as the last half. More precisely, it is assumed that p(t) = p(t-Tp/2) te [Tp/2, Tp) k It follows that if p (t) is the basic pulse with arrival time k - Tp, pk(t) = p[t-( - T)], then pk(t) satisfies p p(t) = pt- Tp/2) t e [tk- Tp/2, Y

-186An example of a pulse p(t) which satisfies this translation property is shown in Fig. 5. 4. p(t) 0 T Fig. 5.4. A pulse which satisfies the translation property As a consequence of this assumption it is seen from Eq. 5. 5a that kk(Xk k = -0 Sc) = (Xk k 0 = Tp /2 Sc) Thus at time t the odds ratio density 0k(Ok Xk-1 Okl) depends on the observation only through the past carry-over odds Pk-1 and a single likelihood ratio. Denote this likelihood ratio by 1 tk k 1 t k kW(k) = exp N x(t)p(t)dt - 2 [p (t)2 dt 0 tk-T/2 otk -Tp/2 (5. 9) Then, it follows from Eq. 5. 6 and 5. 9 that the update equation for the odds ratio density may be written as

-187~1 9, ~~= T ~k(Xk) a- k_1 k (1-a) k (xk) k a + k (5 11) One last result will be necessary in order to describe the m = 2 Bayes decision rule. Specifically, we must have a description of the probability measures of the likelihood ratio k^(xk) that are induced from the probability measures of xk by Eq. 5.9. In Appendix K it is shown that there are two such distributions on kk that must be considered, one arising when xk consists of noise alone and one arising when xk consists of noise plus a signal pulse. Here we need only the fact that both of these probability measures can be described in terms of density functions that depend only on the signal-to-noise ratio of the basic pulse. 1 p =T In the following we will denote this signal-to-noise ratio by d2 where T T N p(t)2 dt (5. 12) o 0

-1885. 2 The m = 1 Bayes Decision Device In Section 4.2 we obtained the general solution of the recursive equations for the m = 1 Bayes decision device. There it was seen that the resulting decision device is the same as that obtained from classical detection theory (classical pulse detector). In this section, we will interpret this device in terms of the basic setting of the preceding section. We will then be able to compare both the structure and the performance of this device with the m = 2 Bayes decision device. To begin, recall from Theorem 4. 3 that the response condition for the m = 1 Bayes decision device is given by SIl(dok l ) > WF f II(dk0 k) k E [, Tp) {k = T } p p Now, to interpret this condition in terms of the assumptions of Section 5.1, use Theorem 4. 6 to replace II(dOk k) by p(Ok xk) jI(dOk) Then, use Eq. 4. 20 to substitute for P(Ok ixk) in terms of Ok(xk, k Ok 1). The resulting response condition is f Ok(k' Xk' 0k-1) I(dOk) > WF (5.13) k E [0, Tp) The next step is to note from the discrete case of Eq. 4. 18 with q = 2 that i(dok) = e{} (d k) + e{T /2}(dOk) + e{T }(dOk) Thus we may evaluate the integral in the inequality (5. 13) to express

-189the response condition as Ok(k, Xk' k-1) + Ok(k= Tp/2, Xk Ok-) > WF (5. 14) Next, note that the set of decision times for the m = 1 decision device can be considered as those decision times for the m = 2 decision device that have even indices. (See Fig. 3. 10. ) Thus, we can substitute for 0k(k,' Xk, Ok 1) from Eq. 5. 10 if we allow decisions only for k even and if we realize that xk and thus lk(Xk) and Pk(k, Pk-1) pertain to the observation over the interval [tk - Tp/ 2, tk) Carrying out this substitution results in the response condition k(xk) a Pk-1 + k (k) a (1-a) > WF or equivalently Tk(Pk_, lk) > 0 (5. 15a) where Tk(Pk-l' Ck) = Ik - K(Pk-1) (5.15b) with WFa K(Pk-1) = 1 - a + P (5 15c) The function K(P, k) is illustrated in Fig. 5. 5.

-190K(Pk1) Pk- 1 Fig. 5. 5. The threshold function for the m = 1 Bayes decision device The response condition of Eq. 5. 15 and the updating procedure for Pk of Eq. 5. 11 completely determine the block diagram for the m = 1 Bayes decision device. This block diagram is illustrated in Fig. 5. 6. (1-a) k 1 if k > i(xk) T k= a+ P - Delay 1k-1 a K(Pk- )and k(XJk__\ k a+k pkk-Pk-1 ak1 k even 0 otherwise Fig. 5.6. The m = 1 Bayes decision device

-191The operation of this device is as follows. At time tk the value of the carry-over odds ratio Pk-1' which has been stored in memory from time tk_, is used to determine a value of the threshold function K(Pk_). The decision device then compares the current value of the likelihood ratio k(Xk) with the threshold K(Pk 1) and responds iff kfk) k> K(Pkl) and k is even Finally, the values of Ik(xk) and Pk-1 are used to determine Pk for use at the next decision time. We conclude this section with a few interpretive remarks on the m = 1 Bayes decision device. First, note from the response condition WFa ak) _ K(Pkl) =(l-a) + Pkthat the decision at time tk can be considered as a likelihood ratio test against a threshold that depends on the information prior to time tk_l through the statistic Pk- k- 1). Here, the likelihood ratio lk(xk) is a measure of the evidence in the current observation Xk that signal energy is present if the interval [tk - Tp/ 2, tk), whereas Pk-1 (Xk-1) is a measure of the evidence based on the past observation xk, that the carry-over pulse is present. These quantities and the carry-over pulse are illustrated in Fig. 5. 7.

-192tk2 tk-1 tk k-1 (k-1 ) E'-) Ik(xk) Fig. 5.7. The temporal relation of the carry-over pulse, the previous carry-over odds Pk- (k- 1) and the current likelihood ratio (xk) The form of the threshold function K(Pkl) can be interpreted as follows. The large values of Pk-1 indicate strong evidence that the carry-over pulse is present. For these values, K(Pk_ ) is small as seen from Fig. 5. 5, and thus little evidence is needed from B(xk) in order to respond. On the other hand, small values of Pk-1 indicate strong evidence that the carry-over pulse is not present. For these values, K(Pkl) is larger, so that more evidence from (xk) is needed in order to respond. The extreme values of Pk 1 are interesting. Suppose first that Pk1 = + oo. Then, the decision device knows with absolute certainty on the basis of xk-1 that the carry-over pulse is present. Now, since the carry-over pulse must be detected at time tk, the decision device should respond regardless of the information present in the likelihood ratio (xk). But, this is compatible with the response condition, since K(Pk1 =+oo) = 0

-193On the other hand, suppose Pk =. Then, the decisionmaker knows with absolute certainty on the basis of xk - that the carry-over pulse is not present. Now, from Fig. 5. 2 it is seen that the only other possibilities are that 0k = T so that no pulse is k p present in the interval tk Tp, tk), or that 0k = T /2 so that a pulse arrives at time tk. But, at time tk these two possibilities can be distinguished only on the basis of whether or not there is signal energy in the interval [tk - Tp/ 2, tk). Thus, from classical detection theory, we would expect that the optimum decision device would respond iff Q(xk) exceeds the threshold Pr[0k = Tp 0k 0] WF Pr[ 0 = Tp/2 9 Ok 0] p k But, this is also compatible with the response condition for the m 1 Bayes decision device, since, by substituting for the conditional probabilities from Eq. 5. 2, we conclude that Pr[ Sk = Tp Ik / ~] F Pr[t 0k=T I 0 O] k p k F Pr[Ok=Tp/2 ok =OJ w ka F 1-a K(Pk = 0)

-1945.3 The m = 2 Bayes Decision Device In this section we present the results of a numerical solution for the m = 2 Bayes decision device. To obtain these results, we have interpreted the recursive equations of Section 4. 4 in terms of the assumptions of Section 5. 1. The resulting equations were then solved on a digital computer. The details of this procedure are found in Appendices K and L. To begin, we first draw on Theorem K. 3 of Appendix K to conclude that, as in the m = 1 case, the test function Tk depends on the total past observation xk only through Pk-1 and k. Thus, in the m = 2 case we may also write Tk(~k' Tk) = Tk(Pk_ 1' k' k) Next, we state two basic properties of the numerical solution for Tk(Pkl, lk'7k). These properties have not been established theoretically but have only been verified through the computations. The first of these properties states that, as in the m = 1 case, the m = 2 Bayes decision device can be considered as a likelihood ratio test against a threshold function that depends on the previous value of Pk-1. In the m = 2 case, however, the threshold function also depends on the specific decision time tk and on the past decision ak-1 as well.

-195Property I: Likelihood Ratio Property. For each set of values of the losses WX and WF, there exist two sequences of functions {Kk0 (Pk-l) and Kk1 (Pk-_l) such that Tk(Pk-1' k' Tk = 0) = k Kk (Pk-1) and Tk(Pk-' k' Tk = Tp/2) = k - k(k 1) The above property specifies the form of the response condition for the m = 2 case. These conditions together with the updating equation for Pk allow us to conclude that the general form of the m = 2 Bayes decision device is as shown in Fig. 5. 8.

ak- I Delay Pk %~~~~~~~~~~~~~~~~~~~~ ak~~~~~~~~~~~~~~~~~~~I a +i k Pk-l a1k ad s d 0 2 <K (Pkl) Fig. 5.8. The general form of the m = 2 Bayes decision device

-197The operation of this device is much the same as the m = 1 decision device. At time tk the value of the previous carry-over odds ratio Pk-1 and the previous decision ak- are both stored in memory. The previous decision ak- is used to determine which of the two threshold functions, Kk ( ) or Kk1( ), is to be used at time tk, and the value of Pk-1 determines the specific value of the threshold function. Then, at time tk the device responds iff it did not respond at time tk 1 and k > Kk (Pk-1) or iff it did respond at time tk and k > (Pk-1) Finally, the value of Pk-1 is used along with the value of k to determine Pk for use at the next decision time. The value of ak replaces ak-l in memory. The second of the two basic properties states that, except for decision times tk close to the terminal decision time tN, the structure of the m = 2 Bayes decision device does not depend on time. More precisely, Property II: Asymptotic Stationarity Property. For each set of losses WX and WF, the two sequences of threshold functions {Kk (Pkl) } and {Kk i(Pk_)} converge through decreasing indices K to limiting threshold functions

-198K (Pk ) and K1 (Pk_1) respectively. The interpretation of this property in terms of the system model of Fig. 5. 8 is that, except for tk near tN, we may replace the threshold functions Kk(Pk1) and Kk(Pk1_) by K (P k-) and 1 K (Pk-). Thus, for these decision times, none of the components in the decision device depend on the particular decision time t. We might point out that it is certainly reasonable to expect that Property II holds in the setting of this chapter, since we have assumed that both the noise process and the prior probabilities are stationary. We now turn to the task of describing the m = 2 Bayes decision device for different values of the losses WX and WF. To this end, we note from Property I that this device is completely specified by the sequences of threshold functions {Kk (Pk-) } and {K k-1) Moreover, by Property II these functions converge to limiting threshold functions K (Pkl) and K (Pkl) Thus, we may describe the stationary behavior in the m = 2 case completely in terms of these two functions. We will defer the discussion of the convergence of the threshold functions to Appendix L.

-199In order to provide a basis for the description of the limiting threshold functions, we recall from Section 3. 6 that the general Bayes decision device can be divided into five special cases depending on the values of the losses WX and WF. For the m = 2 Bayes decision device, each of these special cases results in a distinct form for the threshold functions. Thus we may describe the numerical results by presenting specific examples for each of the five cases. Case I: W (F) < W F > 0 - X F The Bayes decision devices for these values of WX and WF are respond-once decision devices which satisfy the inhibit rule and which are independent of the loss WX. Typical threshold functions for this case are shown in Fig. 5. 9 and 5. 10. There are several features of these functions that we shall take special note of here. First of all, it is noted that K (Pk1) = + oo for all Pk 1 This fact is a consequence of the inhibit rule property. To see this, we note that in the m = 2 case the inhibit rule says that the decision device cannot respond at time tk if it has just responded at time tk_1 Thus we must have Tk(Pk 1' k'Tk = T /2) k - K (Pk-l) < 0 for all Lk and Pkl.1 But this is true for all jk iff we take K'(Pk ) = + co for all P 1

-200I (.\Pk-l) W\\ =1 WF'1 d iK2 - I Dp.3 d3g 5.d = Dp $o Cs 3F 3 T\

-2015 -I —I (Pk- 1) K (Pk-) = 4 W = 1 WF = 2,3- \ d2=1 D =.3 I I I 1 2 3 Fig. 5.10. Threshold functions for Case I; dT2 1 Dp -.3, W =, WF 2

-202-'0 Next, it is noted that the threshold function K (Pk_1) approaches + oo as Pk-1 approaches 0. In the following paragraphs it will be seen that this property holds for all respond-once decision devices. To gain some insight into this property, consider the situation when Pk-1 = 0. Now, from the preceding section we recall that when Pk-1 = 0, the decision device is assured that either no pulse is present at time tk, or if a pulse is present at time tk, then t is the first opportunity to detect that pulse. Thus the response condition k > K (P 1 = 0) = +o k - k-1 = +00 says that, in this situation, the decision device will always defer a response decision to the next decision time. A few additional comments on the results for Case I are appropriate here. First of all, it has been verified computationally that the threshold functions are completely independent of the loss WX. This is to be expected from Theorem3. 11 which says that any Bayes decision device that satisfies the inhibit rule depends only on WF and not W. Secondly, we point out that we have not been able * to obtain an analytical expression for the function WX (WF). Nevertheless, by trying different values of the losses WX and WF and noting for which of these values the inhibit rule holds, we have obtained a numerical description of this function. This description appears in Fig. 5.11 for dT2 = 1 and D -.3. T ~P

-203Wx*(WF) Dp.3 dT2 =1 0. 5 1 2 3 WF Fig. 5.11. The function WX*(WF) Case IT: 0 < W < W * (WF), W > 0 X X F F The Bayes decision devices for these losses are respond-once decision devices which do not satisfy the inhibit rule property. Typical threshold functions are shown in Figs. 5. 12 through 5. 15. We make the following observations. First of all, it is noted that for constant WF, K1 (Pk_) decreases with decreasing WX. This says that the decision device with the smaller value of WX will break the inhibit rule more often than the decision device with the

-20425 W =.1 WF =1 20 dT = 1 Dp= 3 15tl / (1Pk 1) \ < K (Pkl) 0.5 1 0 Fi 512 Threshold functions f Case II;

.II as-e Joj suomounj ploqsaJTqL'CLI''S2T. ad' T O'l S'0....... I' I1........... 01I' = i=p aPT T= M Ag = X ( d) X -~~~O | *0

-20625 20 1 / W.=1 WF=2 lWXEW i0 5- < 0^-0.5 Fig 5.14 Threshold functions for se dT =1, Dp=.3, WX.1, WE

-20725 20 =l / 25 W ==2 Wvx F d / dT2=p Dp=3 T1- \ KiO('-l) \ \ ^ --- — l — -- ----'~~1.0 - "^5 KPk-I 0."[yr.^.y

-208larger value of W. This is certainly compatible with the physical X interpretation of the loss W, since large values of WX reflect a strong desire to avoid extra detections and extra detections can occur only when the inhibit rule is violated. Next, it is noted that K (Pk_1) is always larger than K (Pki) and that, although K (Pkl) is finite for some values of Pk-1' it approaches +oo for both small and large values of Pk-1 The interpretation here is that the decision device can break the inhibit rule only when the past information on the carry-over pulse is inconclusive and then only if lk is large. Finally, by comparing Fig. 5. 12 with Fig. 5. 13 and 5. 14 with Fig. 5. 15, it is seen that K0(Pki) is completely independent of the value of W. Moreover, for the same value of WF the function K (Pk-1) in Case II is identical to the function K (P 1) in Case I. We shall return to this point in the discussion of Case III below. Case III: W = 0, W > 0 The Bayes decision devices for these losses are the maximum pulse detection devices. Typical threshold functions for these devices are shown in Figs. 5.16 and 5. 17. There are two observations to be made here. First of all, the function K (Pkl) remains finite for large values of Pk-1 Thus, even if the decision device knows that the carry-over pulse is present, (P,. 1 = + 0), and even if it has detected this pulse at the

-20925 20 WX = WF = 15= d2 1 D =.3 10 K "(Pkl) l 0.5 1.0 1.5 Pk-. Fig. 5.16. Threshold functions for Case 111; dT2= l, Dp.3, WxO, WF=l

-21025 20 Wx = 0 WF = 2 d2 =1 D =.3 T P 15 10 K (P k- 1) 0.5 1.0 1.5 Fig. 5.17. Threshold functions for Case III; dT2 =1, D =.3, W=O- 0, WF =2

-211preceding decision time tk-, it may still respond again at time tk It is to be noted, however, that the extra detection that results incurs no extra loss since Wx = 0 for the MPD decision device. The second and most important observation is that for the same value of WF the threshold function K0(Pk_ ) in Case III is identically equal to the threshold function K0(Pkl ) in Cases II and I. Thus, for any respond-once decision device (Case I, II or III), the threshold function K0(Pk ) does not depend on WX. Hence, the only effect of the loss WX on the respond-once decision devices is to determine the function K (Pkl) The remaining two cases pertain to the respond-and-hold decision devices. Case IV: -1 < WX < 0, W > 0 - X FThe Bayes decision devices for these losses are those decision devices which balance the detection rate RD against the detection duty D. Typical threshold functions for this case are shown in Figs. 5. 18 through 5. 21. There are several observations to be made here. First of all, it is noted that both the threshold function K (Pkl) and K0(Pk_1) depend on both the losses WX and WF. Moreover, these functions decrease with either decreasing WX or decreasing WF. Secondly, both the threshold functions are finite at the origin. Thus, even if the decision device knows with absolute certainty that

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-213WX =-.25 WF = 1 dT2 =1 D =.3 T'(P 3 2 2K \K (Pk_1) 1 - \ K~ > (Pk 1)l I I. I. 1 2 3 Pk Fig. 5.19. Threshold functions for Case IV; dT2 =1, Dp=.3, WX =-.25, WF =1

-2145\ K\ \ WX =-.5 W = 2 d\ \ 2=1 D =.3 T p 1 2 3 Pk Fig. 5.20. Threshold functions for Case IV; dT2, Dp.3, W.5 W 2

-2154 - \WX =-.25 WF =2 dT2 =1 D =.3 \ \ K(k-l K (Pk ^K (Pk- 1) 1 1 2 3 Pk Fig. 5.21. Threshold functions for Case IV; dT =1, Dp=.3, WX =-.25, WF =2

-216time tk is the first opportunity to detect a pulse present at that time, (k-1 = 0), then it will still respond at time tk even if it responded at time tk. This behavior is typical of all respond-and-hold decision devices. Finally, we note that all of the threshold functions satisfy K (Pk- ) < K (Pk-1) for all Pk-1 This condition says that the decision device is more likely to respond at time tk if it did not respond at the preceding time tk1 then if it did respond at time tk_. That this is reasonable can be seen by noting that the response condition "0 kk > K (Pkl) can result only in a detected pulse outcome or a false alarm, whereas the response condition -k > K (Pk-1) can also result in an extra detection outcome. But, in the respond-andhold case the extra detection outcome only results in a gain of A units of detection time, whereas the detected pulse outcome results in A units of detection time plus the gain for a detected pulse. Thus, it is reasonable to expect that the decision device is less likely to respond if it has just responded at the preceding decision time. Case V: W = -1, W > 0 X B es F The Bayes decision devices for these losses are the maximum

-217detection devices. In this case the response condition can be obtained analytically. The procedure is to note from Section 4. 2 that the response condition is the same as the m = 1 Bayes decision device, except that decisions are made at each decision time tk Thus, we may conclude from Eq. 5.15 that K (Pk-) = K (Pk_) = K(Pk_1) WFa (1-a) + Pk-1 An example of these threshold functions is shown in Fig. 5. 22. We might point out that since both of the threshold functions are equal, the MDT decision device does not need to store the preceding decision. Thus, the decision feedback loop can be removed from the general block diagram of Fig. 5. 8. The resulting block diagram is shown in Fig. 5. 23.

-218E I.Pi l) (Pk- I ~\ WXE- -1 WF1 a = dTz =1, pd=.3, =1 DPWF ~.L^*v

k k (1-a) R_ _ RkHT k(l a)k |Delay Pk-1 1 > K(P) 1 ^- a~fk-a + ^P a- k- I k-1 k -I ak = F 5. ca for_ m 2axmudteciotmedeisonk < K(Pdik-c ) Fig. 5. 23. Block diagram for m = 2 maximum detection time decision device

-220In concluding this section, we note that all of the above examples for the five cases were calculated for the signal-to-noise ratio dT2 = 1 and the duty D =.3. In general, the basic form of these T P functions does not depend on dT2 and D, but the functions themT P selves decrease with increasing dT2 or increasing D. The one ~T ~P exception is the MDT threshold function which depends only on D Examples of this behavior appear in Figs. 5. 24 through 5. 27 The decrease in the threshold functions for an increase in D can be P seen by comparing Figs. 5. 24 and 5. 25 with Figs. 5.12 and 5. 18. The decrease in the threshold functions for an increase in dT can be seen by comparing Figs. 5. 26 and 5. 27 with Figs. 5. 12 and 5. 18. 5. 4 The Matched Filter Decision Devices In Section 1. 4 we introduced a matched filter (MF) decision device for both the respond-once and the respond-and-hold case. There we mentioned that, although these devices are suboptimal, they are quite practical from an engineering point of view. In this section, we will adapt these decision devices to the basic setting of this chapter. This will provide a common basis for comparing the structure and the performance of these devices with the optimum m = 1 and m = 2 Bayes decision devices. We begin by recalling from Section 1. 4 that at the decision time tk both the MF devices calculate the quantity

-2214 K (Pk 1) (P k-) + 3 W =1 WF =1 dT2 =1 D =.5 2 1e \ I I I 1 2 3 Fig. 5.24. Threshold functions for Case I; dT2 = 1, D =.5, W = 1, WF

-2225m 4 W =-.5 W =1 dT= 1 Dp =.5 K 0(Pk_1) = =: = 1 2 3 P 1 2 3 pk-1 Fig. 5.25. Threshold functions for Case IV; dT2 =1, Dp=.5, W= —.5, WF=l

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-2245W =-.5 WF =1 dT = 9 D =.3 3 2 K\ (Pk-l) 1 2 3 Pk Fig. 5.27. Threshold functions for Case IV; dT2 =9, Dp=.3, W =-5, WF=l

-225Ctk(x(t"); t" < tk) = x(t) p(t - (t Tp)) dt tkt- T - To interpret this quantity in terms of the assumptions of this chapter, we first break the integral into two parts, and then use the pulse translation property and the fact that tk = tk1 + t/2 to write tk-Tp/2 tk C = x(t) p(t - (tk - Tp))dt + S x(t) p(t - (tk -Tp))dt tk tk-T tk-T/2 P p tk 1 tk = J x(t(t) p (t-(t - Tp) )dt + f x(t) p(t - (tk- Tp) ) dt tkTp k-T /2 p or Ck 1 x(t) pl(t)dt + t1 S x(t)pk(t) dt (5. 16) N0 0 tk-l-T/2 N0 k-T/2 Next, add and subtract d T2 tk 2 = [pk(t)]2 dt 2N0 tk - T 1 tk i (t)] 2No f [p (t)]2 dt + dt tl Tp/2 2No t-T/2 to Eq. 5. 16 to obtain

-226NO f p x(-)p (t)dt [p (t)]2 dt o O tk_1- Tp/2 20 - p I tk 1 k + ( J x(t) p (t) dt- - f [pk(t)]2 dt o0 t- /2 -d /) p t Tp/2 dT2 + 2 or C dT2 No- 2 = ln 2 k-lXk-1) + n (5.17) = in i[k-1 (k-1) kXk) The next step is to replace the response condition C > 3 by the equivalent response condition tk dT2 B dT2 exp-( > exp- -' Now, substitute Eq. 5. 17 into the above expression to write the response condition as lk-1 (k-1) lk(k) ->' or finally, as lk(xk) > P'/kk-) (5. 18) We are now in a position to describe the MF decision devices. To

-227this end, note from Section 1. 4 that the RHMF responds whenever inequality (5. 18) is satisfied. On the other hand, the ROMF device must also satisfy the inhibit rule. Thus, for this device the response condition is Ik) if ak- = kk k) > + o if ak_- = 1 The block diagrams for these devices are shown in Fig. 5. 28.

Lk > k Ik(xk) Delay Ik I(Xk 1) 1 9k> /k- a {kt 0 otherwise (a) ek ) Dekay I (Xkk1-) 1ak > > /k-1...... ak= 0 otherwise (b) Fig. 5. 28. The MF decision device (a) respond-and-hold case, (b) respond-once case

-229We ngliht point out here that tlhese block diagrams are essentially the same as the block diagrams for the m = 2 Bayes decision devices; the main difference here is that the threshold function 3/tk does not depend on Pk-1 5. 5 Performance In Chapter III it was seen that the performance of the FRD decision devices can be described in terms of either the respond-once performance set J'(m) or the respond-and-hold performance set f(m). Moreover, the optimum performance for these devices is described in terms of either of the two ROC surfaces X 9(m) or AX (. In this section, we present a numerical description of 9(m) m these surfaces for the m = 2 and the m = 1 case. Also, we will present a numerical description of the performance for the MF decision devices of the preceding section. To obtain the description of the ROC surfaces, it is sufficient to determine the performance points R(6) = (RX(6), RF(6) RD(6 ) and D(6) = (DD(6), DF(6), RD() ) for the Bayes decision devices. This has been done by simulating the operation of these devices over a long period of time and measuring the resulting values of R(6) and D(6). The details of this procedure

-230are found in Appendix M. The performance description for the MF decision devices has also been obtained by this procedure. Before presenting the numerical results, we will introduce certain bounds on the ROC surfaces. First, we consider the bounds for the respond-once ROC surface AX (2). To this end, we assert that the performance of a respond-once decision device cannot exceed the performance of the classical signal-known-exactly (SKE) decision device with the same signal-to-noise ratio dT2. This follows from the fact that in the SKE paradigmthe decision device always bases its decision on an observation that is influenced by the whole pulse, whereas in the FRD paradigmthe decision device must sometimes make decisions on an observation that is influenced by only part of the pulse. The basic SKE paradigm is shown in Fig. 5. 29. --- Tp --- — Observation — 4 Decision Time Fig. 5.29. The SKE paradigm for signal-to-noise ratio dT2

-231Next, we assert that in the m = 2 case the performance of the optimum respond-once decision device will always exceed the performance of the optimum SKE decision device that bases its decision on only half of the signal energy. The basic paradigm here is shown in Fig. 5. 30. -- Tp T — ----- Observation Decision Time Fig. 5.30. The SKE paradigm for signal-to-noise ratio dT /2 The validity of this assertion follows from the fact that the m = 2 respond-once decision device always bases its decisions on at least half of the signal energy. To interpret the above statements in terms of the ROC surface X Y(2), it is necessary to relate the probability of detection PD and the probability of false alarm PF for the SKE decision device to the rates RX, RF and RD. This is done in Appendix N. There it is seen that, for the SKE decision device,

-232RX = R PF RD = PD It then follows from the above statements that the projection of X9 (2) onto the RX = 0 plane must not exceed the ROC curve for the SKE case with signal-to-noise ratio d 2 and must lie on or above the ROC curve for the SKE case with signal-to-noise ratio dT2/2. Thus this projection must lie completely in the region shown in Fig. 5. 31 below. It is to be noted here that these curves, as well as the remaining curves in this section, are plotted on normal-normal paper. We may use the same argument as above to obtain an upper bound for the ROC surface XA(2). This is done by associating T seconds of detection time to each detection decision and T seconds p of false alarm time to each false alarm decision. It then follows that the projection of the ROC surface, X (2), onto the RD = plane cannot exceed the ROC curve for the SKE decision device with the same signal-to-noise ratio. This latter curve is obtained by noting from Appendix N that for the SKE decision device DD = PD DF PF RD = PD The region containing the projection of A Cf() is shown in Fig. 5. 32.

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-2340.99 DD 0.9 — b& 0.5 0.1 0.1 0.5 0.9 DF Fig. 32. The region containing the projection of XA,

-235We now turn to the description of the performance of the respondonce decision devices. First we consider the m = 2 decision device. Recall from Section 3. 6 that the m = 2 Bayes decision devices can be classified into three special cases. The Case I decision devices satisfy the inhibit rule and thus their performance plots as a curve in the RX = 0 plane. This curve has the property that it represents the smallest values of the detection rate RD. The Case III decision devices are the MPD devices. These devices do not satisfy the inhibit rule and thus their performance plots as a curve in the region RX > 0. This curve has the property that it represents the largest values of RD Finally, the Case II devices are those devices that achieve a trade-off between RD and RX. These devices also do not satisfy the inhibit rule. Their performance constitutes the rest of the ROC surface A,(2) in the region RX > 0. To illustrate the performance in the m = 2 case, we have plotted the inhibit rule curve and the projection of the MPD curve into the RX = 0 plane. The area bounded by these curves can then be considered as the projection of Xy(2) into the RX = 0 plane. These curves are shown in Fig. 5. 33 for d = 3 along with the upper and lower bounds discussed above. In Fig. 5. 33, it is seen that for the low signal-to-noise ratio, (dT2 = 1), the surface kA(2) is essentially independent of RX, since the projection of the MPD performance curve and the inhibit rule

-2360.99 RD // C;R D blll(^~// Fig. 5.33. The projecton o PDnto the RX = 0 or d UL 1, (dT = 1) 0.1 0.5 0.9 RF tnto the Ry = 0 p, ror dT=l, 3.

-237performance curve coincide. Thus, for low signal-to-noise ratios, one can expect little or no improvement in the detection rate RD for increases in the extra detection rate RX. On the other hand, if the signal-to-noise ratio is large, (dT2 = 9), there is a noticeable difference in the two curves, so that in this case one can expect a substantial increase in RD for an increase in RX Next, we consider the performance of the m = 1 R-O decision devices. In this case the ROC surface Xy(1) degenerates to a curve in the RX = 0 plane, since these devices are incapable of making extra detections. Figure 5.34 illustrates X, (1) for dT = 1, 3 along with the upper and lower bounds. The main point to be made here is that these curves lie only slightly above the lower bound and, as can be seen by comparing Fig. 5.34 with Fig. 5. 33, they lie substantially below the projection of X(7(2) Thus, as is to be expected from Theorem 3. 9, there is a substantial increase in the optimum performance for an increase in the number of detection opportunities m. Finally, we consider the performance of the ROMF decision devices. As discussed in the preceding section, these devices also have two decision opportunities per pulse, (m = 2), and they also satisfy the inhibit rule so that their performance plots as a curve in the RX = 0 plane. Furthermore, it should be noted that since these devices are suboptimal, their performance can be expected to depend on the duty D p

-2380.99 RF 0.3 Te RC 0.1 0.5 0.9 R F Fig. 5.34. The ROC curve ^O(i)

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-240This is seen to be the case in Fig. 5. 35, where we have plotted the ROMF performance curves for dT = 1, 3, and D =. 1,. 5, along with the upper and lower bounds. It should be noted from Fig. 5. 35 that the performance of the ROMF device for D =. 1 shows a substantial improvement over the p performance for D =.5 for the higher signal-to-noise ratio. P We conclude the respond-once case by comparing the performance of the three different R-O decision devices for dT = 3. This is done in Fig. 5. 36. In this figure it is seen, as expected, that the ROMF performance curves lie below the projection of XA7(2). On the other hand, these curves lie above X y(1). Thus we have an example of a suboptimum m = 2 device whose performance exceeds the performance of the optimum m = 1 devices. We turn now to the performance of the respond-and-hold decision devices. First, we treat the m = 2 decision devices. Recall from Section 3. 6 that the R-H Bayes decision devices have been divided into two cases, Case IV and Case V. The Case V decision devices are the MDT devices, (Wx = -1). These devices achieve the maximum values of DD for fixed DF. The Case IV decision devices are those devices that sacrifice detection time for an increased RD Their performance constitutes the rest of the ROC surface Ay(2) To illustrate XJ(2) we have projected the MDT performance curve and the curves for the Bayes decision devices for WX = -. 5, Wx = -1.,

-2410.99' RD Projection of X2) 0.9 ROMF (D =.3) y/^ ~ ROMF (D =.5) 0.5 p 0.1 l I l I 0.1 0.5 0.9 RF Fig. 5. 36. The performance of the R-O decision devices.

-242and W = 0 into the RD 0 plane and the D = 0 plane. These curves are shown in Figs. 5. 37a, b and 5. 38a, b for dT = 1, 3 Here it is seen that the MDT has the largest values of DD but the smallest values of RD for fixed DF. On the other hand, the curve associated with WX = 0 has the smallest values of DD and the largest values of RD for fixed DF.This is to be expected from the results of Chapter III. We might also point out here that the projection of the WX = 0 curve into the RD = 0 plane passes through the point RD = 1 DF =.5 DD =.5 This can be verified analytically by noting that this point is the performance point of the decision device that responds every other time. (See Section 3. 4.) Next we consider the performance of the RHMF. As in the respond-once case, this device is a suboptimal m = 2 device whose performance depends on the duty D. The projections of the perp formance curves are shown in Figs. 5. 39 and 5. 40 for dT = 1, 3, and D =.1,.3. By comparing these curves with the curves in Figs. 5. 37 and 5. 38, it is seen that there is a substantial degradation in the performance

-2430.99 DD 0.9 WX =-1I Wx = - 0. =0 0.1 I....I 0.1 0.5 0.9 DF Fig. 5.37. The projection of X, (2into the R = 0 plane (a) dT = 1.

-2440.99 DD Wx =-1 0.9 X WX = - 5 Wx = O 0.5 0.1 0.1 0.5 0.9 DF Fig. 5.37. The projection of AX (2)into the RD = 0 plane (b) dT. = 3.

-2450.99 RD 0.9 WX = 0 WX U =_.5 0.5- p W (a d =0. 0.1 I I 0.1 0.5 0.9 DF Fig. 5.38. The projection of X (2\into the D = 0 plane^' (a)d~ = 1.

-2460.9 Xx w -.5 0.9 WX =-1 0. 5 0.1 0.1 0.5 0.9 DF Fig. 5.38. The projection of nto the DD = 0plane r(2 (b) dT =3.

-2470.99 DD 0.9 Fig. 39 The projections of the performance cures for the into the = plane. 0.4 0.1 0.5 0.9 DF Fig. 5.39. The projections of the performance curves for the RHMF into the R = 0 plane.

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-249of the RHMF, although, as in the respond-once case, the least amount of degradation occurs for the small value of the duty D. P

CHAPTER VI SUMMARY AND CONCLUSIONS In many physical situations it is desired to detect the presence of randomly occurring pulses as they occur. In this study we have defined and analyzed two classes of decision devices that function in this setting. The respond-once decision devices seek to respond at a single instant of time within a prespecified time after the arrival of each pulse. These devices are applicable in a situation where it is desired to take a single action for each pulse that occurs. The respond-and-hold decision devices seek to respond at each instant of time at which a pulse is present. These devices are applicable in a situation where it is desired to take a continuous action during the time when each pulse is present. To describe the performance degradation due to noise at the input of the R-O and the R-H decision devices, we have defined two loss functions. The loss function for the R-O decision devices measures the performance degradation in terms of the number of detected pulses, the number of false alarms and the number of extra detections. The special R-O decision device for which the loss for an extra detection is zero has been referred to as the maximum pulse detection device. The loss function for the respond-and-hold decision devices measures the performance degradation in terms of -2 50

-251the number of pulses that are detected, the amount of detection time, and the amount of false alarm time. We have referred to the special R-H decision device whose performance depends only on the amount of detection time and false alarm time, and not on the number of pulses that are detected, as the maximum detection time device. To the author's knowledge, no formal theory exists for the R-O and the R-H decision devices described above. In psychophysics, some work has been done to characterize the human decision-maker in the respond-once setting, while in systems engineering, it is generally agreed that the matched filter decision device described in Chapter I will function reasonably well in either setting. Nevertheless, no effort has been made to obtain optimum decision devices and to characterize their performance. This problem, which we have referred to as the free running detection problem, is the subject of this study. To analyze the free running detection problem, it has been necessary to do three things: define the basic decision model, characterize the performance of both classes of decision devices, and obtain the optimum decision devices. The methods used in carrying out these tasks and the results are summarized in the following paragraphs. In order to provide a solid theoretical foundation for the basic model, the definition of the FRD problem formulated in

-252Chapter I has been interpreted in terms of the general decision theory model 4 0E, L). This was done in Chapter II. There it was seen that the action space,I, could be defined as the set of binary N-tuples, if we added the restriction that the decision device be allowed to make decisions (change the value of the response function) only at equally spaced times on the time axis. The separation between these decision times is an important parameter, since it determines the number of decision opportunities available to detect each pulse. The state space for the FRD problem was defined in terms of the pulse arrival times. The procedure was to assume that either the pulses can arrive at any time as long as they do not overlap, or else to assume that they can arrive only at discrete, equally spaced times. In the continuous arrival time case, we assumed that the time between the completion of one pulse and the arrival of the next pulse can be described as an exponentially distributed random variable. In the discrete case, we assumed that this inter-arrival time can be described as a geometrically distributed random variable. The next step was to define the distribution of the arrival time of the first pulse, so that the pulse rate (the expected number of pulses per unit time) was independent of time. Finally, we defined the parameter space in terms of a sequence of random variables, 0 = (0 1'..' N), so that the ith random variable 0i describes the pulse arrivals relevant

-253to the decision device at the ith decision time. As a consequence of these definitions, it was seen that the sequence 0 could be described in terms of a stationary, first order Markov probability law. To complete the description of the decision model, we applied the formal definitions of Chapter I to interpret the loss functions in terms of the action space and the parameter space. It was seen that the loss functions for the R-O and the R-H decision devices could be expressed in terms of a single canonical form. The means of describing the performance of the FRD decision devices was developed in Chapter III. The procedure followed there parallels that of classical detection theory. Specifically, a detection rate RD, a false alarm rate RF, and an extra detection rate RX were defined for each R-O decision device. It was then possible to characterize the performance of the whole class of R-O decision devices in terms of the performance set 9. Similarly, it was seen that it was possible to describe the performance of an R-H decision device by the detection rate RD, a detection duty DD, and a false alarm duty DF. This gave rise to the performance set 6P as the means of describing the performance of the class of R-H decision devices. The next step was to establish some basic properties of the performance sets JSand A. First, it was shown that both

-254-.9P and SP'lie in triangular regions in IR3. Next, we introduced the concept of a randomized decision rule to show that both performance sets are convex. The convexity property was then used to define the concept of a chance plane for both performance sets. Finally, we extended the notion of the antipodal decision device from the classical detection problem to the FRD problem in order to obtain a reflection property. With the above properties at hand, we were able to verify two conjectures introduced in Chapter I. First, it was seen that any FRD decision device had the property that the decision at a particular time t' depends on the previous decisions only through the time of the last response. This principle was introduced in Chapter I as the most recent response rule. Next, we proved that the only decision devices that never make extra detections are those devices that never make more than one response within an interval of duration equal to the pulse duration. This principle was introduced in Chapter I as the inhibit rule. In the final sections of Chapter III, we used the performance sets 09and.9to define a notion of optimality that is independent of the loss function and the prior probability laws. There it was seen that this optimum performance could be described in terms of two ROC surfaces, one for the R-O decision devices and one for the R-H decision devices. Moreover, we were also able to show that this notion of optimality and the notion of Bayes

-255optimality are equivalent. This last fact together with the geometrical properties of the ROC surface enabled us to verify certain conjectures introduced in Chapter I. First, it was shown that, if the loss for an extra detection is sufficiently large with respect to the loss for a false alarm, then the corresponding R-O decision device satisfies the inhibit rule. Moreover, the structure of the decision device is then independent of the particular value of the extra detection loss. Secondly, we were able to show that, for those R-O decision devices that do not satisfy the inhibit rule, an increase in the detection rate can be obtained at the expense of an increase in the extra detection rate; the maximum values of the detection rate occur for the MPD decision device. As a final result, it was seen that the performance of the R-H decision devices also reflects a trade-off. Specifically, an increase in the detection duty comes at the expense of a decrease in the detection rate with the maximum values of the detection duty occurring for the MDT decision device. In Chapter IV, we completed the theoretical portion of this study by obtaining a description of the optimum Bayes decision devices. The first step was the derivation of a general expression for the Bayes decision rule at the decision time tk in terms of the Bayes decision rule at time tk+1. From this expression it was possible to verify the configuration of the Bayes decision device

-256that was introduced in Chapter I. Next, we considered the structure of the MDT decision device and the structure of the decision device that is allowed one decision opportunity per pulse. It was seen that both of these decision devices are equivalent to the classical detection theory device which responds if a pulse is present at the current time. In the remaining sections of Chapter IV, we added the assumption of conditionally independent observations to obtain an updating procedure for the Bayes decision device in order to replace the past observation memory with a memory of fixed size. This is certainly a necessary requirement of any decision device that is to be operated over long periods of time. It should be mentioned, however, that the resulting structure of the Bayes decision device is still considerably more complicated than the structure of the matched filter decision devices. In Chapter V, the theory of the preceding chapters was applied to obtain specific examples of FRD decision devices. In particular, we considered the m=1 Bayes decision devices (m is the number of decision opportunities per pulse) and the m=2 Bayes decision devices. The structure of the m=1 devices could be obtained analytically from the results of Chapter IV, but the structure of the m=2 devices had to be obtained numerically. For both of these devices it was seen that the decision at the decision time tk is made by comparing a likelihood ratio Jk

-257with a threshold that depends upon a certain statistic of the past observations. The threshold for the m=2 devices also depends upon the preceding decision, but the threshold for the m=1 devices is independent of the preceding decisions. The performance of the Bayes decision devices considered in Chapter V was calculated by simulating their operation over long periods of time. It was seen that the ROC surfaces for both the m=2 and the m=1 devices exhibited the properties developed in Chapter III. These properties were most apparent, however, for the higher signal-to-noise ratio. For example, the performance of the m=2 MDT devices showed a substantial improvement over the performance of the m=2 inhibit rule devices for a high signal-to-noise ratio but essentially no improvement for the low signal-to-noise ratio. As a final example, we calculated the performance of the matched filter decision devices in the m=2 setting and noted that their performance was inferior to that of the Bayes decision devices. It might be emphasized, however, that since these calculations were made only in the m=2 setting, the question of the performance degradation of the matched filter devices for large values of m remains open.

APPENDIX A THE CLASSICAL DETECTION PROBLEM AS A GENERAL DECISION PROBLEM In this appendix the classical detection problem is interpreted in terms of the general decision model. To begin, we identify the elements yI, 0, L. Consider first the action space d. There are two possible decisions in the classical detection situation; decide that a pulse is present or decide that no pulse is present. Denote the first of these alternatives by "1" and the second by "0". Then the action space,e is given by 4= {a;ae 10,1}} Next, consider the parameter space O. Two possible situations can occur; either signal is present or no signal is present in the observation interval. Denote the set of parameters which influence the observation when signal is present by OSN and denote the set of parameters which influence the observation when no signal is present by ON. (In many situations, 0N will consist of a single point indicating the event "no signal present".) Then, the state space 0 is given by 0 = OSN U ON Now consider the prior probability law II(' ) on O. -258

-259In general, this probability law can be written in the form II(d0) = II(dO I SN) Pr[SN] I (0) { SN} + II(dO I N) Pr[N] I (0) tN} where n(dO I SN) is the prior probability law on 0SN given that signal is present, I(dO | N) is the prior probability law on ON given that no signal is present, Pr[SN] and P[N] are the prior probabilities of the events "signal present" and "no signal present"' respectively, and I (0) is the usual indicator function defined by I (0) = ( 0' le'te 0' Finally, consider the loss function L (, *). In the classical detection problem, this function has the form illustrated in Fig. A. 1.

-260N N NN SN a = LC LM a= 1 LF LD Fig. A.1. The loss function for the classical detection problem where LC, LM, LF and LD are the losses for a "correct rejection", a "miss", a "false alarm" and a "detection", respectively. Next we calculate the Bayes risk of an arbitrary decision rule 6e ). By definition, r(6) = E [L(6(x), )] x, 0 or r() = E {E [L(6(x), 0)] 0 x1O For the decision rule 6, define the set, Ac X, by A = x; (x)= 1 Then the Bayes risk can be written as

-261r(6) = Pr[N] [LF f P(dxl0)IT(d I N] + L J f P(dx I)II(d0| N) ON A A + Pr[SN] L f f P(dx l)II(dOSN) + LMf CP(dx I0)(d01SN) +OSN A OSN A The usual probabilities of detection and false alarm, P and PF, are D Fr defined by P = f r P(dxl0)II(d0 SN) (A.la) OSN A P = r P(dxI0)II(d0 IN) (A. lb) ON A N In terms of these probabilities, the Bayes risk can be written as r(6) = Pr[SN] (LD - LM) PD + Pr[N] (LF - LC) PF (A. 2) + Pr[SN] LM + Pr[N] LC Finally, we shall derive an expression which defines the optimum Bayes decision rule. We have r(6) = E [L(6(x),0)] x, 0 = E [E [L(6(x),0)]

-262By a well known argument in classical detection theory, the Bayes rule 00 6 is that rule which associates with xc X the a (x)e.~ satisfying E [L(a0(x), 0)] min E [L(a=, 0)], E [L(a=1, 0)] OIx xx OIx or 1 if E [L(a=1,0)] < E [L(a=0,0)] 6(x) = Ox 0 otherwise The quantity E [L(6(x) = a, 0)] is known as the posterior risk of the eIx decision rule 6. A direct substitution into the condition E [L(a=1,0)] < E [L(a=O,0)], OIx OIx results in LF I (dl1x) + LD f I(dl1x) N SN < LC Il(d01x) + LMf In(dl1x) N SN or (LF - LC r n(d0lx) > (LM L- n(d0~x) (A.3) SN N

-263where II(d lx) is the posterior distribution of 0 given x. An alternate expression for the response condition can be obtained from the inequality (A. 3) by expressing I(d0lx) in terms of the observation distributions and the prior probability distribution. Namely, f P(dx 0) n(d 0SN) 0N (LF - LC) Pr[N] sN______ __ _ L_ Pr[N] f P(dx 10) II(d IN) - (LM- LC) Pr[SN] N To obtain a more usable form for the above response condition it is necessary to place additional restrictions on the probability laws. One restriction that yields a familiar result is to assume that there is some parameter 00cE E such that P(dxl0) << P(dxI00) for all e 0. ( << denotes absolute continuity; see Appendix C.) We may then write the response condition as f (x 0)P(d0ISN) SN l LF: - L Pr[N] f J (x 0)P(d0 N) - D Pr[SN] N where (xl0) is the likelihood ratio P(dxle) D(xl|) = P(dx|_ )

APPENDIX B DERIVATION OF THE PRIOR PROBABILITY LAWS -- PART I In this appendix, the conditional probability law on 0k given 0k-1'...' 01, is calculated. As a consequence of this calcultation, it will be seen that II(d0) satisfies the Markov property I(doklok-, * 0 1) = = I(dOk 0k-l) We proceed as follows. From the definition of 0 = (0,..., 0 ) in Section 2.3, it is noted that the random variable 0k depends on whether or not a pulse arrives in the interval [tk - T, tk). The conditioning event 0k1 = Ck-1'..., 0 1 = c1 depends on pulse arrivals in the interval [-(Tp- A), tk). This basic temporal relation between 0k 0k-,..., 01 is shown in Fig. B. 1. I I-I. ~ -I I I I T- A 0 t1 t2 tk-T t-T tk- t P 1 2 kp kk 0 Fig. B. 1. The temporal relation between k, k-l'...' 0 1 - 264

-265Suppose first of all that no pulses arrive in the interval [-(T - A), tk ). Then, Ok-1 =- = = Tp. Now, if apulse arrives in the interval [tk - T tk), then its arrival time must be equal to r1 and, by the definition of 0k, 0k = - (tk- Tp) = 1- (tk- + -Tp) 1 -tk 1+ (T - A) Thus the condition 0 < c < T is equivalent to the condition k- p 1 < C+tk-l-(Tp-A) On the other hand, since it is assumed that no pulses arrive in the interval [-(Tp - A ), tk ) the arrival time 771 must also satisfy i > tk-i Thus, for the continuous case we may write Pr[0k< c < Tp 0k. =1 = Tp] (B. la) = Pr[ 1< c -(T p- )+t.tk ]< r) ]

-266For the discrete case Pr[0k = c < Tp|0k- =... =01=Tp] (B. lb) Pr[ 1 = C-(T p-A)+tk- tk-l < 77] Next, suppose that at least one pulse arrives in the interval [-(Tp - A), tk ). Denote the arrival time of the last pulse to arrive in this interval by ok. We consider first the case cre[tk - Tp, tk ) as shown in Fig. B.2. ~ — Sk-_I k k-l- Tp t-Tp tk- 1 tk Fig. B. 2. The case a ce[tk - T, tk 1) Here, it is seen that 0kle[A, Tp) and that 0k is related to 0k-1 by Tk fk- Thus, for the continuous case,

-267Pr[0k < c < Tpl0k-l ck-l 1=C11 0 0 < Cki-1 (B.2a) ~1 0 > Ck1 - and, for the discrete case, Pr[k = c < Tp0kl =Ck-,..., 01C1 0 c / ck-1 (B. 2b) C1 =CklNext, suppose acE[tk - Tp, tk T ) as shown in Fig. B. 3.,14 - 1 o+1 Fig. B.3. Thecase a [t1-T tk -T) k-ip' k p In this case, 0k _1 [0,A). Now, if a pulse arrives in [tk-Tp, tk) then its arrival time is given by "^+1 = a^ + r7+1~

-268Substituting this relation into the definition of 0k results in 0k -= + U71t+ - (tk - T) +i + 7 T -tk-l+ [T -A] Thus the condition 0k < c is equivalent to the condition 7q+1 - < C [T -A] +tk-l- 9c On the other hand, the condition that ac is the last pulse to occur in the interval [-(T -A), tk- ) is equivalent to 1 +1 > tk-1 - r Thus, for the continuous case, we have Pr[0k < c < Tp|0 k = Ck-'. 01 =C1 (B. 3a) = Pr[7q +1 < c-[Tp-A] +tk - O 7qf+1 > tk - (xQ ] and for the discrete case Pr[Ok=c < Tp|0k-l=Ck- 0 1=c 1 (B. 3b) Pr[ 1+1 = c - [Tp-A] +tkl - I 1+l > tk- - 1 Q] The last possibility to consider is the case a e[[-(T tk1Tp ). This case is shown in Fig. B. 4.

-269-< —---------.......................+1 No O*k- j i I ---— /l tkj 1-Tu ( tk_ -T k t I- tk-T tk1 a +l tk k-j-i p Q k-j p k1 p k p k- 1 +1 k Fig. B.4. The case au e[-(T- A), tk 1-Tp) As shown here, ue[tk-j 1-Tp, tk -T ), for some j > 1, so that k-j-l p k-j p 0k j.[ 0 A) and 0k j+1 =* = 0k- = Tp. Now as before the condition ok < c is equivalent to the condition 7,+1 < c - [Tp -A] + tk-l- af and the condition that au is the last pulse to arrive in [-(T1 - A), tk-1) is equivalent to the condition?U+1 > tk-l- a Thus, in the continuous case, Pr[0 <c< T 0=... =0 = T 0 -=ck 0 -=c Pr[-k < c < "pk p' k-j+ p k-j'''' 1 1 (B. 4a) Pr[ r1 < c -[Tp-z] +tkl-|-1'1V+1 > tkl - ar]

-270and,in the discrete case, Pr[ = c < Tp0k-1=.. =0k-j++1 T k-j kj'' 1 (B. 4b) =Pr[t+l = c - (Tp -A)+ tk_ - e + > tk- - A ] We are now in a position to calculate the conditional probability laws. The continuously arriving pulse case is considered first. First suppose that 0 kl[[A Tp). Then,from Eq. B. 2a Pr[k < c < Tp l =Ck-...' 1= 1 0 c < Ck — A (B.5) 1 c> Ck-lNext, suppose that 0k1 e[0, A). The equation of interest here is Eq. B. 3a. To begin, note from Fig. B. 3 that tkl - Ua is related to k-1 by t -1- = T - k k-i p k-i Now substitute for tkl - af from above into Eq. B. 3a and set k-1 = ck-. The result is

-271Pr[0k < c < Tplk- =ck..- 0 =c] C1 Pr[1+ c - [Tp-A] +Tp - Ck-l7I l T - k-] Next apply Bayes rule to write the right hand side of the above equation as Pr[T - ck l< 7+l< c -ck- + ] Pr[ 1+l > Tp - k- 1] Finally, from the density in Eq. 2. 8 of the text, it follows that for ck-l e[0,A) Pr[r,1l < Tp- c k-] and Pr[T - Ckl1 < 11< c k-l+ A] - a(c - (ck -1+T -)) 1- e (c- P if c > ck+Tp-A k-i p 0 otherwise Thus, for 0k- = k-l [' )'

-272Pr[lk < C0k-1 k = Ck-l,. l] (B. 6) 0 c < Ckl+[T -A] - a(c-(ck_ +T -A)) =! 1- e p c k+T -A < c < T k-1 p p 1 TT <c p Finally suppose that k_1 = T. There are two possibilities. Either k-2 =... = = Tp, or there is some largest index j > 1 suchthat 0k je[0,A) and 0k j+l= = k- T. Equation B. la applies in the first case and Eq. B. 4a applies in the second. In the first case, the probability on the right hand side of Eq. B. la is determined from the density in Eq. 2. 9 of the text,and,in the second case,the probability on the right hand side of Eq. B. 4a is determined from the density in Eq. 2. 8 of the text. In either case we have Pr[0k < c0k-1 = T 0ok-2 = k-2' 01= 1] (B. 7) r0 c < T -A - p -a(c-(T -A)) = 1-e P T -A < c < T p - p i1 T <c This completes the derivation for the continuous case. The conditional cumulative distributions are given by Eqs. B. 5, B. 6 and B. 7.

-273Furthermore, since the right hand sides of these equations do not depend on the values of 6 1 0 k-2 we have shown that the Markov property holds. We turn now to the discrete case. In this case we have assumed that the separation between the allowable pulse arrival times, v, is related to T by p qv = Tp for same integer q. Moreover, we also have mA =T P Now we will also assume that A can be divided into an integral number, s, of intervals of length v. Thus, we may write A = s v As a last bit of notation we shall write c and ck 1 as c = fv and ck_ = jv First of all we consider the case ck c[ ATp), or equivalently the case je {s..,..., q-1. Equation B.2b applies here. The result is that Pr[6k= -vI/k-1 =iv,... 1 =cl] 0 ft j - s (B. 8) 1 I =-j-s

-274Next consider the case 0kle[O,A). In this case je {0,..., s-}. If we reason in precisely the same way as in the continuously arriving pulse case for 0-k_- [ 0, A), then we conclude Pr[0k =Ivlkl =. V, I01] (B.9) 01]k-l'^ *- e^ (B.9) 0 < q-s-j = (l-a) a (q) j+q-s < <q as-j = a3. =q Finally consider the case Ck_1 = T. In this case, j =q. Again the reasoning is the same as in the corresponding continuous arrival time case. The result is Pr[ok = 2 v|0k = j *... 0 1] (B. 10) 0 < q-s ( -a) a-(q-s) q-s< <q |a =q The conditional probability law in the discrete case is given by Eqs. B. 8, B. 9 and B. 10. Again this law satisfies the Markov property since the right hand sides of these equations depend only on the value of Ok

APPENDIX C DERIVATION OF THE GENERALIZED DENSITIES FOR THE PRIOR PROBABILITY LAWS In this appendix we obtain generalized densities for the prior probability laws of Section 2.3. The procedure will be to find a-finite measures i (dOk) such that II(d0k ) << t/(dk ). As a preliminary we review the definitions of absolute continuity and a- finite measures. Let w and /i be two measures defined on the same measurable space (Y,WR). Then w is absolutely continuous with respect to M if given any measurable set A eC such that W(A) =0, then co(A)=0. The measure AJ is said to be a -finite if there exists a countable partition of measurable sets Aie^S, i= 1, 2,..., such that u(Ai) < oc for all i. We return to the problem at hand. Consider the discrete arrival time case. In this case the relevant probability laws are given by Eqs. 2. 17 and 2. 19. These laws assign positive mass only to the points 0k = f v; f =0,..., q. Thus, if we define iu(d k) to be the measure which assigns unit mass to each of the points 0k = v; 2 =0,..., q, then clearly these laws are absolutely continuous with respect to i (d0k). (/(dOk) is often referred to as the counting measure.) Moreover, if A is any measurable set then -275

-276Pr[Ok Al0 k-l=j] Prok= - Vk-lO = jV] f Pr[Ok = vOk_1 = jv] (dOk) A and similarly Pr[0keA] = f Pr[0k= ^vlOk-l =jv](d k) A Thus, by the Radon-Nikodym theorem, we can take the densities P(0k= 2V0k-1 = jv)' P(0k =.v) to be the functions P(k = v) = Pr[0k = v] (C. la) and P(Ok= v lk-1 j) = Pr[Ok = v k1_ j] (C. lb) Next we treat the continuous case. In this case the relevant probability laws are given by Eqs. 2. 16 and 2. 18. Consider first the unconditional distribution function of Eq. 2. 18. This function defines a probability law that consists of a mixture of a probability law that is absolutely continuous with respect to Lebesque measure on [0, T ) and a probability law which assigns unit mass to the point k =T. Thus, we may define 4u(dOk) by

-277i(ddk) = dOk + jT }(dok) P p -where d k is the Lebesque measure and e{Tp (dok) is the measure which assigns unit mass to the point 0k = T. It is then easy to conclude from the Radon-Nikodym theorem that the generalized density P( k) with respect to Y(dOk) is given by [Tp+( al)] -1 k [0 Tp) P(k) = (C. 2a) (a )[T +(a )] -1 =T Finally, consider the conditional probability law defined by Eq. 2. 16. Suppose first that 6ke [0, A) or ke {Tp. In this case the conditional probability law is again absolutely continuous with respect to /i(dOk) = dOk +{ T} (d0k) and it is easily seen that we may take ( k ) to be the unction P(B, |,._ ) to be the function

-278P(eklk-1) = (C.2b) o0 ki[k-1_+Tp-A T ] -ar (sk-[sk_+T -A]) are kP e [Ok 1+Tp-A, Tp) OklE [O,) ae 0k k-1 p p k-1c- [0'l - (A - k ) k p e 0k = Tp o Okr[Tp-A Tp] -a (Ok-[T -A]) 8~a e Pk E[Tp' k-l p -a A e Ok=Tp Ik = T This function is illustrated in Fig. C. 1 below. P(kl okl) T-A Tp Ok T - TP k (a) (b) Fig. C. 1. The density p(0,| ( k ) for O, e [O,A)U{Tpl (a) ki,_u,)e ) b) k-1_, = T

-279Next suppose that 0 k- [A, Tp). In this case, the probability law defined by F(ck ck-1) assigns unit mass to the point 0k = Ck-1 - A. That is, if we define the measure E{c (dOk) to be the measure which assigns unit mass to the point c, then for 0k l1 [a, T ) f(dOkl k-l) {k1- AL (d0k) Now the difficulty here is that it is not possible to find a a - finite measure p(dOk) which does not depend on 0k 1 and such that e (dOk)<< /(d k) for all 0k e [ALT ) k A( k k Al Thus, in this case, it is not possible to find a generalized density P( kk1 ) = II(d0kl0kl) /L(dOk) for which / (d0k) is independent of 0k,1. The results of this appendix can be summarized as follows. Define the measure II(dOk) by 11(dOk) = l[0 T ) (dk) + e T}(d k) (C.3a) where

-280[, (d ) d k in the continuous case (C. 3b) q-1 E.j }(dsk) in the discrete case =0 Then, in both the continuous case and the discrete case there exists an unconditional density p(0k) such that fn(dOk) = p(0k) i (dok) (C.4) The density P(0k) is given in Eqs. C. la and C. 2a. In addition, for 0kE [0, A)U { T}, there exists a conditional density p(ok 0k 1) such that P(okk-lkl) (l(d k) 0ke [0, A)u{Tp fI(dokl0k-l) -= (C. 5) e{ 0k A dlk (dk) k Tp The density p (O k0 L, ) is given in Eqs. C. lb and C. 2b.

APPENDIX D DERIVATION OF THE PRIOR PROBABILITY LAWS -- PART II In this appendix we show that for both the continuous arrival time case and the discrete arrival time case the unconditional density Pk (k) is independent of k. Now the most direct approach would be to simply perform the necessary computations separately in each case. However, an alternate procedure is to first derive an expression for the unconditional density that is valid in both cases and then evaluate that expression to obtain the desired result. This procedure has the added advantage that it automatically provides a result that is fundamental to the proof of Theorem 4. 6 in Chapter IV. Thus, this alternate procedure will be followed here. To begin, it is noted that since IT(dO) is Markov with a stationary transition density, it is only necessary to show that TI(d02) can be described by an unconditional density p2( ) which is equal to the unconditional density p1(.). We proceed as follows. The unconditional distribution II(d 2) is related to II(d02 01 ) and f(d2 ) by nI(d02) = f (d020 1) T(d01) (D. 1) Now from Appendix C we have -281

-282I(d 1) = p(01) /i(d 1) (D.2) where l (dl ) = o[0, T (dS1 ) +e Tp(d ) 1 and dO 1 in the continuous case 1[0, T (ds0)1) q-1 j 1 z jv(dO ) in the discrete case Now substitute Eq. D. 2 into Eq. D. 1 to obtain n(d 2) = f P( ) n1(d02 01 )A(d1 ) (D.3) 01 The basic procedure to be followed here is to derive a relation of the form f n(d 2) = f [ ](d2 ) E A for all measurable sets A. Then by the Radon-Nikodym theorem we can identify the bracketed quantity on the right hand side as the density P2(02) with respect to the measure i(d02). This procedure will be used to obtain desired result as a special case of a more general theorem.

-283Theorem D. 1. Let f(01, 02) be any non-negative function that is,(') measurable in 01 for each 02. Define the measure w(d02) by w(dO2) = f(02, 01) II(d02 01)t(do1) (D.4) O1 Then the Radon-Nikodym derivative of co(d02) with respect to I(d02) exists and is given by w(d 2) wIi(d02) I= 0) Sf f(02, 01)P(02O01)u(d01) IT -, T ] ~ (0,)u T p Tp (D. 5) + I (02) f( 2' 01 = 2 + A) [0, Tp -A) where o, 02-(Tp-A)] 2 Tp -A Tp) O1( 2) [0, A) 02 = T Proof. We begin by breaking the integral in Eq. D. 4 into two parts to obtain

-284w(dO2) = f(02' 01) I(d021 1) [O, A) U Tp} (D. 6) + f (02, 1) II(d22 0 )#(d 1) [ Tp) Now note from Eq. C. 5 of Appendix C that for 02c [0, A) U {Tp}, nl(d 2 0 1) is described by a density p( 2 0 1) with respect to iL(d 2). Thus we may write nT(d0 201) = P( 0201) lM(d02) o [O, ) U {Tp Also from Eq. C. 5 fl(d02 01) = e lO A(d02) 0 [A T p) Substitute these expressions for II(d02 101) into Eq. D. 6. The result is w(d02) = f f(02' 01) P(e2 01) 1 I(d).(d2) [0 A)u Tp } + f f( 2' 1) E 0 1 - d (d a 2) (dO 1) [a, Tp) Now let AC 2 be any measurable set. Then 2

-285f w(d02) = A [ f(02, 0 1) p(o02 1)(d0 1) (d 1) A A [0~ A) U {Tpi (D. 7) +S f(02, sl)EI0l-A}(d02) (d 1) A [A,Tp) In Lemma D. 3, which is proven at the end of this appendix, it is shown that the second integral in Eq. D. 7 can be written as f I (02) f(02,' 1 = 02+a)(d02) A [O T -A) P Thus we can rewrite Eq. D. 7 as J w(d02) (D. 8) A = f f(02, 0 1) p(02 1) (d 1) + (02) f(02, 0 1=02+ ) (d 2) A [0,A) U {Tp [0, Tp-A) Then since Eq. D. 8 holds for any measurable set A, we may take the Radon-Nikodym derivative w(d02)/p (d 2) to be the function w(d02) gj(dd2) - f( 2, e1) p(e02e01)(d1)+ I (02) f(2, 0 2+) [0, A)U {Tt [0, T - A) pocompleteth proo, we (D. 9) To complete the proof, we examine in greater detail the first

-286integral on the right hand side of Eq. D. 9. This integral can be written as r f(02, 01) P(2120 1) (d02) [0, A) (D. 10) r f(e,, sl)P(e^ 6^1)1(dBl)+ J f(02' Sl)P(B 1 d1)M( 01) [O,) ) The second integral in Eq. D. 10 can be evaluated immediately as follows: f f(02, 01P) P(0201)L(d01) f f(82, 1) p(02 j1) T (dOl) ~Tp} g{ p} P f(2' 01-=Tp) P(21 0 =Tp) Now note from Eqs. C. lb and C. 2b of Appendix C and Eq. 2. 17 of Section 2.3 that for 1 = T the density p(2|01 =Tp) is non-zero only for 02 satisfying T -A < 0 < T. Thus we may write this second integral as S f(02, 01) p(02 01)(d01) = I (02) f(02 01 Tp) p(02 01 =T) IT I} [T -A, T ] (D. 11) Next consider the first integral in Eq. D. 10. Here we note from Eqs. C.lb, C.2bandEq. 2. 17 of Section 2.3 that for 0 [0, ), p(02j 1) is non-zero only for 02 satisfying

-2876 <0 <T 1 - 2- p This region is shown as the cross-hatched area in Fig. D. 1. 02 02 = 1+ T-A 72 0 1 p A_0 Fig. D. 1. The region in which P(0201)' 01E [0, A), is positive From this figure it is clear that the integration with respect to M (dO ) need be performed only for 01 satisfying 0 < 01 02 -(Tp 02e[Tp-A Tp) and 0 < 01 < A 02 =T -- 1 2 p Thus, if we define the subset 01(02)c 01 by [0,2 2-(Tp-A)] 2E[Tp-A Tp) p p 01(02) - [0,) 02 Tp then we may write the first integral as

-288I (02) f f(2' 01)P(02 1)j(d 1) (D. 12) [Tp-,Tp] 01(a2) Now substitute Eqs. D. 11 and D. 12 into Eq. D. 10 to obtain f f(02, 01) p(02 01 ) (d 1) [0, A) u Tp = I (02) 1 f(2' P(2 0l) (d) [Tp -A Tp] O1(O 2) + f( 2' 1 iT ) P( 0 -1Tp) or f( 2, 01) P(0 206)1i(d01) (D. 13) [0, )u ITp = I (02) f f(e2, 0e) p(o02 e )(de ) [Tp - Tp 1(2) + ( f(02' ) P(o 02J 1)(do 1) Tp I Substitution of Eq. D. 13 into Eq. D. 9 and combining the integrals over 01(02) and {T p into a single integral completes the proof. The result of interest in this appendix is provided by the following corollary.

-289Corollary D. 2. P2(0 2) = I (02) [ P(f \ )pde=T I2 ( L P (02101) pl(0l) /(d 1)+p(02 lOrTp) Pl(01TTp [Tp-A Tp] (0 ) + I (02) p1(02+A) (D.14) [0, T -A) Proof. This result follows immediately from the above theorem by taking f(0O, 1) =pl(01) and then rating that T(d02) = c(d 2). To compute the density p2(02) from Eq. D. 14 we first need the density p1( 1). In the continuous case it is quickly concluded from Eq. 2.9 in the text and the definition of 01 that 1( 1) is given by (T + a ) -1 O < 01 < T P -P p1(01) ( + (D.15) La-\-1 -1 0 T a +(Tp+a ) 01 -T In the discrete case we appeal to Eq. 2.12 in the text to conclude that [q + a/(l-a)] 0 < 01 < T 1(01) - (D. 16) [a/(1-a)] [q+a/(l-a)]1 0 = T p

-290In both cases we have K1 0 < 0,1 < T P2(01) = (D. 17) K2K 0 =T where the definitions of K1 and K2 are provided by Eqs. D. 15 and D. 16. Now substitute for p( 01) from Eq. D. 17 into Eq. D. 14 to obtain K1 02 [0 Tp-A) P2(02) = K1 P(o02 01)L(dol) + P(02 1 =Tp) K2 0 1(02) 02 LT p p (D. 18) Now for 02e[0, Tp -A), p2(0) = K which is the desired result. The next step in the derivation is to show that for 02e [T - A,T ), the bracketed quantity is equal to unity, for then it follows that for all 02eLO, Tp), p2(02) = K1. Finally the derivation is completed by showing that for 02 = Tp, the bracketed quantity is equal to K2, for then we have p2(02 = Tp) = K1K as desired. We shall treat the continuous case first. For notational

-291convenience we shall denote the bracketed quantity in Eq. D. 18 as J(02). Suppose first that 02e[T -A, Tp). Then substitute for P(02 | 1) from Eq. C. 2b of Appendix C into the expression for,(02), and note that for the range of interest, (d02)= d02. The result is 02-(Tp -A) -a( 2 -(0 1+T )) -))a( 2 -(T -)) -(2)) 2-(T -A))-A)) = l-e +e 1 Next suppose 02 = T. Then, if the expression for p(02 = Tp 0 ) p p 1 from Eq. C. 2b is substituted into the expression for Hi (0 = Tp), the result is A -a(A-0 1)aA (02-=Tp) -= f e dO + K e -) + K0e — = (1e + = K2 since -1 a1 = K2 2~~~~

-292Now consider the discrete case. Suppose first that 02E [Tp -A, Tp). Then by substituting for p(02j01) = Pr[02 01] from Eq. 2. 17 in the text and noting from Eq. C. 3b of Appendix C that q-1 g(d0) = 2ld J =1 cjjv}(d 2) we obtain j=- -(q-s) =(02) = (-a) a (j+qs) + K2(1-a) a (-) j=0 Q -(q-s) = (-a) a (q-s) (a-1 + a-(q-s)+ 1 j=0 -(q-s) + 1 -1 ( -a) a -(q-s) 1 - (a + a -(q-s)+l = (1-a ) a + a (1 -a -a-(q-s)+l 1 a-(q-s)+ = 1 Finally suppose 02= T. Then the substitution for P(02 = T 01) from Eq. 2. 17 into the expression for'6(02 = T ) yields P

-293j=s-l s-i S ~(O2) = E aS J + K2 a j=0 s+l - a z. (a-1 + a j=0 s (1 - a-S) aS+l -1 1-a (1 - a-1) 1 - a a -a = K 1-a 2 This completes the derivation of p(02). To complete the logic of this appendix we need to prove the lemma used in the proof of Theorem D. 1. Lemma D. 3. Let f(02, 01) be any function which is measurable with respect to i(d0 1) for each 02 and let A be any measurable subset of [0, T). Then A L P 10 - Al (d0 2) f(02' 0 1)M(d )] A [A, Tp) (f I (02) f( 0 1=02 +A)(d02) (D. 19) [0, T -A) P Proof. Write the left hand side of Eq. D. 19 as

-294T) [A f(02, 1) eE }(d0 ) 2 (dO ) [Al T Now consider the inner integral. We have 0( 2 - 1 -z' 1) 0 - A S f( 02, ~1) E1 1-A(dO2) - [A 2I 1otherwise or equivalently f f(2, 0 1) e:l( A 2(d02) = I(A+}(0l)f(02 =' O1) where A+A =!y'|y' y+A for ye A Thus, the total integral can be written as f ITA+A}(O1) f(02 0 1 -' 0 1)(d01) [A, Tp) = f f( 02 - A, 01) / [0 T )(d 1) (D.20) [h Tp) n (A+A) pw where we used the fact that on [0, Tp), Il(do1) = [, T)(d1 ). Now make a change of variable by letting

-29502 1- 1Then we have [A, Tp) [0, T -A) A+A - A so that [A, Tp)n (A+A) -[0, T -A) n A Furthermore since i [O, T ) is either the Lebesque measure or the p counting measure it follows that for any set A/C [0, T- A) p [0 T )(Al [0 T (A) Thus we may write the integral in Eq. D. 20 as f (02 01 =2+ A)[0,T )(d02) [0, T -A)n p = S I (2) f(02, 01='2+A) (d'2) [0, Tp-A) which is the desired result.

APPENDIX E EXPRESSIONS FOR THE RATES r, r AND r In this appendix we calculate the rates rp, rN and r The temporal relation of the random variables 0. is shown in Fig. B. 1 of Appendix B. It is evident from this figure that N (0) can be written as N-1 Np(0) = I (0i) +I (N) i=m [o~ ) {N6o [0,T )} Taking expected values and dividing through by T results in EINp(O)} r = p T N-1 } E E I (0,) +I ( N) i=m 0 itE[O,)j ) {0 Ne[0, Tp)J or -296

-297(N-m) Pr[i0eO, A)] + Pr[ 0N[0, T)] r= (E. 1) V T Next, from Eqs. 2. 18 and 2. 19 of Section 2. 3 we note that in both the continuous case and the discrete case we have Pr[ ONeO, Tp)] = m Pr[Oi [O, A)] (E.2) Now substitute Eq. E. 2 into Eq. E. 1 and write T as NA. The result is (N-m) Pr[0iE[O, A)] + m Pr[Oie[O, A)] r. = p NA or Pr[o0e[O, A)] r = P (E. 3) p A Finally, divide numerator and denominator by m and note that Tp =mA. Then the resulting equation for r is Pr[ e[O, T )] r = (E.4) p T Next expressions for rN and rS are derived. From the definitions of NN and NS in Section 3.2, it is seen that NN(0) and N( 0) can be written as

-298N NN() = I (0i) {0. =T 1 p N NS(O) = 1 I (0i) {0ic [0O Tp)f' p Dividing by T and taking expected values results in N Z Pr[0 = T ] i=1l i p rN = T N Z Pr[0 i[0, Tp)] =T S or N Pr[O =T ] Pr[0 =T ] N NA A(E. 5) and NPr[0e[0,T )] Pr[0e[O0T )] = -_ N_-_ — - (E.6)

APPENDIX F THE PROOF OF THE MOST RECENT RESPONSE RULE In this appendix we prove that each performance point in the set 9? can be obtained by a decision device that satisfies the most recent response rule. Theorem F. 1. If (Y1, Y2, y3) e jT(m), then there exists a 6e q (m) such that y1 = Rx(6), Y2 = RX(6) and y3 = RD(5) and 6i (xi, ai- ) = 6i(xi Ti) Proof. Since (yl, Y2, y3) c YT(m) implies that there is at least one receiver 5 which has y1 = RX(6), Y2 =RF() and y3 =RD() it is sufficient to show that rx, rF and rD can be written in terms of a decision rule whose component rules can be expressed in the form ai = (, To this end consider first the expression for rD(6): E fi (i, ai ) i N xI V {a.=l, T. < 0 i 0ie[0,Tp)} rD(6) =- i9 T -299

-300The ith term in this sum can be written as P1 = E I (7i a. 0i P'i. i i) X0 ) x, i1 Ti< il SiE[O, T )l But a. = 6.(., a. 1i' i - 1.4 and Ti Ti(a- 1) Ti (i-1) Thus, the indicator function in the above expression depends only on x. and 0.. We thus have 1 1 PD = E fI (r., a, 0.) i ( LI i i) i' i {ai 1, Ti < i' 0ie[O, Tp) E E (E (Ti, ai, 0i) 1 i-1 i 1xl i- i a= Ti ~< i' 0iE [O, TP Now, since T. takes on one of the m values, 0, 1,..., m-, the space of observations up to time Ti-1' X can be partitioned into the sets Xi -l i-llt= ij=A j =0,..., m- 1 Now, for each x. define the response set Ai(x, Ti= jA) - x= xIa= xl Xi-ei Xi Xit

-301Note that in terms of this set, the decision rule can be written as 01 xi Ai(X-_1', i) i' ai-l'.., al) =4 1 xi iAi- (-1' i We are now in the position to evaluate the indicator function. Fix.e [0, T hen, for x E X' such that jA > i0, Ti > i', sthat p. i-i i-I 1 1 I (ri, ai, i) 0 lai=l, 1 Ti<, ei [0 Tp)' On the other hand, for xi. Xi such that jA < 0i, Ti < 0., so that 1 x.EA'r. =jA) 0 1 xic Ai-i-l' i ) {ai =1, i < Oi',~[0~, Tp)} 0 x1i/ Ai(X, TjA) Thus, we can write

-302P I (, (i a 0.) P(dx X,. 0) t L X. i ii- i p I i [o p [X i ai=L 0E T )t ii x P(dxi-|0 II (d0i) =S~0 [ C il 1 IP(dx X 1 ) ] P(dxxiI. e )le(d ) 0ie[0,Tp) T j)jA< A0( T )1 1 (F. la) The ith term in the expression for rX(6) can be determined in the same way, the only difference being that instead of summing over j ) jA < 0i we must sum over j ) j A > 0i. The result is F m- 1r-1 f i X P i A) (F. lb) Finally, the ith term in the expression for rF(6) can be written as P = f T i S 1) ] i i() n(d i) - i [ =T i) njj - 1di [~i- 1' p X i-l I A) (F. lc) This completes the proof since

-303X EAW 1' I 0.x f A (W, -1) implies that 6i (X~= 6.(, ) ii, -l) i(x i, 1 and thus 6. depends only on the time of the most recent response r. 1 1

APPENDIX G THE DERIVATION OF THE RECURSION RELATION FOR THE BAYES DECISION RULE In this appendix the recursive procedure for determining the Bayes decision rule is derived. The Bayes decision rule is obtained by finding a 60 that satisfies r(6o) = inf r(6) (G, 1) where E L(6(x),0) r(6) = xO (G. 2) We begin by developing an expression for r(6). From Section 2.5 we have N L(a-,') = L.(r a. 0) i=1 i 1 1 i1 where from Eq. 4. 1 1- - A decision rule 6 is an n-tuple 6=(6i,., 6N) where is a function of the form -304

-305a. = (x,. 1 1 1 where a. = (a,..., ai). Now consider the function L(6(x), 0). From the above definitions it is clear that N L(6(x),0) = L i(Ti, 6i(xi, ai), 0i) (G.3) i=1 where T! -T(6 (^X. T.(G. 4) Substituting Eq. G. 3 into Eq. G. 2 and distributing the expected value operator over the sum results in N r(6)= Z E X 6 a(^. 0/T X 0 i(Ti' i( i' i-)' i}/ Now consider the ith term in the above sum. From Eq. G. 4 it is seen that Ti depends on the past decisions only through 6i-1 Therefore Li(, *, ) depends only on. Moreover, since 6. depends only on the observation up to time i' xi, the random variable Li (*,,) depends only on xi and 0. Therefore, the expected value need be taken only over X. x O.. Thus, we may write N r(a) = Z r (-) (G. 5) i=l

-306where ri( ) = E {L(Ti, i(i ai), i)} (G. 6) X., 0. 1 1 With Eq. G. 5 for r(5) at hand, we may proceed with the derivation. We seek 5 = (61 6' ) such that'.,' N First note that by a well known property of the infinium the above minimization can be carried out over individual components one at a time. That is, inf r(6) = inf inf r(6 6N ) 6 61., 5!'..., 6k N Now substitute for r(bl,..., 5N) from Eq. G. 5 into the right hand side above. The result is N inf r() = inf inf L ri(6i) 6 61' k-1 k'., 6N i = or

-307k-h1 O N _ inf r(6) = inf Z r(6 )+ inf r(6) (G. 7) 657b i=6 =k 6k-l k N where 56k = (5., 5k) k-16 1( ) 6 k-1 [+ This last equality follows from the fact that, r(6.) does not depend i= on 6k,..., 5N Now define the function Gk( kl) to be the second term in Eq. G. 7. Specifically, 6 [k ri(i j' NN Then, in terms of Gk(. ), Eq. D. 7 becomes k- 1 inf r(6) = inf r( )+ G(6k 6 61 *. i=1 B1, Next we obtain a recursive relation satisfied for the functions Gk(' ). From the defining Eq. G. 8 we have = N Gk( k-1) inf inf r i(5i) 6k k+l' 5 P N Li=k ~N inf r ()+ inf r (5,) 5k k+k But

-308N 5 inf ri(6i) Gk+l (k) 6k+1, *** N i=k+l Thus, Gk("k-1) inf rk(k) + Gk+(6k) k Equation G. 9 provides the basis for determining the Bayes rule 6~ and the minimum Bayes risk r(60). First 6N is determined from GN(N-) = inf rN( N) N nf E ILN(N 6N(N' N-)' ON) N XN' N or GN(6N1) = inf E{ LN(TN N(XN' aN-l)' ON) N xN aNXN N (G. 10) The procedure is as follows. Recall from Eq. G. 4 that N -= (ON-l(XN-1' N N-1

-309so that for f- fixed, T is a fixed function of the observation XN-1 Moreover, aN-1 depends only on xN_1 through 6 N-(xN-1)=aN-1. Thus the inner expectation in Eq. G. 10 depends only on xN and 6N. It then follows by the standard argument of Appendix A that the 6N that minimizes Eq. E. 10 is given by 1 if E - LN(TN' 1, ON) < E ILN(TN,O, 0N)t 0Nl N -N xN -~ (xN, aN-) - 0 otherwise The function 65 can be written in a more convenient form in terms N of indicator functions as follows. For each XN- e XN_1 and TNe 0,..., m-1n define the set of current observations AN(XN-' TN) XN by AN(xN-1I' TN) N E ILN(T N, 0, ON) ON XN E I x L 1N(,rN, 0N)} > 0 =) l~l^^N^N-^ =^(GC.11) = {XN16N (XN' aN-1 ) (G. 11) Then 56 (, ) can be written as (N aN) = (x (xN) ~ -'- A [i:- J —'~(X

-310or 6N ( NN) A= (XN TN) N) (G.12) with the last equality following from the fact that 6N ( *, * ) depends on the past decisions only through the function T Next we obtain an expression for GN(6N_1). First it is noted that since 6N (XN, TN) minimizes GN(6N ) we have GN 1(6N) =1 E E LN( 7N,6N (xN, N), N) XN 8Nl N X 0 NN X N -f E I LN(?^N, N 1O - X1 XN ON1xN XNNo wE LN(T N'N (XN TN 0 N) P(dxN txN -1) Now use the fact that Now use the fact that

-311E LN(TN'6N (XN TN) 0 NX N N N lxAN(xN j EN n N-X AN(xN_ N) where N( N-' 1N)= AN(XN-' N) to write GN(6N_1) as G =0 NN' ) N P(d XNXN-l (N (G. 13) xN- 1 f x IAJJ N-i L AN(XN1, TN) where RN(XN' TN, j) E ILN(TN, jN) NI(XN The function 6N 1 is determined next. This function is defined by GN_(6N_2) = inf [rN-l(N-l) + GN(6N1)] (G. 14)

-312Substitute for rN-1('N1) from Eq. G. 6 and for GN(6 1) from Eq. G. 13 into Eq. G. 14 to obtain GN 1(6N_2) inf E E I LNl( N -1Nl N-1( N-l'aN-2), N-1) N N-1 O1 N-1IXN-1 N-1 + E R Jlo "N(x-N [6 a T f (G. 1 5) RN- 1(X N- 1 N- 1' N- 1(XN- 1' aN-2)) 0N-i EN-1 XN N N-N- 1(N- 1 N-2) 1 + ^ RN('NX N-1' -l- N-1]' ) P(dxa-2) A Nf- I(.-NN r~ N".- a x,lN 1 NkAPxN[6N(xNaN-)TN-1)

-313Now apply the standard argument to Eq. G. 15 to conclude that N-l(XN-' aN-2) is given by 0 X (N' 1 (G 16) 6N-1(XN-1' N-2) N-1 N-1N-) (G. 16) if RN 1(XN-,' N-1' 0) > RN- (xN_1'TN- 1'1) SO otherwise As before we may express 6 in terms of indicator functions by AfeyrN-1 defining AN- 1(XN-2,'N- 1 by A Ni-xN2,'N-i) = IxN161'N- = N- 1 N-2 N- 1 N-1 N-1(N-1 N — 1) and we may use the sets kA (XNN, TN-1 to express GNN_('-2 as GN- 1(6N_2) 1 =0 E C f RN-l(XN-lN'N-li N ) P(dXN-llXN_-l) XN-2 LN-21 AN-2(XN- 2' TN- 1) With the basic character of the recursive procedure established we may conclude that the kth decision rule,6(x, ak l) is given by

-314(xkakl) k(k,k) (G. 17a) Rk(xk, Tk, 0) < Rk(xk Tk 1) = 0 otherwise where Rk(xk, Tk k) (G. 17b) = E _ {Lk(kk' j k)} 0klXk 1 D+ Z Rk+l(Xk+:,1Tk+l [Jik k1]' ) P(dkl xkl) Ak+l(xk, Tk+l[ik, Tk-1]) and where Ak+l(Xk', k+1) Xk;6k+1 (Xk+, k+ ) = (G.17c) A formal proof to this effect can be obtained by the method of induction. The procedure is to assume the result holds for 6 ~...' and then to show that the result holds for 56 by using precisely the same argument as used to obtain 6 ~. We shall not N- t t go through the steps of this argument here.

APPENDIX H DERIVATION OF THE UPDATING EQUATION FOR THE ODDS RATIO DENSITY In this appendix, the theorems of Section 4. 3 are proven. We begin by first proving three lemmas. The first lemma establishes a formula which is useful in proving the second and third lemmas. These last two lemmas are then used to prove Theorem 4. 4. Lemma H. 1. If x = (x1,..., xN) is conditionally independent for each 0e 0, then, for j < i < k, P(dx..., dxi [k,..., ) f P(dxl,, dxi01,... i) n(dOj_l...,do0k'..., j) O. x... xO j-1 1 Proof. By Bayes law, (dk',...,do X) P(dx,..dx i P(dxl...., dxilk,...,O) = In(d0,,k" dOj) JK 3 (H. 1) Write n(dk,..., dojxi) = S n(dOk,..., dslx ) Ojlx... xO1 -315

-316or (d k,., d jlxi) (H.2) P(dx |.., dxi0k **,' 1) II(dk,..., dO 1) P(dx,...,dx.) o.jX... xe 1 i But P(dx,...,dxi Ik,..., ) = f P(dxl,...,dxk lk,.., 0 1 Xi+lx... xXk Xi+x... xXk k X X i+lx. Xk k,' *', 0i) i II P(dxQ 0 0,,1) or P(dx...,dxi [,...,01) = P(ddx1,.,d10 i... 0 (H. 3) Substitution of H. 3 into H. 2 gives

-317n(d k,... 0,d j ) P(dx l,..,dx ( i,.' 01) n(dk,..., dO l) ~Ojlx. x.xO1 P(dx,..., dxi) P(dx,dx.) - x 1' 1 Substitution of H. 4 into H. 1 and cancelling the appropriate terms yields the desired result. Lemma H. 2. If x (x x is conditionally independent for each 0, then (dxkdkli) = P(dxk k0 0 -)n(d ) n(dO k- xkl) 1" k0 x k" j — l I x'We begin with Bayes law: We begin with Bayes law:

-318l(d_ dO Ix) =P(dx...,dxklk 0 k-1) I(dk, dok-) n(d0,,d0 k_ 1xk).... k' k-1IR XP(dxl,..., dxk) P(dxk klk - kl' k' k-1) P(dxl, dXk 1 1 0 k- 1) r I (dok' d 1 P(dxkl xk_ 1) P(dXl ~~, dXk_ 1) or (d d x - k kP(dxklXk k' k- l)IIk'1 dok-ll k-1) n(dOk,dOk_llxl,...,xk) -- dkk P(dxk xk 1) (H. 5) Now, consider P(dxk x k_-l,' ok' ) k- 1) = (dxl, ~, dxkI 0k' k-l)/ P(dxl,., dxk l0k, 0k-1) Apply Lemma H. 1 to the numerator and denominator of the above. The result is P(dxk xk-,1' k' k-1) (H.6) k P(dxl.. dxk0k,..,, 01) 1n(d0k_2,..,dsO 1 k 0k-1) k-2. P(dx"'',dXk_1J0k_.,- 0 1 l)I(dk_2',d1 0k' k-1) k-2 But,by using the conditional independence of x1,...,XN, the numerator of (H. 6) can be written as

-319P(Xk k' ek-1) f P(dxl",.. dxk-1 0k-l' 011) k-2 n(dOk_2,..., dO k 0k1) Substituting into (H. 6) and cancelling gives P(dxkxkl, 0k' k-i) = P(dxk k, kl) (H. 7) Finally, substitution of (H. 7) into (H. 5) completes the proof. The above two lemmas have used only the assumption of conditional independence. The third lemma also requires the Markov property. Lemma H. 3. If x = (x1,...,xN) is conditionally independent and 0 is Markov, then fl(dok(Oklxkl) = (dOkl(k-l) Proof. We have _ I(dO,dO 1xk ) k' k-l[ 1 P(dx,.., cdxk_ 1 | 0 k- ) II(dok, dOk-l) = P(,dx:.., d._A- tn I. i K _)

-320P(dxl,., dxkl1k, 0k-l) - POXI,.,x |I k 1) k k -1 The proof is completed if it can be shown that the bracketed quantity is equal to unity. Applying Lemma H. 1 to both the denominator and the numerator of the bracketed quantity results in f p(dxl,.., dxklIOkl,.., O l) f(dOk2,.., dO 1' Ok-i) f P(dxl dxk_l. k-l'' 1) n(d ok-2' d 0,.., k'0 k- ) Q 2x.. x 1 J-2 1 This quantity is equal to unity if (n(d k-2',..., dO'k- 01) = (dOk_-2,., dO 0k) But this fact follows easily from the Markov assumption on 0. This completes the proof. We are now in a position to prove Theorem 4. 4. Theorem 4.4. If x = (x1,..., xN) is conditionally independent and if 0 is Markov, then P(dxk _ k{e 1) n(dokl ) = f k k1 n (dOk 0 k-) (dk X xk-l ) )0 P(dxkl Xk l) k-1i

-321Proof. From Lemma H. 2 we have n(dokXk) = f n(dOk,dOk1J k) k-l P(dkl Ok, k-_) o P(dxklxl,., xk_) "(dk'k-,li -l) k-1 Now, "(fk,- k-l x -i) (d klk-1'X k ) n( _ k-1lxk-l) = I(dkl k1) II(dOk-1 Xk1) Substitution completes the proof. Next we prove Theorem 4. 5. To this end it is sufficient to prove that the function P(dxk j[k' 9k-1) (Xk IXk- 1' 0k'k- 1) P(dxk Xk 1) exists. The theorem then follows as a corollary to Theorem 4. 4. Proof. Consider first the distribution P(dxk Ixk- 0k' 0k ). Clearly, this distribution is absolutely continuous with respect to P(dx k 1il), since the conditioning of P(dxklxkl) by k, 0k-1

-322cannot take away any sets of measures zero. Thus, the Radon-Nykodym theorem can be applied to P(dxklxk_ 1, k' 0k-1) to obtain the function (XkXkt _l, k' k-l) satisfying f P(dxkk-l' k' 0- f k-l'= k''XkXkk-1) k(dxk xk-l) E a for any measurable set A. But, in Lemma H. 2, it is shown that P(dxkl k-l k' 0k-_) = P(dxklk, k- 1 so that S P(dXklk, k-1) fA (xk xk-1 k' ok-l) P(kdxk 1) A A for all measurable sets A. Thus the function f(xk I1kl, k' 0k-1) is also a Radon-Nykodym derivative of P(dxk 0k, 0k-1) with respect to P(dxkl xkl) This completes the proof. Next we prove Theorem 4. 6. Here, it is desired to show that II(dOklxk) << (dk) for all xk Xk and to determine the density pk(Ok xk) with respect to u(dOk). The proof is by induction on k. First, for k=l we have from Bayes law P(dx I01) I(dO I xl) IP(dXs) I( 01 Now, using the fact that II(d 1) = p(G l)#(d 1)'

- 323n(d 111) = P(dx2) p(O 1) (d1) Then, if A is any measurable set of uc (.) measure zero, then A also has measure zero with respect to n(dO 1Jx1). Thus, II(d 1 x1) << A(d01) for all xle X1, and moreover the density p1(0O1xl) is given by P(x1 1 ) P1( 1 1X) P(dx) P( 1 Next, we establish the induction step. Specifically, it is assumed that nl(d k-1 X k-l) << (d k-1) for all xk_ 1 Xk-1' and it will be shown that I(dOk[xk) << (dOk) for all xk Xk. We begin by noting from Theorem 4. 5 that r(doklxk) - f (xklXk-l' i k' ok-1) l(dOk0k-1) x n(dOkl lk-1) k-1 Now, by the induction hypothe sis, we may replace I(dOk 1 Ixk ) by P(0_k- 11Xk-_ )(d0k- 1) in the equation above to obtain

-324II(dOk]Xk) = f [f(Xkl X k-l' k'0k-l)P(0k-1k-1)] k-1 I(dk 0k 1kl)(dak- 1) Now apply Theorem D. 1 in Appendix D for the general index k and with the function f(k0,k_) replaced by [2(xklxkl 0k, 0k-) Pk- 1(0k-i lXkl)] to conclude that the density Pk(0kik) existsand is given by P(Ok Xk) (H.8) -I (0k) f kVlk(XlXk-l' O' Ok-1 = Ok +A) -11 = k+AJXk-1 1[o T -T)t + I (Ok)T (Xklxk-l' k' Ok-1) P(OklOk-1) p(Ok-_1lXkl) [ P p p k-1e k- (k) U Tp A(dok_1) We conclude this appendix with the proof of Theorem 4. 7. Proof. First substitute for p(0klxk) and p(0k = Tpxk) from Eq. H. 8 into Eq. 4. 30 in the text to obtain

-325Ok(k k) I(^k) f (Xk- Xk-l' Ok' 0k-1)P(Ok Ok-1)P(Ok-l Xk-l 1)ul(d k-1) I[T P' -,Tp I6 [, k-(Tp-A)] U {Tp} + Tpk- (xk-l' k' k-1 )P(k-l= k -1 Xklxk-1 k = Tp,' pk-1)PP(k = {Xk-1 )(d k-1) k. e [O, A) u Tp Now divide numerator and denominator by Pk 1(Ok 1 = Tplxk l) The result is 0k(Bklxk) = (H. 9) I T(k) f - (klxk-1l k Ok-l)P(Ok l k- l)Ok-l(Ok-_l k-l)=(d k-l) L[Tp-,Tp)P 0kl E [0, 0k- (Tp- )] U ITp} + ( k) (xk'xk-' k' Ok- 1 = k+ ) 1(k =01 k+ l xk- ) + {[0, Tp-A)} - - I (xklxkIl, k = Tp k-1)P(0k =Tp lk-l)ok-l(ok- Ixkl)(dok-l) k-1 E[O,A) U ITp Equation H. 9 is in the desired form except for the term I(xxk Jk,' k- ) Now, the idea is to break out the dependency on Xk_1 This can be done as follows. By hypothesis, P(dxkk, k-1) << P(dx, k=T, l,_=T )

-326Therefore, by the Radon-Nykodym theorem, there exists a function (xk0k, 0k-l) such that P(dxk 0k, k-_1) P(xk10k, k_1)= (xklOWkk1) = kP(dXk Tk 0 k - Tp ) On the other hand, P(dxk k =TpOk =T T) << P(dxkxkl) by Theorem 4. 5. Therefore, there also exists a function Q (xk(xk, 0k = T 0k- = Tp) such that =(dxk 0k =T 0k- =T ) p' k-1 p (x|Xk-, -T=TpTp' k- = Tp ( kk ~~P(dxkxkl) Then,by the well known chain rule for Radon-Nykodym derivatives, we have f(xkl1X, l0k'0k1)i = -kl o k-1> k( kXk- W B X kk l) P(dxk x_ -1) P(dxk 0k, 0k-) P(dxklk =Tp 0k-1=Tp) P(dx O0 =T 0 =T k|k p' k-1 Tp) P(dxklxkl) or

- 3271(Xklxk-1' V k k-l) (xkl 0k' k-l) (H. 10) *(XklXk1 Okk- = T 0 k -T ) Now substitute for (xk IX,'k' k-1) from Eq. H. 10 into Eq. H. 9. Next, note that, since the factor (xk, k = Tp = T) does k p'k-i =T ) does not depend on 0k and 0k-1 it cancels out. The result is the desired expression.

APPENDIX I EXPRESSIONS FOR THE ODDS RATIO DENSITY In this appendix, the assumption of additive noise is applied to the updating equation of Theorem 4. 7 to get specific expressions for the odds ratio density. We begin by considering the function P(dxklo k', k)) (Xk1Okk1) k k' k-)1 = P(dxT ) This function is normally viewed as a function on the current observation space Xk with the pulse parameters 0k', k-1 held fixed. But, from Eq. 4.21, it is seen that in order to calculate 0k(k, xk, Ok- )' the current observation xk is held fixed and the parameters kl, 6k are allowed to vary over certain subsets of Ok_ x O. Thus, to implement the update procedure, it is necessary to determine the behavior of (xk 0k, 0k1) for the values of 0k-1,k that appear in the integrations in Eq. 4. 21. Specifically, 0k E [Tp - Tp), k-1 e [0, k - (Tp-^A)] {Tp } (I. 2a) ok=Tp, k- E[OA)U TpJ (I.2b) and -328

-329kE[0OTp - ) k-1= k+ (I. 3) To evaluate (xk| 0k-1) on the sets defined above, it is necessary to specify in more detail the probability measures on the right hand side of Eq. I. 1. It may be recalled from Section 2. 4 that, for additive noise, the observation process is defined by x(t) = n(t) + s(t, 0k' 0k-) tc[tk_,tk) (I.4) The function s(t, 0k, 0k- ) is given by Eq.. 2.21. This equation can be simplified by denoting the pulse arriving at time tk - T by k p pk(t) = p(t -(tk-T )) (I. 5) Then, we may write s(t, 0k, 0 k_) as s(t, 0k' 0k-1) (I. 6) I (0k-1) pk(t-(kl- A)) + I (0k) p (t-Ok) {0k_i E [01 A)' I E T )} {0k- 01 [A3{Sk 6jO~Tp)} Our first task is to restrict s(t, 0k, 0k-i) to those values of Sk and 0k-1 satisfying Eq. I. 2. If these values are substituted into Eq. I. 6, then s(t, 0k_,1 0k) can be written as

-3300 "k_-le {OU{Tp} k =Tp pk t-(tkl-A)) k-1e (0,A) k T k s(t, k,' k_) (t- k) 0k-ie o{0i{Tp} ke[Tp- Tp) pk (t-(0k-lA)) +pk(t-k) -k E (0, 0k(T -A)] Ek[Tp -A'T k k-1 e(~k pi\) k Tp ) (I. 7) where we have excluded the parameter value k-1 =0, since this value does not result in a pulse contribution to the interval [tkl,tk). The function s(t, 0k', 0k ) is illustrated in Fig. I. 1 for the values of 0k and 0k-1 of Eq. 1.2.

-331-T tk-T tk- t Ki1 p k p k-i k P P (a) l-T t-T t/ l k-1 p k p tk-1 tk tk-1 -Tp tk-T tk1 tk (c) Fig. I 1. The function s(t, 0k' k-1) (a) 0k-_1 (0 A), = T (b) 0k-i-=T- 0.klt-A T p ) (c) /k-lE0 EOk- (Tp -)] 0 E[T -A Tp) In the above figure, it is seen that the observation xk in the current observation interval [tkl,tk) is influenced either by a pulse that is present at the decision time tkpk(t-Ok), or by a pulse that is not present at the decision time tk, pk(t-(Ok - A)), or both. In the following, we shall refer to the pulse pk(t- 8k) as the "current pulse", and we will refer to the pulse p(t - (Skl - A)) as the carry-over pulse.

-332Next, we consider the parameter values of Eq. I. 3. If these values are substituted into Eq. I. 6, the result is s(t, Ok' k- ) = pk(t- k) 0k-1 k'k A ke [OTp-A) (. 8) This case is illustrated in Fig. I. 2. k-1=0 k+A tk_- p tk- tk- Fig. 1.2. s(t k, 0k-);0k-e[ Tp -A) 0k=k- +A We are now in the position to describe the function (xkl 6k, 0k-1). First, substitute for s(t, 0k, 0k_1) from Eqs. I. 8 and I. 9 into Eq. I. 4. The result is

-333x(t) = n(t) k_lO {OIU{T } 0 = T ~~n(t) ^k-i1 p k p k n(t) + p (t-(_l-A)) Ok-eE (0,A) 0k = T n(t)+pk(t-ek) kk e }U{T } 0 E[T -' Tp ) - k-. p k p p or 0k-l k+A' Ok[OTp -A) n(t) + pk(t- (Ok —A)) +p (t-Ok) Okl[, [0k(Tp)]- O k [T -pT' (I. 9) Now make the following definitions. Define P(dxkl N) to be the distribution of the process x(t) = n(t) (I. 10a) Define P(dxk |k-1' Sco) to be the distribution of the process x(t) = n(t) + pk(t- (0k1 - A)) 0k-le (,A), Ok = Tp (I. lOb) Define P(dxkl( k' Sc) to be the distribution of the process k x(t) = n(t) + p(t- 0k) 0kl {0} U {T } 0k [Tp -,Tp) p kp p or k-1 Ok + [O,Tp - A) (I. lOc)

-334Define P(dxkl 0k-1', k S, S ) to be the distribution of the process x(t) = n(t) + pk(t- (0 - A)) + pk(t- 0k) (I. lOd) k-i k k-l E ( 0k -(T - A)] 0kE [T - A T ) Then, we may write (Xk Ok, 0k-_l (I. 11) 1 Ok E J01 0U {Tp} k =T t(xkl0k- 1 Sco) 0(0) k=T =- (xk 0k' Sc) 0 k.l{O}u {Tp} Ok e[Tp-ATp) k-1 k k p or Okl= k+A 0kc[0 Tp-A) (Xkl0k',k 0-k Scc) Ok_-1 (O 0k-(Tp A)] 0k [Tp-ATp where (XklOk-l'Sco) =P(dxkOk1,Sco)/ P(dxkN) (I. 12a) ((xk|0k,Sc) = P(dxk|0kSc)/P(dxlN) (I.12b) and (XkOk- 1 Ok Sco S c) = P(dxkl k-l kScoSc)/P(dxkN) (I. 12c) The functions, (Xk Ok_, Sco), (xk k,Sc) and f (xk Ok_-1' k S C,Sc) are "ratios" of distributions describing the observation xk when signal is present to the distribution describing the observation xk when noise alone is present. In classical detection theory, these functions are

-335usually referred to as conditional likelihood ratios. In this context, (xkl k-1 S ) is the likelihood ratio of the carry-over pulse, (xkl0,S ), is the likelihood ratio of the current pulse, and 2(xklOk, k-, Sc Sc) is the likelihood ratio of both pulses. Using the above results, we may now obtain the desired expressions for the odds ratio density. First, we write the update equation for the odds ratio density Ok( k, k, Ok I) in terms of the likelihood ratios. This is done by substituting Eq. I. 11 into the update equation, Eq. 4.21, and evaluating the integrals over the set (k =T). The result is Ok(k' Xk'Ok- 1) (I. 13) (Xkl Ok'Sc) Ok-1(Ok-1 = ok+' xk-1' Ok-2)' k-1' Xk-l' -A) D Oke [O Tpf (Xk k' Sc) P(0kl k-l) Ok-1( k-1'Xk-l' Ok-2)' l(dOkl) {k-1 = ~ + S (xk k'y k- 1 ScowS ) P(k Sk-l) ok-I(2k-'Yxk-l' Ok-2) A(d k-l) k-1 6 (O k-Tp - A)] + l (Xk OkS c) P(0k k-1 = T) D k[O, Tp - A) where

-336D = f P(0k =Tp Ik-1) 0k-l(k_-1, k_, Ok-2))/(dOk-l) {k-1 i } + S (xk _,Sc) p(kk Tp k-1 k-1k-lk- Ok-2) kk-1 (0, ) p k-1 p + P(Ok=TpI0k.i=Tp) The next step is to substitute for p(Okj 0k-1) from Appendix C. This is accomplished in the continuous case by substituting P(Ok 0k- 1) from Eq. C. 2b and integrating with respect to the measure,u(dk) = dk + {Tp (dok) and in the discrete case by substituting p(0k = 0k- 1 =ji^) from Eq. C. 1 b and integrating the measure with respect to q (dk) j Z v(d k) In the continuous case, the results are

-337Ok( k XkOk- 1) = (I. 14) (X1k ok Sc) at ea Ok- l(k- 1= k + AXk-1' Ok-2) x A... —0 L (Xk Ok-,Sco) e Ok-( k-'k-( k-2) dk-l+ 1 kE[O, Tp-A) a(Tp- ok) ae 0k- (Tp- A) a 0 (Xk k' ok-lScoSc) e k-_l(Okl Xk-_l k-2)dOk-l+ (xk k'Sc) L'(x { klk-1 Sco)) e k-l(O k-'Vk-k-2)dk —1 k E[Tp-A Tp) For the discrete case

-338Ok(Ok = v, xk) = (1.15) (Xk 1 k = VS ) aS. 0k-l(0k-l =k+s Xk-' Ok-2) D e {0,..., q-s-l} (l-a) a-[q-] e(Xklok=|,SC)Ok((0k=0, O_0 2) (xk 0k = V Sc) k-1 ( = Xk- l' k- 2 j=- q-s) + l(Xk IOk=V Ok-l=j'S coSc) j=l a-j Ok-1(Ok-_ =j, Xk-l' k- 2) + (Xkk 0k=v, S ) D 2 e {q-s,...,q-1} where S-1 D = k-l(0 k-l= Oxk-l ok-2) + S[(xki k-l= jv Sco) a- Ok-l(k_ iv,XklO'k2)] + 1

-339In concluding this appendix we will consider the special case of white Gaussian noise. In this case, it is well known (Ref. 10) that the likelihood ratio of Eq. I. 12c, l(xklkO, k 1,So Sc) can be written as tk tk expt N o t x(t) s(t, k', k_1) dt - 2N s(t, Sk' Ok-1) dt ~ 1 0 ^1 Jtk- (I. 16) where N is the noise power per unit bandwidth and s(t, ok' k) = p(t - (ok - A)) + pk(t- ) (1.17) kk' k-I k-. k Now, by definition, the pulses pk(t -(k 1- A)) and pk (t - k) do not overlap, so that k k p (t-((k-1A)) p (t-k) = 0 for t [tkl,tk) It then follows that s2(t, 0k, Ok 1) = p(t -(0k_1- A)) + P (t- k)2 (I. 18) Thus, we may substitute Eq. I. 17 and Eq. I. 18 into Eq. I. 16 and separate the terms involving pk(t - (k 1 - A)) and pk(t - 0) to obtain

-340tk tk exp {N1( f x(t) pk(t-(k_ l-A)) dt- 2N f pk(t-(k -A))2 dt t~ -1 o tk 1l tk- d tk tk x expi I x(t) pk(t- )dt- 2N ( / P tk2 dt N o tk_2 O t k But, in the white noise case, we also have (Xklk-l'Sco) = tk tk 1exp fk x pk))t 1 f pk(t_ (o 2-A)d expi N t x(t) pk(t- (Ok-1-i)) dt- 2N (t- (k —)) d t0 -10 tk-1 (I. 19) and (xklOk, Sc) tk tk exp {N1 S x(t) p (t- (Ok) dt 2N S p (t- ok) dt) (I. 20) o t -1o tk-1 Thus, we may conclude that (X\,., 01 o, S, Sc) = (xl ok,_1, S) (^xk 0o, Sc) (1. 21)

-341The update equation in the white noise case can be obtained by substituting Eq. I. 21 into Eqs. I. 14 and I. 15.

APPENDIX J THE PROOF OF THEOREM 4.10 In this appendix, we prove Theorem 4. 10. To do this, we must show that Tk(Xk' k) - Tk(Ok-' k' ) (J. 1) Rk(Xk' k' k) = -Rk(O k' ak' k) Ak(Xk 1 Tk) = Ak(Ok- 1 Tk) Proof. The proof is by induction on the index k. For k=N we have RN(XN' N' aN' 0N) = LN(7N' aN' N) =RN(ON' TN'aN, N) Thus TN N0(XN'N, N = RN(,NaN=,N)- RN(X NNNaN N 1,N = RN(O N'N, aN = 0,N) - RN(ONN' aN =1, N) = TN(ON' TN' aN, N) and -342

-343TN(xN'7) = TN(XNTN' 0N) ON(O )N' ON- 1) (dON) N = f TN(ON' TN'N) N( N' XN' ON-l) (d0N) f TN(U(ON- XN)' TN N) U(ON-1' XN) (d N) = N(ON1' XN' N) and finally AN(xN, N) = XN TN(XN' N) > O XN IN(ON-1'XN'N) -> 0 AN(ON 1' N) Now assume Eq. j. 1 holds for indices k+l,..., N. Then, Ak+l(k' k) = Ak+1(Ok' k) and Rk+l(xk+l' k+l'ak+l' k+l) = Rk+l(Ok+l1Tk+l' ak+l' k+ ) = Rk+l(U(Ok,xk+l),rk kak,+l, 0k+l

-344Substitute these quantities into Eq. 4. 8 for Rk(Xk, Tk, ak' 0k) and note that the right hand side that results depends on xk only through k. Thus, Rk(k'7 k' ak' k) =Rk(Ok'k' ak' 0k) We may then conclude that Tk(xk, Tk) = Tk(O k- 1' Xk k) and Ak(xk- 1' k) = Ak(Ok- 1' k) by using precisely the same reasoning as was used for k = N. This completes the proof.

APPENDIX K THE RECURSIVE RELATIONS FOR THE m = 2 BAYES DECISION DEVICE In this appendix we adapt the recursive equations of Chapter IV to the basic setting of Section 5.1. We begin by obtaining an expression for Tk(Okl, xk, Tk) by applying the assumptions of Section 5. 1 to the formula for Tk(Ok _, Xkk) of Section 4.4. This is done in the following theorem. Theorem K. 1. Under the assumptions of Section 5. 1 a Tk(Ok- 1' Xk, k) (K. 1) N [(1-a)+ [1-(l+WX) aN1] PN-1 k=N k [(1-a) + [1-(1+WX) ak] Pk-] WFa 0 0 + (k Pk1 +a) [(F (Ok, ak=O) - Fk (Ok,ak=l)) Pk(F (Ok, ak=) -Fk(Okak=))] k < N where Fk (Ok, ak) (K. 2a) 1 Rk+ (k+l' 7k+'t ) P(dk+l +1 = 0sc) t= Ak+(Ok (Tp/2) ak) -345

-346and Fk (Oak) (K.2b) 1 t ((-a) Rk+l(Ok+l' k+l' t' Tp/2) P(dxk+lk+l = 0, S ) k(Ok, Tp/2 ak) + a Rk+l(Ok+l 7k+l t, Tp) P(dxk+lIN) and P(dxk+lS) = P(dxk+l k+ 0, Sc) (K. 2c) and Rk+ ( k+ l' 0k+l +lk+l) (K. 3) Lk+(T. a ) + F (O l'a+l0 k+ /2 Lk+l(' k+1 ak+ k+1) + k+ ) k+l p Proof. From Eqs. 4. 32a and 4. 31b we have Tk(Ok-' Xk' Tk) = Tk(Ok_,xk'rk, k) Ok(Ok' 0Xk 0k) -(dOk) k

-347Now, since 0k takes on only the values, 0, T /2, and T, we may evaluate the above integral with respect to the measure i(dOk) = E{O(d k) + e{T /21(dOk) + ET (dOk) to obtain Tk(OklXk,'k) (K. 4) kTk(OklX k,' kk =0) Ok(Ok =O,Xk'Ok-1) Tk(Ok k-1', k'k 0k =Tp/2) Ok(Ok = Tp /2xk Ok) + Tk(Ok-l xk'Tk' Ok = Tp) x 1. Next, substitute for 0k(k, k', Ok' )k from Eq. 5. 10 to obtain Tk(Ok- 1 xk Tk) (K. 5) =k a -[Pk-1 Tk(Ok-l'Xk' k' 0k =0) +(1-a) Tk(Ok 1, Xk Tk' k = Tp/2)] +Tk(Ok_l Xk Tk' k =T) From Eqs. 4. 32b and 4.26, we have

-348T ko x ) (K. 6) Tk(Ok- 1, Xk Tk k (K.6) Rk(Ok' k' ak = 0k) - Rk(Ok'k' ak = 1, k) where Rk(Ok' Tk ak k ) = L(Tk ak 0k) + Fk(Ok' 0k l (K.7) and where Fk(Ok' k+l0k) (K. 8) \ (t O t Rk+l(Ok+l, k+lak+lt, k+l) P(dxk+l k+' Ok)) k+l k- k+l tAk+1(Ok,Tk+l) II(d0k+,l 0k) Denote the integrand in Eq. K. 8 by Bk+l(0k, + ak+l' 0 k+l' k k). Then, Fk(O k k+l' ek)= Bk+l(Ok' Tk+' ak+1 k+1' ak) II(d k+l1 k) k+l (K. 9) Next, use the discrete conditional probability distribution with q=2 to evaluate the above integral. The result is

-349Fk(Ok' k+1 0k) (K. 10) Bk+1(Ok' rk+' ak+l,' k+1 = 0, ) Pr[0k+l= Ik] + Bk+1(Ok' k+' ak+l' k = Tp/2, 0k) Pr[0k+l =Tp/2 0k] + Bk+l(Ok'rkk+l+l' 0k = Tp 0k) Pr[ k+ = Tp 0k] Now, substitute for Pr[Ok = T p/2 0k] from Eq. 5.2 into Eq. K. 10 to obtain Fk('Tk+'0 k) (K.11) Bk+ l(Ok' k+l' ak+lk = k+ Tp/2, 0k) (1 - a) Bk+l(Ok' k+' ak+l' k+1 =T P0 k)a 0k= ~ Tp Bk+ l(k' k+l'ak+l' k+l=, Ok+l Tp/2) k = Tp/2 k p The next step is to evaluate B *,*,*, To this end, consider k+e nt sp sv, ). To this end, consider the distribution P(dxk+l 0k+l, k1 ). This is the distribution of the process x(t) = n(t) + s(t, 0k+1, 0k) t [tk, tk+1) where from Eqs. I. 7 and I. 8 in Appendix I

-350pk+l(t) 6k+l= ~kTp/2 s(t, k+l', k) = 0 0k+1 =Tp, Tp p (t-T/2) 0k1 /2 0k = 0 Tp p P k+I p k p But, by the pulse translation assumption of Section 5. 1, p k+(t) = k+L p (t - T/2) for t E[tk,tk+1). Thus, the process x(t) can be written as x(t) = n(t) + p (t) Ok+l = = Tp/2 k+1 p k - p and 0k+l = T/2 k'0 0 T (K. 12a) and x(t) = n(t) 0k+l Tp, k = 0 Tp (K. 12b) Now, from Eq. I. lOc, it is seen that the distribution of the process of Eq. K. 12a is P(dx.1 k+l = 0,S ) and from Eq. I. lOc, it is seen that the distribution of the process of Eq. K. 12b is P(dxk+l N). Thus, if we write P(dxk+l S) = P(dxk+ll k+1 =- Sc) then we may substitute for P(dxk+l k,1', 0k) into the definition of

-351Bk+l(O' k k+l ak+l' k+l' 0k) and then substitute into Eq. K. 11 to obtain Fk(Ok' k+l' 0k) = (1-a) Rk+l (Ok' k+ k+l k+=t,Ok+ Tp/2) P(dxk+llSc) t=O t Ak+l (Ok' k+1) + a t Rk+l(Ok+,l'k+l'ak+l = t' k+1= T ) P(dxk+ lN) Ak+ 1(k'rk+l) for k+l = O, Tp, and Fk(Ok' rk+l' k =Tp/2) t=o I k+l(ok+l( +'Tk+la+l = k+1 = T) P(dxk+llSc) tA+l(Ok, T+1) k+1 k k+1 Finally, define F (Ok') and Fk(Okak) by (Ok' ak) = Fk(Ok' k+l' k+l =) (K13a) Fk (Ok,ak) = Fk(Ok'k+1 k+l =T/2) (K.13b) and thus establishes Eqs. K. la and K. lb.

-352Next, note that Eq. K. 3 then follows immediately from Eq. K. 7 and the definitions of Eq. K. 13. It remains to verify Eq. K. 2. To do this, substitute Eq. K. 7 into Eq. K. 6 to obtain o o Tk(Ok-l'xk Tk ak) [Lk(rk, 0, 0) - Lk(Tk, 1,0)] + [Fk(Ok, ak =) - Fk(Ok,ak = 1)] 0k -0 [Lk(k T /22) -k(rkk 1,Tp/2)] + [Fk(Ok, ak=) - Fk(Ok a 1)] Ok Tp/2 [Lk(Tk o Tp) -Lk(T, 1 Tp + [Fk (k' ak=0) Fk (k ak= )] Ok =T k p Now, from Eq. 2. 27 with m = 2 Lk(kak=,k) = 0 and

-353Lk(Tkak = 1, 0k) Tk 0 |-1 Tk =0 k 0 WX 7k = Tp 2 < - ~1 Sk~~ = Tp/2 WF k= Tp or, equivalently, Lk(rk, ak = 1, 0k) - 1 -(1 + WX) ak-l] k = O - -1 Sk = T /2 WF hk - T Thus we may write

-354Tk(Ok- 1' Xk' kk (K. 14) 0 0 [1- (l+W )al] + [Fk(Ok ak =0)- Fk(Okak = 1)] 0k= ( ~1 ~+ [Fk(Ok'ak =0) - Fk(Ok,ak = 1)] ok =Tp/2 -W + [F+k (Ok, ak =0) - Fk (Ok ak = 1)] k Tp Now, substitute Eq. K. 14 into Eq. K. 5 and rearrange to obtain Tk (Ok- 1' Xk, k) k a-l[(l-a) + [1 - (+ Wx) ak k] P -- ] WF + { a'(1-a) (F1(Ok,ak=O) - Fk(Ok,ak =1)) + fka 1 k-1 (Fk(Okak=0) - Fk(Okak = 1)) + (F(Ok ak =)-FO (Okak= 1))} or

-355a Tk(Ok-I'xkTk) (K. 15) = k [(1-a) + [-( - 1+l)ak] l] -WF a ( kPk+a) (Fk (Ok ak=O) - F ak =1)) (1-a ) k 1k + k Pk + a) (k Fk (Ok, ak 0)= 1) Finally, note from Eq. 5. 11 in Section 5. 1 that (1-a) 2k p = k a+k Pk Substitution into Eq. 5. 15 results in Eq. K. 1. This completes the proof. Next, we examine the dependence of the functions Tk(Ok_ 1'XkT k) and Fk (Ok ak) and the set Ak(Ok_, rk ) on the odds ratio density k. In Section 5. 1, Ok is written in terms of Pk-1 and k. Thus it seems reasonable to expect that Tk(.,.,.), Fk (, ) and Ak(-,.) can also be written in terms of Pk-,' k A statement to this effect constitutes the following theorem. Theorem K. 2. We may write

-356Tk(Ok- 1rXkk) = Tk(Pk- 1,' kTk) (K.16a) Fk (Okak) FPk ak (K. 16b) Rk+1(Ok+1 k+1' ak+l, k+1) = (K. 16c) Rk+l(Pk' k+' k'Tk+l' ak+1' 0k+1) A proof to this theorem can be obtained by induction on the index k. The steps of this proof parallel almost exactly the steps in the proof to Theorem 4. 10 in Appendix J and shall not be repeated here. Next, note that rather than take the response set Ak(Pkl, ak-1) as a subset of the observation space Xk we may instead take the response set as Ak(Pk-l1rk) = {kTk(Pk-1I k'a k- 0) (K. 17) This follows from the fact that the response condition Tk(Pk_-, k,ak1_) > 0 depends on the current observation xk, only through the likelihood ratio tk tk k k 1 f pk 2 S (xk)= exp f x(t) p (t) dt - -2N [p (t)] dt k N _0 ~ tkTp/2q 4 t -T /2

-357Thus the condition ke Ak(Pk-' k) = {YXklTk(Pk-1 k(xk),ak- ) > ~} is equivalent to the condition Ike Ak(Pkl,'k) = klTk(Pk,-1 kak I) > 0} This fact results in the following theorem. Theorem K. 3. We may write f ( )P(dx N) f= ( )f(kI )dfk (K.18) Ak(Ok- 1' k) Ak(Pk- l'k) where the density f(2kIS ) is obtained from the relation k = exp{zk} (K. 19) where zk is a real valued normal random variable with E[zkl] = ~dT /4 (K.20a) and Var[ZklN] = dT/2 (K.20b) and dT2 is the signal to noise of the basic pulse

-358T dT2 1 P 2 d N - [p(t)] dt (K.21) o 0 Proof. It follows immediately from the statements preceding the theorem that f ( )P(dxl ) = f ( )P(dk ) (K. 22) Ak(Ok-_ 1 k) Ak(k- 1' k) where P(dk S ) is the distribution induced from P(dxk ) by the mapping in Eq. 5. 9. Now, write the argument of the exponent in Eq. 5.9 as tk tk k(Xk) = N x(t) pk(t) dt 2 f [pk(t)] 2 dt Zk(xk) N 2N 0 t - T/2 0 t - T/2 k p k p In the case of additive white Gaussian noise, it is well known that zk is distributed as a normal random variable with parameters E[zkI] =- +d2/2 (K. 23a) and Var[zk N] = d (K. 23b) where

-359tk d2= N [pk(t)] 2dt N t- Tp/2 k p The next step is to relate d2 to the signal-to-noise ratio dT2 of Eq. K. 21. First, it is noted from the fact that pk(t) = p(t-(tk-Tp)) 2 that dT2 can be written as tk T2 N [pk(t)]2 dt t - T k P or tk - T /2 tk 2 1 _ k 2 1 2 d =N S [p (t)] dt + N- ] dt t - T ~ tk - T /2 k p k p Novg apply the pulse translation property of Eq. 5. 8 to the first integral in the above expression to conclude that

-360tk tk dT N [Pk(t)] dt + N1 [pk(t)]2 dt t-Tp/2 ~ tk-Tp/2 tk =2 tk [pkt)12 dt tk -Tp/2 or dT2 =2 d2 (K. 24) Finally, substitute for d from Eq. K. 24 into Eq. K. 23a, b to get Eq. K.20. The proof is then completed by noting that, since zk has a random variable, Bk must have a density function. Thus, we may replace P(dlkN) in Eq. K.22by f( k(I)dik to get Eq. K. 18. This completes the proof. In the next appendix, we describe the computer algorithm used to solve the recursive equations developed here.

APPENDIX L THE COMPUTER ALGORITHM FOR OBTAINING THE m = 2 BAYES DECISION DEVICE In this appendix the computer algorithm used to obtain the description of the m = 2 Bayes decision device is discussed. From Section 5.3 and Appendix K it is seen that it is necessary to solve for the response sets Ak(Pk-1' k) {k;Tk(Pk-1' Bk' k) >0} where a Tk(Pkl, Xk k)L.1) k l-a) + [1 - (1 +Wx) ak_] Pkl] -WFa + (k Pkl+a) [(Fk(Pkak=0) - Fk0(Pkak=1)) + Pk k (Pk ak) - Fk(Pk, ak=1))] k < N |N[il-a) + [1- (l+Wx) aN_lN-]Pk = N -361

-362and where Fk(Pk, ak) (L.2a) 1 E S 1 k+l( k+l, (T /2) a' t, ) f(k+l S) dk t=o t'-~ k~l + p k k k +l k+l t= l t k+l(Pk,(Tp/2)ak) Fk (Pk ak) (L.2b) S {(1-a) k+l(k+, (T/2) ak, t T/2) f( ) +l(Pk Tp2 ak) + a R+l(Pk+, (Tp /2) akt,T) f( k+l N) dfk+l and Rk+l(Pk+l Tk+l' ak+l' k+l) (L.3) L(Zk+l' ak+l k+l) + Fk+l (Pk+l +ll ) k+1 =0, Tp L('k+l' k+l' Ok+l) + Fk+l1 Pk+l1 ak+l) 0k+l = T /2

-363These equations are to be solved recursively, beginning with the index k = n and proceeding through decreasing values of k. This procedure is indicated in Fig. L. 1. set k = N \ ii calculate F (Pk -1 Sk' Yk) from Ea. L. 1 from Tk L.k( k_2, kn L.. calculate I- Ak(Pk-' ki) Eqs. L. 2 and L. 3 Fig. L. 1. The recursive procedure

-364To carry out the computational procedure indicated above, it was first established that, for each value of Pk-1 > 0 and Tk = (T /2) ak1, there exists at most one root of the function ak-1 Tk(Pk-l' Ik' k)' Moreover, this root, to be denoted by Kk (Pk) has the property that Tk(Pk-l' k''k) T for fk Kk k(P k1) This fact was established by carrying out the computational procedure a stage at a time and checking the function Tk(Pk-1,'k' Tk) at each stage. As a consequence of the above fact, the response set Ak(Pk_l, Tk) can be written in the form P Tak-1(L.4) Ak(Pk-l''k ={ k; { k > Kk ( k-l) (L. 4) Thus the likelihood ratio property of Section 4. 3 is established. ak- 1 The function Kk (P k-) was obtained at each stage of the computation by a subroutine that calculates the root of the function Tk(Pk' lk' kT ). This subroutine operates by first determining kki k'" 1 kK

-365if a root lies in the interval [0, Ku], where K is equal to the fivestandard deviation point for the random variable zk under the condition "signal present". If no root is present in the interval [0, K], Kk k l(Pk 1) is set equal to K. If a root is present, then the subroutine proceeds by a standard interval-bisection algorithm to locate -6 that root to within 10 of its true value. The procedure for calculating Fki(Pk, ak) is as follows. First, the values of Fk+1 (Pk' ak), which have been stored from the previous stage, are used to determine R+1(.,.,.) according to Eq. L. 3. These values are calculated for T k+e {0, Tp/2 }, ak+l1{0,l}, 0k+l{0, Tp} and for Pk-1 = (.05) j;j=l,..., 100. Then the values of Rk+1(.,.,.) and the threshold function ak-1 K Pk l) from the preceding stage are used to calculate Fk (Pk ak) according to Eq. L. 2a and L. 2b. The integrals in these equations are evaluated using a seven-point Gaussian quadrature routine and a seven-point Gaussian-Hermite routine. The values Rk(.,.,.,.) in the integrand of these integrals are obtained by a three-point Lagrange interpolation procedure from the values of Rk+(..,.,.) at the points Pk+ = ( 05) i, i 1,..., 100. Finally, the values of Fk (Pk, ak) are stored for use at the next stage. To complete the computations in one stage of the recursive procedure of Fig. L. 1, it is necessary to calculate Tk(Pkl, k' 7 k) according to Eq. L. 1. Actually, this computation is carried in the

-366root-finding routine using the stored values of Fk (Pk' ak) In conclusion, we point out that it was only necessary to carry out the computations for six stages in order to obtain a reasonable.k-1 approximation for the limiting function K (P This fact K Pk-1' This fact was established by calculating the difference Kk+ (u) - Kk (u); i=0 1 and terminating the calculations when this difference became less than 10.

APPENDIX M THE SIMULATION PROCEDURE In this appendix we describe the procedure used to obtain the ROC surfaces discussed in Section 5. 5. We begin by recalling the basic definitions of the normalized rates RX, RF and RD. Specifically RX = rx/r RF rF/(rN/m) RD = r rp where E {N (6(x),0)} r T Now, use the equalities Pr[iE [0,Tp)] D r = ___ -P p T T P P 1 - Pr[ e [0, Tp) 1 - D rN/ m =- T T P P T = mA; T = NA to write E {NX( (x), 0) } m RX= N D-367

-368E{NF(6(x), 8) m R "" m RF L N -D p [E{ND(6(x), 0) m %D N D Now the point here is that, for a fixed number of decision opportunities per pulse, m, and a fixed pulse duty, D p E N (6(x), o) R (6) of E ^ -^- N Thus to determine the rate R (6) for a specific decision device 8 it is only necessary to determine N(O6(x), 0) To do this, we recall that Np (6(x), 0) represents the total number of occurrences of the event j after N decision times when the decision device 6 is used and the observation is x = (1,..., xN) and the signal is described by 0 = (081.'' 0N). Thus, we may interpret the quantity N(65(x), 0) N as the relative frequency of the event I in N decision times. Now, 0 0 if N is large and x and 0 are typical realizations of the observation sequence x and state sequence 0, then it is reasonable to expect that this relative frequency approaches some limiting value

-369N (6(x), 0) N and that N(6(x), 0 N (6a~), 0~) E _ N N Indeed, this is the underlying philosophy of the law of large numbers. Unfortunately, it appears quite difficult to establish this result theoretically due to the complex nature of the random variable N (6(x), 0). We have, however, tested the validity of this result experimentally by calculating NB (6(x), 9 ) for different values of x and 0 and noting that for N > 8000 the quantity N ( (x0), 0) N remains constant out to the fourth decimal place. We turn now to the question of how Np (6(x0), 0 )/N is calculated. First, note that all the decision devices of interest in this chapter can be represented as shown in Fig. M. 1. In this figure, Qk(xk) = - [zk(Xk)] where zk; k=l, 2,..., are Gaussian random variables with a k(xck) |_ Decision a k(Xk) Devic ee f f e ii i a r Fig. M. 1. The generic form for the decision devices of Chapter 5

-370mean and variance that depend on the signal content of the interval [tk - Tp/2, t). More precisely, if we note that signal is present in the current interval iff 0k = 0, T /2 and signal is not present k p in the current interval iff k = Tp then we may apply Theorem I-3 of Appendix I to conclude that E[zk S] =E[zk k = 0, Tp/2]= dT2/4 E[zk N] = E[zk k = T] = dT /4 Var[k = dT2/2 0 Now the point here is that if we have a realization 0 of the sequence 0 = (0p.., N) then we can calculate a realization I of the sequence i = ( 2,' 2.' IN) by using a Gaussian random number generator to calculate zk with mean and variance determined by 0k. We can then use a particular decision device to determine the sequence of decisions (a1, a2,..., aN). Finally, the number of occurrences of the event 3 can be determined according to the table in Fig. M. 2. ak 1 ak 0k =0~ k Tp/2 k = T 0 0 miss --- --- 0 1 detection detection false alarm 1 0 1 1 extra detection detection false alarm Fig. M. 2. The relation of the events 1 to the decisions a, - and ak and the state 0,

-371It remains to describe how the realization of the sequence = (01,. " ON) is calculated. To this end, we note from the transition matrix of Eq. 5. 2 that 0 is described by a state diagram as shown in Fig. M. 3. I a A m- ~1-a Fig. M. 3. State transition diagram for the random variables Sk The interpretation of this diagram is as follows. If Sk = Tp/2j k p j = 0, 1, 2, then 0k+ takes on the value indicated by the arrow emanating from the appropriate node with the probability shown next to that arrow. For example, if 0 = 0 then /2 with probability 1 - a k+ 1 = NT with probability a

-372Thus, if we have a sequence of random variables, bl, b2,.., bN, such that -a j = 0 Pr[bk =j] a j = 1 then we can determine 0 = 01,.. 0N recursively by Tp/2 bk 0 0 =0 k k 4T bk =1 P k 9( b =o0 9k = T/2 Sk k k p Tp/2 bk P ~ k p 4T bk 1 p k The sequence, bl,..., bm, can be obtained from a uniform random number generator.

APPENDIX N THE RATES RX, RF AND RD FOR THE SKE DECISION DEVICE In this appendix we relate the probabilities of detection and false alarm, PD and PF, for the SKE classical detection decision device D F' to R%, RF and RX and D and D. The basic idea here is to view the SKE device as making successive independent decisions over contiguous intervals of time. More precisely, we divide the time axis into N equal intervals of length T and assume that the decision device makes decisions at the end of p each interval. We may then consider the total decision of this device as the vector a = (a1,..., aN), ae {0, 1}, where ai represents the decision at the end of the ith interval. In addition, one may describe the total signal by the vector 0 = (1', "" 0 N), i0e OS U ON, where OS, 0 ON represent the signal parameters and the noise parameters which affect the observation over the ith interval. (See Appendix A.) It is to be noted here that we have assumed that the noise is independent from interval to interval and that any signal associated with the parameter 9i is completely contained within that interval. We now appeal to the basic definitions in Section 2. 5 for an extra detection, a false alarm, and a detected pulse to write NX -o -373

-374N ND = ~ I(ai' ai) {ai= 1, OS} 1 S and N NF = I (ai.i) F =1 f {ai = 1, ON Then, from the definitions of rX, rD and rF in Section 3.2, rx=0 N Pr[ai = 1, OS] rD = N T i=1 p and N Pr[ai =, ON] rF = T i= l p i=1 p Next, since the probabilities in the above sums are identical, r= 0 Pr[ai = 1, Os] Pr[0 ] rD = T r[ai = OS] T P P and Pr[ai = 1, ON] Pr[ON] rF TPr = ra 1 ON] T P P or finally RX = O

-375RD = rD/Pr[ S]/Tp = Pr[a.= 1 1OS = P and RF = rF Pr[ N]/T = Pr[ai lN = P To relate PD and PF to DD and DF we simply appeal to Eq. 3.13 with m = 1. The result is DD = PD and DF PF

-376REFERENCES 1. Egan, J. P., Greenberg, G. Z., and Schulman, A. I. "Operating Characteristics, Signal Detectability, and the Method of Free Response." JASA, 1961, Vol. 33, pp. 9931007. 2. Lucas, P. A. "Human Performance in Low-Signal-Probability Tasks." JASA, 1967, Vol. 42, No. 1, pp. 158-178. 3. Luce, R. D. "A Model for Detection in Temporally Unstructured Experiments with a Poisson Distribution of Signal Presentations." Journal of Mathematical Psychology, 1966, Vol. 3, pp. 48-64. 4. Ferguson, T. S. Mathematical Statistics: A Decision Theoretic Approach. New York: Academic Press, 1967. 5. Feller, W. An Introduction to Probability Theory and Its Applications. Volume 2. New York: John Wiley and Sons, Inc., 1966. 6. Lehmann, E. L. Testing Statistical Hypotheses. New York: John Wiley and Sons, Inc., 1959. 7. Breiman, L. Probability. Reading, Massachusetts: AddisonWesley Publishing Company, 1968. 8. Birdsall, T. G. "The Theory of Signal Detectability: ROC Curves and Their Character." Doctoral dissertation. Ann Arbor: The University of Michigan, 1966. 9. Hilborn, Jr., C. G. and Lainiotis, D. G. "Recursive Computations for the Optimal Tracking of Time- Varying Parameters." IEEE Transactions on Information Theory, Vol. IT-14, No. 3, pp. 514-515. 10. Van Trees, H. L. Detection, Estimation, and Modulation Theory, Part I. New York: John Wiley and Sons, Inc. 1968.

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UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA - R & D (Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified). ORIGINATING ACTIVITY (Corpo rate author). 2a. REPORT SECURITY CLASSIFICATION Cooley Electronics Laboratory Unclassified University of Michigan 2b. GROUPnAnn Arbor, Michigan 48105 3. REPORT TITLE THE FREE RUNNING DETECTION PROBLEM: THE DETECTION OF RANDOMLY OCCURRING PULSES AS THEY OCCUR 4. DESCRIPTIVE NOTES (Type of report andinclusive dates) Technical Report No. 209 - 036040-5-T 5. AUTHOR(S) (First name, middle initial, last-name) Richard M. Heitmeyer 6. REPORT DATE 7e. TOTAL NO. OF PAGES 7b. NO. OF REFS September 1971 410 10 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) N00014- 67-A-0181-0032 036040-5-T b. PROJECT NO. c. 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report) d. TR 209 10. DISTRIBUTION STATEMENT Approved for public release; distribution unlimited. 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Office of Naval Research Department of the Navy ___________________ Arlington, Va. 22217 13. ABSTRACT ABSTRCT This study provides an extension of classical detection theory by defining and analyzing two classes of decision devices that both operate in real time to detect the presence of each pulse in a sequence of randomly occurring pulses while that pulse is still present. The respond-once device seeks to detect each pulse once and only once whereas the other device, the respond-and-hold device, seeks to respond at each instant of time at which a pulse is present. The problem of obtaining the decision devices that are optimum in the above setting is the free running detection problem. It is assumed that all pulses are identically shaped and known up to their arrival times and that it is possible to observe these pulses only through an observation that contains additive noise. It is also assumed that the decision devices are allowed to make decisions on the presence or absence of pulses only at discrete, equally spaced points in time. The mathematical model is constructed in terms of th general decision theory model providing the theoretical basis for describing the performance of both classes of decision devices and for obtaining decision devices that are optimum with respect to the Bayes criteria. The performance of the respondonce decision device is described in terms of the average number of pulses detected, the average number of false alarms, and the average number of times a pulse is detected more than once. It is shown that the detection rate can be increased for the same false alarm rate by increasing the extra detection rate. The performance of th reponad-a d-hold decision devices is described in. terms of the detection rate, duty, DD ORM 147 (PAGE 1) U NOV 651 ^ /0__UNCLASSIFIED S/N 0101 -807-681 1 Security Classification A-31408

UNCLASSIFIED Security Classification 14. K OLINK A LINK B LINK C KEY WORDS _____ _ ____ ____ ROLE WT ROLE WT ROLE WT Random pulses Optimum detection Free response DD,?Ov^s,1473 (BACK) UNCLASSIFIED S/N 0101-807-6821 Security Classification A- 31409

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