THE UNIVERSITY OF MICHIGAN OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR ON CERTAIN STRATEGIES OF SIGNAL DETECTION USING CLIPPER CROSSCORRELATOR (Single Signal Size) Technical Report No. 128 3674- 1-T Cooley Electronics Laboratory Department of Electrical Engineering By: G. P. Patil Approved by:-~.J~ P. Cota T. G. Birdsall ORA Project 03674 Contract No. Nonr-1224(36) Office of Naval Research Department of the Navy Washington 25, D. C. October 1962

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iv FOREWORD AND BACKGROUND v ABSTRACT vi ACKNOWLEDGMENT vi 1. INTRODUCTION 1 2. BINOMIAL STRATEGY OF SIGNAL DETECTION (BS) 3 3. INVERSE BINOMIAL STRATEGY OF SIGNAL DETECTION (IBS) 6 4. EQUIVALENCE OF BS AND IBS 8 5. EFFICIENCY OF THE CLIPPER CROSSCORRELATOR 6. MODIFIED BINOMIAL STRATEGY OF SIGNAL DETECTION (MBS) 11 7. MODIFIED INVERSE BINOMIAL STRATEGY OF SIGNAL DETECTION (MIBS) 13 8. EQUIVALENCE OF MBS AND MIBS 14 9. TABLES 15 10. A SEQUENTIAL STRATEGY 27 10.1 Sequential Probability Ratio Test 27 10. 2 SPRT when y, 6 are Determined by Wald's Approximation 29 10. 3 ASN for the SPRT 32 10. 4 Comparison of SPRT with MBS 33 11. CONCLUDING COMMENTS AND REMARKS ABOUT FURTHER WORK 34 REFERENCES 35 DISTRIBUTION LIST 36 iii

LIST OF ILLUSTRATIONS Figure Title Page 1 Values of efficiency for C. C. C. (Based on Table I, using n. ) 21 2(a) Efficiency of C. C. C. (BS) using actual a, 3, n. 25 2(b) Efficiency of C. C. C. (MBS) using actual a, 3, n, under signal plus noise. 25 3 Termination boundaries for clipper crosscorrelator. 30 g3 0 0 4 Range of a and, when y=1 -a and 6 =. 30 a0 5 Termination boundaries for p = 0. 67, a = 0. 00098, and B = 0. 09871. 33

FOREWORD AND BACKGROUND The problem of evaluating the performance and the efficiency of the type of detection receiver known as a clipper crosscorrelator has been studied by a number of people in the acoustics and engineering fields. The majority of these studies determine the signalto-noise ratio at the output of the receiver. By the very nature of the problems considered in these studies a great many approximations are usually made. In this report Dr. Patil studies a specific detection situation and treats the performance of the receiver in detail, rigorously, and without approximations (beyond the assumption that the input samples are independent). The correlator studied crosscorrelates against a local reference signal. Four versions of operation of the clipper crosscorrelator are considered. The standard operation of a detection device, integrating the receiver input over a fixed time, is studied under the title of "binomial strategy. " A variation of this called the "inverse binomial strategy" operates the receiver accumulator until a fixed threshold is exceeded. Since both of these operations involve observing the output of an accumulator which has a nonnegative input, modifications are in order which quicken the time of decision when the decision is a foregone conclusion. Such quickening, or decreasing the time necessary to reach a decision, is done without effect on the primary measures, the error probabilities, and leads to an increased measure of efficiency. The results of these studies are presented in both tabular form and graphs. These were calculated by Mr. Cota, who has also added a final section for comparison, which treats the output of the clipper crosscorrelator with a double threshold comparator, following the techniques of Wald's sequential analysis. This section has been added to show the following comparison: in the binomial strategy, inverse binomial strategy, and their modifications the error limits were considered as primary objectives, and the time necessary to reach a satisfactory decision was considered a secondary objective; that is, time was minimized only if its minimization did not affect the error probabilities anticipated on an observation-by-observation basis. In sequential analysis the three variables are considered as primary variables, though not of equal weight, and the average time is minimized subject to the over-all or average error probabilities. T. G. Birdsall

ABSTRACT We consider in this report the problem of signal detection using clipper crosscorrelator when the signal of single size and the Gaussian noise are known exactly. We develop strategies in order to meet the requirements dictated by the gravity of the "false alarm" and of the "miss. " Four such strategies are suggested which arise in a very natural way, and their interrelations are studied. Efficiency of the clipper crosscorrelator in relation to the usual crosscorrelator is defined and investigated in the setup as described. Associated tables and charts are given. ACKNOWLEDGMENT The authors wish to take this opportunity to express their sincere thanks to Mr. T. G. Birdsall, at whose insistence these investigations in the area of signal detection were started and with whom they have had several instructive discussions. The authors' thanks are due to him also for the fitting foreword that he has very kindly and willingly written for the investigations and results contained in this report.

1. INTRODUCTION As mentioned in the:foreword:and'background, we consider in this report the problem of signal detection using clipper crosscorrelator when the signal of -singlesize is known exactly. Without loss of generality let the signal size s be positive. The general problem of signal detection is to decide the absence or:presence of a possible signal on-the basis of a certain number of observations made with, possibly, some noise in the background. Let the random sample of size n of independent observationslbe X1, X2,.. Xi,.. X n. Under usual assumptions, and under noise alone, let X1 q(O, 1); i. e., X. is normal with zero mean and unit standard deviation when noise alone is operating. Further let it be assumed that under signal plus noise -X. - r77(s, 1); i. e., X. is normal with mean value s and standard deviation one when signal of positive size s is:present in addition to noise. 1 Now, the clipper c'rosscorrelator is a device which, instead of recording the magnitude of each observation Xi, records for purposes of simplicity only the count c(Xi) of the observation X.. To be specific, c(Xi) = 1 if Xi > 0 = 0 if X. < 0 Using these unit and zero counts as basic sample data, strategies can be developed for signal detection purposes.'As is well known, the solution to-a dichotomous statistical decision problem traditionally involves the recognition and reconciliation to the two types of errors known'as!a-~Frror hand,-error. In the problem urder -consider'iation: a —ror takies the form of "false alarm" and the T-error means "miss. " The sizes a and: of the a -error -and 3-error in making decisions based on a strategy are measured by the chances of committing such errors under such a strategy. -Different strategies can imply different sizes of the a -error and the A-error, thus bringing out more effective or less effective roles of the different strategies.'The physical interpretation is that s2 is the signal-to-noise ratio at the input to the clipper.

Depending on the gravity of the situation for the problem at hand one may specify the sizes of "false alarm" and "mistaken miss. " On the basis of the observable counts c(X), c(X2)... c(X.)... recorded by the clipper crosscorrelator we suggest in this report certain strategies of signal detection which can meet the "false alarm" and "mistaken miss" requirements, and further consider the interrelations between such strategies.

2. BINOMIAL STRATEGY OF SIGNAL DETECTION (BS) As an example, one can think of the following strategy, which makes use of the n total count C = Z c(X.) obtained from a sample of size n. The strategy requires specificai=l 1 tion of a "detection count" d to make the following decisions: (i) if the total count C > d, conclude that the signal is present; (ii) if the total count C < d, conclude that the signal is absent. We propose to call such a strategy a binomial strategy of signal detection (BS) for reasons which will be apparent in the course of the following discussion. The main problem involved in the BS is the problem of choosing suitable sample size n and the corresponding detection count d. As has been stated before, n and d are chosen so as to meet the a and A3 requirements. Note that under noise alone for whatever n may be Prob (Xi) 1} = Prob Xi > 0 under noise alone} i = 1, 2,..., n. Also X1, X2... Xn are independent. Therefore under noise alone C - B(n, 1) i. e., the total count C based on a random sample of size n is a binomial random variable with 1 parameters n and p = -. To be specific, Prob TC = r r = 0,1,2..., n. Further, under signal plus noise with a positive signal of size s, Prob(c(Xi) 1} = p (Xi> 0 under signal plus noise} = I(s) = p (1)

t2 where: C(s) 1 -2t e dt. -00 Also, X1, X2... Xn are independent as before. Therefore under signal plus noise C - B(n, p) i. e., the total count C is now a binomial random variable with parameters n and p. To be specific, Prob{ C = r} = (n)pr(P)nr r=0,1,2,..., n. Note that p > 1 under signal plus noise whereas p = under noise alone. 2 2 Further let Pro n n k n-k Prob C > rr: r (k)P (1-p) r =0,1,2,..., n k=r = B(n, r,p) (2) As a consequence of BS one can see that the size of false alarm a = Prob {C > d under noise alone} = B(n, d, 1). (3) Similarly the size of a miss is obtainable as i = Prob {C < d under signal plus noise} = 1- B(n,d,p) (4) where p is given by (1). Thus, one has the following two equations: B(n, d, ~) = a (5) B(n, d, p) = 1-3 (6)

to be solved for sample size n and detection count d for specified false alarm of size a and miss of size 3. The solution of (5) and (6) is the pair of n and d required for BS corresponding to specified a and P3. Table I for n and d for different sets of triples of a, A and p = ( (s) extends over the range of detection interest;.01 < <.90, 107 < < 10-2 tends over the range of detection interest;. 01 < j3 <. 90, 10 K a K 10

3. INVERSE BINOMIAL STRATEGY OF SIGNAL DETECTION (IBS) It is possible to think of some "tolerance count 6, " and to go on recording the individual counts c(Xi) on the clipper crosscorrelator until the observed over-all count i c(Xi) accumulates to 6. Let the number of individual counts (sample size) required to accumulate the tolerance count 6 be denoted by R. As might be expected, one can think of the following strategy for signal detection. Specifying some "tolerance sample size" by 71, one makes the following decisions: (i) if the required sample size R < t/, conclude that the signal is present; (ii) if the required sample size R > 71, conclude that the signal is absent. We propose to call such a strategy an Inverse Binomial Strategy of Signal Detection (IBS) for reasons which will be apparent in the course of the following discussion. The main problem involved in the IBS is the problem of choosing a suitable tolerance count 6 and the corresponding tolerance sample size 71. As has been stated in the introduction, 6 and 77 are chosen so as to meet the a and A3 requirements. Note that as before the probability of a unit count is p = p(s) under signal plus noise and p = 2 under noise alone. Individual counts are independent. Therefore under signal plus noise R -d B(6,p) i. e., the sample size required to accumulate the over-all count to 6 is an inverse binomial random variable with parameters 6 and p. To be specific, Prob (R = r} = (1) p6(1-p)r6 r = 6, 6+1,. where: p = <A(s).

Similarly, one has under noise alone R d B(6, 2) Further let Prob R < r = () p (1-p)Xx=6 = I(6, r, p) (7) As a consequence of IBS one can see that the size of false alarm a = Prob R < qr under noise alone} = I(6, 7, 1) and the size of mistaken miss / = Prob R > r1 under signal plus noise} = 1- I(,q, p) where: p = c(s). Thus, one has the following two equations 1 I(6, 7r, 7) = a (8) I(6,, p) = 1- / (9) to be solved for 6 and r7 for specified false alarm of size a and miss of size 3. The solution of (8) and (9) is the pair of 6 and r7 required for IBS corresponding to specified a and 3.

4. EQUIVALENCE OF BS AND IBS As is apparent, BS and IBS are quite different in approach and character. The following observation, however, brings out the connection between the parameters n and d of BS and the parameters 6 and r) of IBS and, later, the equivalence of the two strategies. Note that the event E1 of requiring more than r7 counts to accumulate the over-all count of 6 is equivalent with the event E2 of obtaining a total count of less than 6 in qr counts. Therefore Prob(E1) = Prob(E2) i.e., 1- I(6, r, p) = 1 - B(r7, 6, p) therefore I( 6, I, p) - B(71, 6, p) (10) where: I and B are defined by (7) and (2), respectively. Equation 10 is a very interesting identity and was established by Patil (1960) in a slightly different form. Using this identity it can very easily be seen that 6 = d and r7 = n (11) where BS and IBS are derived to meet the same a and /3 requirements. This follows immediately on comparing the pair of equations (5) and (6) with the pair (8) and (9). This shows that tabulation for the parameters of either BS or IBS is enough; there is no need of two separate tables for BS and IBS. In fact a much stronger relation can be established between BS and IBS. We can show that they are equivalent; i. e., for every possible sample data on the clipper crosscorrelator both BS and IBS come to an identical conclusion regarding presence or absence of the signal. It is very curious to observe that, though totally different in outlook and character, BS and IBS turn out to be equivalent strategies. That they are equivalent easily follows from the equivalence of the events E1 and E2 mentioned in this section, together with the established result that 6=d and r/ = n. Although equivalent, the BS and IBS differ from one another in one major aspect in practice. BS requires that a fixed number n of counts be taken, whereas the number of counts to be made in order to apply IBS is a random quantity.

5. EFFICIENCY OF THE CLIPPER CROSSCORRELATOR It is evident that the clipper crosscorrelator does not utilize the entire information on an individual observation. It makes either a zero or a unit count on it, depending on whether the observation is positive or nonpositive, and ignores the magnitude of the observation. Naturally we expect that it be less "efficient" than the crosscorrelator which does take into account the magnitude of each observation. In other words, in order to nmet the same a and 3 requirements, the sample size n required for the clipper crosscorrelator is expected to be larger than the sample size N required for the crosscorrelator. We define the efficiency of the clipper crosscorrelator by the ratio of N to n. From the discussion in Section 2, we know how to obtain the value of n for some a and p requirement. But how do we obtain the value of N? Following the usual analysis with the crosscorrelator we derive here an expression for N. Table I lists N, and the efficiency of the clipper crosscorrelator for different sets of values of the triple a,: and p. Now, on the basis of the sample X1, X2,..., XN of size N, the crosscorrelator uses the sample mean XN as the statistic. Choosing a "detection point" D, the crosscorrelator has the following strategy for signal detection: (i) if the sample mean XN _ D, conclude that the signal is present; (ii) if the sample mean XN < D, conclude that the signal is absent. The sample size N and the detection point D are chosen so as to meet the a and, requirement. Not e that XN (O0, A) under noise alone whereas XN 1(s, 1) under signal plus noise where: s is taken to be positive without loss of generality. As a consequence, the size of the false alarm

a = Prob XN > D under noise alone} = 1- <>(D,/-N) and the size of the miss / = Prob {XN < D under signal plus noise} = [(D- s) v/N] t2 where: C (o) = V/A f e dt. Thus one has the following two equations ~ (-N) = 1- a D [(D- s) IN] = These are the parametric equations of the normal ROC curve. They can be solved for sample size N and detection point D for specified a and 3. Writing FE for the solution of < (x) = e the equations become D,/ = F1-a (12) and (D- s) ~ = F' (13) Subtracting (13) from (12), we have S Jiw = F1 1-a Fj therefore N =1a ) ( ) (14) S S because F 1- = -F. One may note that FE is nothing but the E -point of the standard normal distribution for which extensive tables are available. Incidentally, D can be obtained as D 1- (15) 10

6. MODIFIED BINOMIAL STRATEGY OF SIGNAL DETECTION (MBS) The following discussion will bring out that there is scope for improvement in the BS by taking in practice only as many counts on the clipper crosscorrelator as are essential for the purpose of making the decision on presence or absence of the signal. For example, if one finds that the first d counts are all unit counts, the conclusion under BS of the presence of the signal is clear and certain, and there is no need to observe any more counts, although such an observation would be demanded by sample size n under BS. For that matter, one can stop taking sample data as soon as one has secured an over-all count of d and conclude the presence of the signal as under BS, even if one has not yet exhausted all the required n counts demanded by BS. Similarly, if one finds that the first n - d+1 counts are all zero counts, the conclusion under BS of the absence of the signal is clear and inevitable. There is no need to observe any more counts because there is no possibility that the over-all count will become even d, as is required for the contrary conclusion, even if the n counts required under BS were completed. For that matter, one can stop taking sample data as soon as one has secured n - d+1 counts, and conclude the absence of the signal, even if one has not yet exhausted all the n counts. In view of the above discussion we propose the following strategy, to be called the Modified Binomial Strategy of Signal Detection (MBS). (i) Choose n and d as required by the BS to meet a and P requirement. (ii) Conclude that the signal is present as soon as the number of unit counts is d, and stop taking sample data. (iii) Conclude that the signal is absent as soon as the number of zero counts is n-d+l, and stop taking sample data. The advantage of MBS over BS lies in the curtailment effected in the total number of counts required to make a conclusive decision regarding presence or absence of the

signal. Whereas sample size for BS is a fixed quantity n, sample size required for MBS is a random quantity. The Average Sample Number (ASN) can be seen to be ASN = E Y' ( 1)p (1-p)Y- + Z y (y-1) (1-p)cpy- (16) y=d d-1 y= Cwhere: c = n-d+1 and p is the probability of a count to be unit. Note that y' y(d-1) p (1,p)y y=d d n (y) pd+1 (lp)y-d P y=d d = - I(d+1, n+1, p) p = B(n+l, d+, p) (17) p where I and B are defined by (7) and (2) respectively. Similarly, E y(n1) y (1-)C = 1CC B(n+l, c+1, l-p) (18) y=c Therefore, as a consequence of MBS, d n-d+1 ASN = dB(n+1, d+1, p) + B(n+l, n-d+a, l-p) (19) p 1-p Substituting p = and p = p(s) gives the value of ASN under noise alone and under signal plus noise, respectively.

7. MODIFIED INVERSE BINOMIAL STRATEGY OF SIGNAL DETECTION (MIBS) One may discover that IBS can be improved on the same lines as BS has been improved to MBS. We propose the following strategy and call it the Modified Inverse Binomial Strategy of Signal Detection (MIBS). (i) Choose t) and 6 as required by IBS to meet the a and: requirement. (ii) Conclude that the signal is present as soon as the tolerance count 6 accumulates, and stop taking sample data. (iii) Conclude that the signal is absent as soon as the number of zero counts is ql-6+l, and stop taking sample data. The advantage of MIBS over IBS lies in the curtailment effected in the number of counts required in all to make a conclusive decision regarding presence or absence of the signal. It is easy to see that the sample size for both IBS and MIBS is a random quantity. It can be seen that c00 r-1 6 p)r-6 6 ASN under IBS = Z rp (20) r=6 6pi where: p is the probability that a count will be one. One can show also that under MIBS ASN = n r_ r-1 6 r6 n r-1 r-( ASN: Z rp + r 1) ( p(-p)r- + (21) r=6 r= where: 5 = T) - 6+ 1. Proceeding on the same lines as in the previous section, we have under MIBS ASN = _ B(7+1, 6+1, p) + 1- B(l+l, Wr-6+2, l-p) (22) P 1ep Substituting p = and p = p(s) gives the value of ASN under noise alone and under signal plus noise, respectively.

8. EQUIVALENCE OF MBS AND MIBS When we recall from Section 4 that 6 = d and 7 = n and compare equations (19) and (22), giving ASN under MBS and MIBS, it becomes clear that both MBS and MIBS have the same ASN. In fact a much stronger relation between MBS and MIBS comes out immediately when we compare the descriptions of the MBS and MIBS as given in Sections 6 and 7. It is very clear, though very curious, to see that MBS and MIBS are equivalent. It may further be said that unlike BS and IBS, MBS and MIBS are identical in practice. The only place they differ in is the starting viewpoint.

9. TABLES The following tables are calculated for: a (false alarm probability) = 107 10, 10,..., 10-2 p (miss probability) =.9,. 5,. 1,. 01; p =. 52,. 54,. 56,. 59,.62,.67,.76,.84,.92,.98. The triples (a, 3, p) are chosen so that 10 < n < 1000. Two sets of tables were consulted, both of which yield a and for given values of p, n, d. "Tables of the Binomial Probability Distribution" (Ref. 4), is a set of 7-place tables tabulated for n = 2(1) 49; d = 1(1) n. This set of tables was used for a = 10 5, 10-6 10-7 whenever possible (i. e., whenever n < 50). Notice that these tables given only one-place accuracy for a = 10 "Tables of the Cumulative Binomial Probability Distribution" (Ref. 5), is a set of five-place tables tabulated for n = 1(1) 50(2) 100(10) 200(20) 500(50) 1000, d = 0(1) n. This set of tables was used for a = 10 102 10-3 10, and, when n > 50, for a = 10-5 -5 Notice that these tables give only one-digit accuracy for a = 105. -6 -7 No values were computed for a = 10, 10 when n > 50. Table I. n, d, and r of C. C. C.(B. S. )for given a, a, p. This table is calculated in the following manner. The desired values of p are all available in the tables. For a given a, 0 pair there are several values of n for which two values of d yield a, / pairs that "straddle" the given a, P pair. For example, if p =. 67, a =.01, a =. 5, the consulted table gives:

n d a _ 46 31.01295.45307 32.00568. 57754 47 31.01999.37349 32.00931.49415 For each value of n, 3 and d are calculated (by linear interpolation) for a =. 01. In this case, for n = 46, 3(. 01) = 3 =.50357, d (. 01) = d = 31. 406. For n = 47, 3(. 01) = P =.48635, d(. 01) = d = 31. 935. Since the desired value of / is 3 =.5, we conclude that 46 < n < 47, N dl = 31. 406, d2 = 31. 935. For the purposes of calculating 7) = n, we calculate (by linear interpolation) n for /3 =. 5 and denote it by n. Linear interpolation is not used, however, for values where the consulted tables give only one-place accuracy, or where An > 10 (An being the difference between tabulated values of n). In these cases the values of n are given which yield the closest (straddling) a, i pairs. Table I. n, d, and 7 of C. C. C. (B. S.) for given a, /, p. p / a n n or n 77 or d1 d2 (BS) ~52.9 102 684 650.635 355 356 700.636 381 382.54.9 10- 925 900.630 506 507 950.638 532 533.9 10- 510 500.634 285 550.634 311 312.9 10-2 170 170.630 100 101 180.633 106 107.5 10-2 841 800.635 433 434 850.635 459 460 ~ 56.9 10-5 615 (600) (. 618) (353) (650) (.623) (380).9 10-4 410 400.632 237 238 420.633 248 249.9 10- 226 220 133 134 16

p a a n n or n 71 or dl d2 (BS) 240.629 144 145.9 10-2 75 77.0.621 48.7 49.8.5 10-4 950 950.635 532 533 1000.632 559 560.5 10-3 659 650.633 364 365 700.633 391 392.5 10-2 374 360 202 203 380.633 213 214.1 10-2 900 900.634 485 486 950 511 512.5 9.9 10-5 270 (260) (.664) (164) (260) (. 608) (165).9 10-4 180 180.618 115 116 190 121.9 10 3 99 100.620 65 66 lio 71 72.9 10-2 33 34.6.608 24.3 24.9.5 10-5 551 (550) (.640) (325).5 io04 418 420.630 248 249 440 259 260.5 10-3 290 280 166 167 300.629 177 178.5 o-2 164 160 95 96 170.628 24. 3 24. 9 ~ 1 10-5 931 (950) (.619) (542) -4.1 10-4 758 750.627 426 427 800 453 -3.1 10-3 579 500 311 312 600.631 338 339. i lo-2 394 380 213 214 400.632 223 224 -3.01 10-3 888 850 470 471 900.633 496 497.01 10-2 655 650.634 355 356 700 381 382 62.9 i0- 5 150 (140) (95) (96) (150) (.642) (101) (150) (. 517) (102) 17

p 3a An n or n 77 or dl d2 (BS) (160) (107) (108).9 104 100 100.614 68 69 110 74 75.9 10-3 55 57.7.607 39.9 41.2.9 10-2 19 20.2.578 15.7 16.3 ~ 5 10-5 306 (320) (.640) (198) (320) (.607) (199) ~ 5 10- 232 220 138 240. 624 149 150.5 10-3 161 160.624 99 100 170 105 106 ~5 10-2 91 92.3.628 57.7 58.8 i1 105 517 (550) (.616) (326).1 10-4 420 420.628 248 249 440 259 260.I 10 322 320.628 188 189 340 198 199.1 10-2 219 220.628 127 128 240 138 139.01 10-5 730 (750) (.626) (434).01 10-4 615 600.629 346 650 372 373 ~01 10- 493 480 274 275 500.631 285.01 102 364 360.630 202 203 380 213 214.67.9 10-5 72 (76) (57) (78) (. 594) (58) ~9 i0 48 52.4.586 39.7 40.9 ~9 10- 27 29.3.577 23.2 23.9.5 0-5 148 (150) (.630) (101) (150) (.579) (102) ~5 10 i4 12 110 74 75 120.607 80 81.5 10-3 78 81.4.606 54.3 55.5 ~5 10-2 44 46.2.605 31.4 31.9 ~ 10-5 250 (260) (.639) (164) (260) (.610) (165).1 10-4 203 200.618 126 127 220 138

p p a i n or n ar or dl d2 (BS) ~ 1 10-3 i55 150 94 95 160.620 99 100 ~1 10-2 106 100 62. 1 110.589 67. 7.01 10-5 352 (360) (.579) (221) -4 ~01 10 296 300.622 182 183 320 193 194.01 10-3 238 240.622 144 145 260 155 156.01 10-2 175 170 100 101 180.626 106 107.76.9 iO-6 38 45.4.532 38.5 39.0 9 10-5 28 (33.9) (.526) (29) (29.5).9 10-4 19 23.4.509 20.7 21.3 -3.9 10 11 13.3.493 12.4 13.0 5 i0-5 57 (63. 1) (. 578) (48) (49) -4 ~5 10-4 44 48.8.568 37.1 37.8.5 10 30 34.1.562 26.6 27.0.5 10o2 17 18.8.576 14.5 15.0 1 i 10-5 97 (100) (72) (110) (. 574) (78).1 10-4 79 85.6.586 59.4 60.7.1 10-3 60 64.7.593 44.8 45.9. 102 41 43.9.595 29.6 30.2.01 10-5 137 (150) (.580) (102) -4.01 10-4 116 120.597 80 81 130 86 87 -3 64..01 10-3 92 97.9.601 63. 64.7 01 102 68 71.9.603 45.2 46.4.84.9 10-7 25 (32) (.477) (30) (33) (31).9 10-6 20 27.4.445 25.7 26.4.9 10- 15 20.4 442 19.5 20. 0 ~5 10- 36 45..506 38.65 39.0.5 10-5 29 36.3.507 30.9 31.6 -4.5 10- 22 27.5.509 23.4 24.0 -3 16 3.5 10-3 16 19.4.498 16.7 17.3 ~ 1 10-5 49 (54) (42) (56) (. 553) (44) ~.1 10-4 40 46.3.547 35.9 36.6 1~ ~54 59 3.

A p n a n nor n or d d2 (BS) ~~~~~~-3 ~.5 26.4 27.0 1 10 31 34.8.556 26.4 27.0.1 10-2 21 23.7.556 17.6 18.2.01 10-5 69 76 57 78.566 58.01 10-4 58 65.6.563 47.1 48.4.01 10-3 47 52.4.566 37.6 38.8.01 10-2 36 37.9.578 26.0 26.7.92.9 10-7 12 (23) (23) (24) (.354) (24) 5 10-7 22 (33) (.416) (31) ~ 5 10- 6 1 27.6.5 10-6 18 27.9.411 26.4 25.7 -5 28.8 29.6.5 iO5 15 24.4.377 28.8 29.6 -4.5 10-4 11 17. 1.410 16.3 16.9.1 10-7 33 (47) (41) (48) (.455) (42).1 10-6 29 39.5.467 34.2 34.9.1 10-5 24 33.7.446 28.8 29.6 - 4 2 7..1 10-4 20 27.1.467 23.4 24.0 -3.1 10-3 15 20.3.478 17.3 17.9 i. i-2 10 13.3.497 11. 1 16.8.01 10-5 35 44.4.496 36.3 36.9 -4.01 10- 29 36.6 505 29.5 30 -3. 9 3. ~.01 10- 23 29.0.512 23.2 23.9.01 10-2 17 21.2.517 16.3 16.9 98.5 o0-7 10 (22) (22) (23) (23) (23) (24) (24) (.297) (24).1 10-7 16 (28) (.350) (27) (29) (28) 14 24.1 0-6 23.4 24..10.1 10 11 20.-8 349 19.5 20.0.01 10 7 21 (36) (33) (37) (.378) (34) 20

-' s.001.002.003.005.01.02.03.05.1.2.3.5 I 2 3 5 10 I I. I i1111 I I I, I,,,,,,,..4.2.54.56.59.62.67.76.84.92.96.52.54.56.59.62.67.76.84.92.96 Fig. 1. Values of efficiency for C. C. C. (Based on Table I, using n. ) Table II. Efficiency of C. C. C. (BS and MBS) corresponding to a, /, p triples. In this table, for each a, 13, p triple, the value of n is selected which yields the a, i pair closest to the given pair. Then ASN is calculated from the tables by (19). The efficiencies are calculated according to the formulae: 71(BS) = -N 77(MBS) = AN n' ASN' Roughly speaking, i7(BS) depends primarily upon 13 (inversely) and secondarily upon a. No such dependence was observed for'qSN(MBS), but for very small n, corresponding to the largest values of a and 3, q7(MBS) is very high. For U7N(MBS), ASNN = 2(n+l-d), to three-place accuracy (except for an occasional error of 1 in the third place).

Table II. Efficiency of C. C. C. (BS and MBS) corresponding to a, /3, p triples. oP 0 a0 n n d a 3 ASNSN (BS) (MBS) (MBS).52.9 10-2 684 650 355.01029.90249 614.636.674.694 -4.54.9 10o 925 950 533.00009.89796 906.640.671.727 -3 10 510 500 285.00100.90355 468.634.678.730 o2 170 180 106.01030.89301 162.633.704. 758 -2 5 102 841 850 459.01075.48591 828.636.653.687 -5 -5 56.9 10 615 600 352 10.89901 564.653.695.787 -4 10 410 420 249.00008.90473 389.635.686.775 -3 10 226 220 133.00117.89708 199.630.697.785 -2 o10 75 78 50.00843.90861 65.1.629.754.846 -4.5 10o 950 950 533.00009.51251 926.637.654.724 -3 10o 659 700 392.00084.48421 680.634.653.718 -0 374 380 213. 01043.48681 366.634. 658. 717 -2.1 10 900 900 486.00895.10718 865.635.661.689 59.9 i0-5 270 260 164 10-5.89906 235.664.735.890 10o i80 190 121.00010.89284 169.624.701.847 -3 10 99 100 66.00089.90777 84.5.625.739.893 -2 0-2 33 35 25.00834.90888 26.3.620.825.986 5 10 551 550 325 10.49896 532.640.661.779 10 418 440 260.00008.49497 424.630.654.766 -3 10 290 300 177. 00108. 47521 287. 631. 659. 763 -2 10o 164 160 95. 01079. 50448 151. 631. 669. 765 -5 -5 I 10 931 950 542 10.10523 916.619.642.719 -4 10 758 750 426.00011.10367 720.633. 659.730 -3 10o 579 600 339.00082.09938 573.633.663.725 -2 10 394 400 224 00933. 10221 378 633 670.715 -3 ~ 01 10-3 888 900 497.00096.00990 842.633.677.705 -2 10 655 650 355.01029.01062 602.634.685.696 -5 -5.62. 9 10 150 150 101 10.89728 130.642.741.963 -4 10o 100 i10 75.00009.89296 93.6.611.718.933 -3 10 55 56 40.00092.90760 44.1.612 778 1.008 10-2 19 20 16.00591.92739 12.8.603.941 1. 206 -5 -5 5 10 306 320 199 10~.50275 307.607.633.796 -4 10 232 240 149.00011.48197 229.625.655. 815 -3 i0 161 160 100.00098. 51688 151.624.662.818 -2 10 91 94 59.00860.47524 87.0.681.236.889 22

17 17 17 p, a n n d'a 1 ASNSN (BS) (MBS) (MBS) -5 -55.1 10-5 517 550 326 10-5.08709 524.616.647.753 -4 6.6 104 420 420 248.00012.09778 398.629 664 763 103 322 320 188. 00103.10511 302.629 666 756 10-2 219 220 128.00904.10867 205.630.677.741 -5 -.01 10-5 730 750 434 10-5.00921 700.626.671.740 -4 -4 10 615 600 346 10.01331 558.629.677 739 -3 -3 10-3 493 500 285 10-3.00977 460.631.686 727 -102 364 380 214.00790.01018 345 632 696 718 ~~~~~~~~~~~~~~~~~~~~~~~~ 10 -5 -5 67.9 10- 72 78 58 10.89822 62.7.594.739 1.103 -4 10 48 54 41. 00009.89647 41. 6 590 766 1. 138 -3 41.6 ~590 ~ 76 1. 138 103 27 29 23.00116.88996 21.4 590.800.222 -5 -5.5 105 148 150 101 10-.49606 141.630.670.945 10-4 112 120 81.00008.50333 112.611.655.916 10 10o 78 80 54.00116.48512 73.6.614.667.910 10-2 44 47 32.00931.49415 42.3.616.685.906 -5 -5.1 10-5 250 260 165 10-5.10114 245 610 647 826 -4.610.64.826 10 203 220 138. 00010.07896 205.618 664 819 -3 103 155 i60 100.00098. 09871 148. 621 671 814 10-2 106 110 68.00837.10538 101. 624.679 795.01 10-~5 77-5 01 105 352 360 221 10.01085 330. 618.674.795 iO-4 296 300 183. 00008. 01235 273.624 686 793 -3.624 ~68.793 103 238 240 144.00117.00954 2i5 625 698 772 -~~~~~~~~~~~~~25 ~ 625 ~1 69 ~772 1-2 175 180 106. 01030.00918 158 627.714 751 76.9 o-6 38 46 39.0000009.8927683 32.4.544.773 1.564 -5 -5 10 28 35 30 10.87754 24.1.551.800 1.607 10 19 24 21.00014.86233 15.7.540.825 1.620 -3.97 i 4 103 11 14 13.00092.88373 7.73.528.957 1.849.5 10-5 57 64 49 10.47351 58.0.588.648 1.176 -45.648 1. 176 104 44 48 37.00011.49086 42.8.577.647 1.154 -3 1. 15 10 30 35 27.00094.47040 39.9.580.657 1.015 -2 ~~~~~~~~~~~~~~. 580. 657 1. 015 10-2 17 19 15.00961.49363 16.1.586. 692 1. 117 -5 -5.1 10 97 110 78 10.08887 102.574.619.957 -4.619.97 104 0.79 86 61.00007.11160 79.2.589.639.974 103 60 66 46.00093.09229 59.8.598.661.940 0-2 41 44 30.01131.08558 38.9.606.686.890 1~~~~~~~~~~~~~~~~~~80 -5 -5.01 10 137 150 102 10.01005 134.580.649.888 -4 10 116 130 87.00007.00720 114.565 10-3 92 98 65.00080.01116 85.4.606.695.873 -2 ~~~~~~~~~~59.4.606.695.873 10-2 68 70 45.01123.00959 59.2.612 23

1) 17 17 p a0 a0 n n d a ASNSN (BS) (MBS) (MBS) -7 -7.84. 10 25 32 30 10.9052782 17.9.477.853 2. 544 -6 o-10 20 28 26.0000015.8479863 17.4.479.771 2.235 -5 10 15 21 20.0000105.8715202 11.5.469.856 2.462 -6.5 1o 36 46 39.0000009.4592471 40.6.523.592 1. 504 -5 10 29 35 30.0000112. 4969924 30. 1. 521. 606 1. 520 -4 10o 22 28 24.00009.47195 24.0.526.614 1.473 -3 10o 16 20 17.00129.40100 17.3.539.623 1. 348 -5 -5.1 10 49 56 44 10.10206 51.5.553.601 1.191 -4 10 40 46 36.00008. 10685 42.0. 553.606 1. 156 -3 10 31 35 27.00094.09542 31.5.564.626 1.097 -2 10 21 25 19.00732.09204 22.1.575.650 1.029 -5 -5.01 10 69 78 58 10.00965 69.0.566.639 1.051 -4 10 58 64 47.00011.00996 55.9.573.656 1.019 -3t 10o 47 54 39.00075.00868 46.4.578.672.974 10o2 36 37 26.01004.01022 30.9.589.705.910 92.9 1 2 24 24. 8648214 10.8.354.787 4.248.92.9 10 12 24 24 1 -7 -7 5 10 22 33 31 10. 4981990 27.0. 416. 508 2.288 -6 10 18 28 26. 0000015. 3905792 24.0. 443. 517 2. 067 -5 10 15 25 23. 0000097. 5911838 22.0. 331. 376 1.379 -4 10 11 18 17.00007.42812 14.8.448.545 2. 017 -7 -7.1 10 33 48 42 10.0859854 44. 8.455.487 1.560 -6 10-6 29 40 35.0000007.0967268 37.2.475.511 1. 583 -5 10o 24 32 28.0000097.1084891 29.6.480.519 1.536 -4 10 20 28 24.00009.06861 25.6.495.542 1.386 10-3 15 21 18.00074.08193 19. 1.504.554 1.323 io0-2 10 14 12.00647.09583 12, 6.520.578 1. 217.01 10-5 35 45 37.0000077.0084345 40.2.507.568 1.267 -4 10 29 37 30.00010.00788 32.6.515.584 1.191 -3 10 23 30 24.00072.00825 26.0.527.608 1. 129 -2 o10 17 22 17.00845.00637 18.5.548.652 1.007 -7 -7.98.5 10 10 24 24 10.3842197 19.2.297.371 3.564 -7 -7.1 10 16 28 27 10.1074660 26.4.350.371 2.450 10o6 14 25 24.0000008.0886451 23.6.357.378 2.231 -5 10 11 21 20.0000105.0653488 19.9.374.394 1.964 -7 -7.01 10 21 37 34 10.0062433 34.6.378.404 1.748 24

.001.002.003 005.01 02 03.05.1.2.3.5 I 2 3 5 10 1.0 I I I I I I I I I I I I.9.7.6_ _.5.4.3 I I.52.54.56.59.62.67.76.84.92.98 p Fig. 2(a). Efficiency of C. C. C. (BS) using actual a, /3, n..001.002.003.005.01.02.03.05.1.2.3.5 1 2 3 5 10 1.0' I I I I I I I I I I I I I I I I I.8.7 - =.6 -.5 - 4 _.3, I.52.54.56.59.62.67.76.84.92.98 Fig. 2(b). Efficiency of C. C. C. (MBS) using actual a, /, n, under signal plus noise. 25

In Ref. 6 a simple expression was developed to be used as an approximation for the clipper crosscorrelator efficiency when =. 50. It was simply (2p-1/s)2. Table III lists this value, and the range q)(BS) at tO =. 50 from Table II. p s (2p-1/s)2 U(BS) at So =. 5 from II very small. 6366 —~54. 1004.6349.636 ~ 56. 1510. 6315. 634-. 637 ~ 59. 2275. 6260. 619-. 640.62.3055.6172. 602-. 681.67.4399. 5974.611-. 630 * 76.7063. 5420. 577-. 588.84.9945.4675. 521-. 539,92 1. 405.3574. 331-. 448.98 2. 054.2184.297 Table III. Comparison of Ref. 6 and some values of Table II. 26

10. A SEQUENTIAL STRATEGY The preceding strategies all consist of choosing, before taking any observations, a sample size n (and a cut-point d) necessary to obtain a given a, 3, p triple. In the case of MBS and MIBS, however, we notice that it may sometimes be possible to make a decision before n observations have been taken. The MBS (as well as the MIBS) is a form of sequential test. That is, after each observation, we make one of the following three decisions: (1) accept hypothesis A (signal is present), (2) accept hypothesis B (signal is not present), (3) take an additional observation. We use two cut-points to make our decision. In the case of MBS, we have 1) if C (no. of ones) = d, accept A; 2) if z (no. of zeros) = n-d+1, accept B; 3) otherwise, take another observation. 10. 1 Sequential Probability Ratio Test A much more efficient sequential test is. the sequential probability ratio test (see Ref. 3). This is an optimum test when the cut-points are chosen correctly. The test is defined as follows. Denote by 0i the probability that xi O0. Testing hypothesis SN against hypothesis N is equivalent to testing the hypothesis 0 = p = (s) against the hypothesis 0 = 1. The probability of obtaining a sample [c(x1), c(x2),..., c(xm)] with Z zeros and C ones is given by P(C,z) = QC(1-Q)z, where C + z = m. (23) Under hypothesis SN the above probability is given by PsN(c, z) = pC(l1p)z (24) and under hypothesis N by 1C 1z 1C+z 1m PN(cz) = 2 2 (25) 27

The likelihood ratio for a sample with the same number of zeros and ones as the given sample is given by PSN(C'Z) C+z2 C Z (26) ) PN(Cz) p (-) (26) The likelihood ratio for the given sample can therefore be computed from (26), and depends on the number of zeros, Z, and ones, C, in the sample. The test is carried out in the following manner: two positive constants y and 5(y < 6) are chosen. After each trial, k(C,z) is computed. If i > 6, judge SN to be true. This is denoted by "A. " If f < y, judge N to be true. This is denoted by "B. " If y < f < 6, take another observation. The values of a and A are fixed by the values of p, y, 6. For practical purposes, it is easier to calculate log l(C,z), after the mth observation, than to calculate k(C,z). The log-likelihood ratio, log k(C,z), is given by log I(C,z) = log [2C pC 2Z(l-p)Z] = C log 2p + z log 2(1-p). (27) The test now takes the form: If log I(C,z) > log 6, A occurs; (28) if log e(C,z) < log y, B occurs; (29) if log y < log B(C,z) < log 6, take another observation. (30) Equation 27 can be written in the form _ log f(m) log 2p C (31) log 2(1-p)- log 2(1-p) If we define log 6 log 2p C and A log 2(1-p) log 2(1-p) (32) log - log 2p 33 B log 2(1-p) log 2(1-p) ( 28

we can write the sequential test as follows: For a given (C, Z), If Z < ZA(C), accept A, (34) if Z > ZB(C), accept B, (35) if ZA(C) < Z < ZB(C), take another observation. (36) Graphically, BS, IBS, MBS, MIBS, and the sequential probability ratio test (SPRT) can be summed up as shown in Fig. 3 (for given a, 3, p). 10. 2 SPRT when y, 6 are Determined by Wald's Approximation Let ao, gO be the design values of false alarm and miss probabilities, respectively. Pick the values of y and 6 using Wald's approximation: 1- ( 6 - 0 (37) and Y 0 1-a (38) T The actual values of a, /3 are not, in general, equal to the values of ao0 f3o' because the test does not necessarily terminate with specific likelihood ratios I = 6 or 2 = y. The fact that m must take on integral values makes possible a "spill-over" in likelihood ratio at the boundaries (6 and y). For all observations terminating in an A decision, the expected value of the likelihood ratio is EA[ 2(C, Z)] = (39) similarly for B decisions EB[(C, Z)] = 1-a (40) It is evident that there is also a possible "spill-over" in the values of a, f3. We shall now calculate these "spill-overs." Suppose that after n observations an A decision is reached; i. e., 2(C, Z) = 2Z+C(1-p)z p > 6, with C+Z = n. (41) On the (n-l) observation, the likelihood ratio satisfied the inequality 29

z z B B n-d n-d I A A C - C d n d BS I BS zZ n-d+l B n-d _L A C C d-I d MBS SPRT MI BS Fig. 3. Termination boundaries for clipper crosscorrelator. I/ 0Q 1-ro Fig. 4. Range of a and wheny = 1 and 6 = o o 30

Z+C-1 C-1 y < 2(C-1,Z) = 2 (1-p)z p <. There will be a real number x (C-1 < x < C) such that 2Z+X (1-p)z p = 6 (42) Dividing (41) by (42) we have ~(C,Z) = 2C-x C-x and since p >, 6 < K(C,Z) < 2p 6. (44) Similarly, if a B decision is reached at C+Z = n, we have (C, Z) = 2Z+C (1-p)Z pC < and Z-(+C Z-1 C y < (C, Z-1) = 2 (l-p) p < 6, and so there is an x (Z-1 < x < z) such that 2x+c (1-p)x p =. Therefore we have (C' Z) = 2Z-x (1_p)z-x and 2(1-p)y < B(C,Z) < y. (45) Equations 44 and 45 set bounds on the likelihood ratio at termination, and these two equations, with Eqs. 37, 38, 39, and 40, can be used to set bounds on a and /: 1- 1-3~ a < a < 2p 0 (46) o 0 and 2(1-p) 1- < < (47) O O

These relationships can be seen graphically on the ROC curve (Fig. 4). The range of a, P is indicated by the hatched region. 10. 3 ASN for the SPRT On page 53 of Ref. 3 the following formula is derived: E(n) 0log k1+... +lo)g (n) E (n) = (48) E0(log l) where: E0(y) is the expected value of y for a given value of 0, n is the number of observations necessary for termination, PsN[C(Xi)] log i = log PSN[c(Xi)] and PsN[c(X)] PN [c(X)] The numerator is the expected value of log f(m) at termination: E0(log L1 +1. + log fn) = E [log k(C, Z)], where C+Z = n P0(A) log 6 + P0(B) log y. (49) The denominator is given by E0(log f) = 0 log 2p + (1-0) log 2 (1-p) The average sample number can thus be written as E(n) (1-/3) log 6 + j3 logy (50) SN(n) p log 2p + (l-p) log 2(1-p) and a log 5 + (1-a) log y EN(n) 1/2 log 2p + 1/2 log 2(1-p) (51) Since Eq. 49 is based on Wald's approximation (Eqs. 39 and 40), there is no loss of accuracy in writing (50) and (51) in the form

(1- 3) log 1 log E SN(n) a 1 (52) SN(n) p log 2p + (1-p) log 2(1-p) (52) a log 13 + (1-a) log 1-a (53) EN(n) 1/2 log 2p + 1/2 log 2(1-p) 10. 4 Comparison of SPRT with MBS In Fig. 5, the termination boundaries of SPRT are compared with those of MBS for the triple p =.67, a =.00098, 3 =.09871. 65 B (MBS) 60 - 50 40 30,.. I0 0 20 30 40 50 60 70 80 90 100 105 Fig. 5. Termination boundaries for p = 0. 67, a = 0. 00098, and 3 = 0. 09871. 33

11. CONCLUDING COMMENTS AND REMARKS ABOUT FURTHER WORK We have considered in this report the problem of signal detection using a clippercrosscorrelator when the signal of known single size is possibly present and with Gaussian noise in the background. As is well known, the solution to a dichotomous statistical decision problem always involves the recognition and reconciliation to the two types of errors known as a -error and P-error. In the signal detection problem the a-error takes the form of "false alarm" and the /3-error means "miss." We have suggested five such strategies which arise in a very natural way and have studied their interrelations. We have also defined and investigated the efficiency of the clipper-crosscorrelator in relation to the usual crosscorrelator. In this report the investigations and results have been all theoretical in nature and yet have been intuitively very meaningful. Associated tables and charts may be useful for application. 34

REFERENCES 1. C. P. Patil, Contributions to Estimation in a Class of Discrete Distributions, Doctoral Dissertation at The University of Michigan, Ann Arbor, Michigan, 1959. 2. G. P. Patil, "On the Evaluation of the Negative Binomial Distribution With Examples," Technometrics, Vol. 2: 501-505, 1960. 3. A. Wald, Sequential Analysis, John Wiley and Sons, New York, 1947. 4. National Bureau of Standards, Tables of the Binomial Probability Distribution, Applied Mathematics Series 6, Issued January 1950. 5. Staff of the Computation Laboratory, Harvard University, Tables of the Cumulative Binomial Probability Distribution, Harvard University Press, 1955. 6. T. G. Birdsall, "On the Extension of the Theory of Signal Detectability," U. S. Navy Journal of Underwater Acoustics, Vol. 11, No. 2, p. 183, April 1961. 35

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