ENGTINEER:ING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR SIGNAL DETECTION AS A FUNCTION OF FREQUENCY ENSEMILE Technical Report No. 86 Electronic Defense Group Department of Electrical Engineering A. B. Macnee By: Florence A. Veniar Approved by: Project 2262 TASK ORDER NO. EDG-3 CONTRACT NO. DA-36-039 sc-63203 SIGNAL CORPS, DEPARTMENT OF THE ARMY DEPARTMENT OF ARMY PROJECT NO. 3-99-04-042 SIGNAL CORPS PROJECT NO. 194B August, 1958

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS ABSTRACT 1. INTRODUCTION iv V I 2. THE 2.1 2.2 2.3 2.4 MATHEMATICAL MODELS The Narrow-Band Scanning Model The Multiple Band Model The Null Hypothesis The Mathematically Optimum Detector 3. THE FIRST EXPERIMENT 3.1 Calculation of Predictions for Multiple Ensemble Performance 3.1.1 By Narrow-Band Scanning Model 3.1.2 By Multiple Band. Model 3.1.3 By the Null Hypothesis 3.2 Comparison of the Obtained Data with Three Hypotheses 3.3 Calculation of the Mathematically Optimum Detector 3.4 Comparison of the Obtained Data with Mathematically Optimum Detector 3.4.1 I as a Function of Ensemble Size 3.4.2 j as a Function of Ensemble Frequency Range 1. 1 2 2 It 4 6 6 6 10 14 14 14 4. THE SECOND EXPERIMENT 4.1 Introduction 4.2 Results 4.2.1 Detection of a Single Signal in Noise 4.2.2 Detection of Multiple Ensembles 4.2.3 Efficiencies 4.2.4 a and P(C) as a Function of Ensemble Size and Ensemble Frequency Range ].6 16 17 17 17 21 21 5. CONCLUSION REFERENCES DISTRIBUTION LIST 29 31 32 iii

LIST OF ILLUSTRATIONS Page Figure 1. Figure 2A. Figure 2B. Figure 3. Figure 4. Figure 5. A comparison of the obtained data with the probabilities of correct detections as predicted by three models. Efficiencies, j' s, for each of the experimental conditions. Each unconnected point represents j for the detection of a single signal; each line cormecting two points, _ for an ensemble-of-two; each line connecting four points, a for an ensemble-of-four. The points on a line represent the signal frequencies constituting that ensemble. Frequencies in cps are noted on the abscissa. Efficiencies, I's, for each of the experimental conditions. Each unconnected point represents j for the detection of a single signal; each line connecting two points, a for an ensemble-of-two; each line connecting four points, j for an ensemble-of-four. The points on a line represent the signal frequencies constituting that ensemble. Frequencies in cps are noted on the abscissa. A comparison of the obtained data with the probabilities of correct detections as predicted by three models. A comparison of the obtained data with probabilities of correct detections as predicted by the narrow-band model and the multiple band model. A line is drawn from the less deviant prediction to the zero line. Efficiencies, a's, for each of the experimental conditions. Each unconnected point represents a for the detection of a single signal; each line cormecting two points, _ for an ensemble-of-two; each line connecting four points, j for an ensemble-of-four; each line connecting eight points,. for an ensemble-of-eight. The points on a line' represent the signal frequencies constituting that ensemble. Frequencies in cps are noted on the abscissa, 9 12 13 22 23 25 iv

Abstract This report presents the results of two experimental investigations of the detection of a signal in noise as a function of signal ensemble size and ensemble frequency range. Signal ensemble size is defined as the number of signals of different frequency that are equallylikely to occur and which the observer must try to detect. Ensemble frequency range is defined as the frequency separation between the highest and the lowest frequency signals included in that ensemble. Data are presented for four observers in the first experiment and for three observers in the second experiment. The results obtained are compared with predictions based upon the null hypothesis, a narrowband observer model, a multiple-band observer model, and the mathematically optimum detector. The possibility is suggested that the models discussed are not necessarily mutually exclusive, but rather complementary in an overall view of detection, v

SIGNAL DETECTION AS A FUNCTION OF FREQUENCY ENSEMBLE 1. INTRODUCTION In 1956, Tanner, Swets and Green reported a decrement in signal detection as the number of different frequencies which the signal might assume increased from one to two. The decrement was not only found to be a function of the uncertainty thus created by the greater ensemble of frequencies, but also a function of the difference between fl and f of the signal. The greater the difference between the frequencies, the greater was the decrement. It was the purpose of the present investigations to explore more systematically detectability of auditory signals as a function of' (1) ensemble size and (2) ensemble frequency range; and further, to determine the extent of agreement between the obtained experimental data and predictions made on the basis of four mathematical models. Two major experiments were carried out and are presented separately in sections III and IV. 2. THE MATHEMATICAL MODELS 2.1 The Narrow-Band Scanning Model This model is based on the assumption that the observer is, at a single "instant" in time, sensitive to only a narrow range of frequencies; that the hearing mechanism requires a measurable amount of time to shift from the observation of one frequency to another, and that, in so doing, it must shift through the intervening frequencies. 2.2 The Multiple Band Model This model is based on the assumptions that the observer can

attend to several frequency bands at once though these bands need not be adjacent, and that the outputs of these bands are linearly combined by the observer. 2.3 The Null Hypothesis The Null Hypothesis predicts performance for the two and four signal ensembles to be no different from average performance on the onesignal-ensemble. 2.4 The Mathematically Optimum Detector The mathematically optimum detector is perfect except for limitations placed upon it externally by the system. In the present experiment, accuracy of the optimum detector is limited by noise and the number of alternative signals which it must detect. 3. THE FIRST EXPERIMENT In the first investigation, signal ensembles of one, two, and 2 four frequencies were used. In the detection of one frequency, d' values were individually obtained for eight signals: 1007, 1057, 1107, 1157, 1207, 1257, 1307, and 1357 cps. In the detection of two equally-likely frequencies the following pairs were used: 1157 and 1207; 1107 and 1257; 1057 and 1307; 1007 and 1357 cps, giving range differences of 50, 150, 250, and 350 cps respectively. In the detection of four equally-likely frequencies the following groups were used: 1107, 1157, 1207, 1257; 1057, 1107, 1257, 1307; 1007, 1057, 1507, 1357 cps; and 1007, 1157, 1207, 1357 cps, giving range differences of 150, 250, 350, and 350 cps respectively. The last two ensembles enabled the experimenter to study pattern effects under constant ensemble size and range. All trials were two-interval forced-choice and were presented 2

binaurally (through PDR-8 headphones) to the observers in the following time order: (1) A warning light to indicate that the trial is about to begin, followed in 0.44 second by (2) Test interval I. - 0.42 second in duration (3) Test interval II. - 0.42 second in duration The signal, which appeared in either of the test intervals, was 0.10 second in duration. (4) Response interval - 1.06 seconds extent. A red light flashes if the observer's response is correct. (5) 0.50 second after the response period the correct response is indicated visually. (6) 1.36 seconds lapse between the end of one trial and the beginning of the next trial. 100 trials constituted a run. Eight runs constituted an experimental session. Voltages of the signals, as measured at the headphones, were variously set (0.0045 v for the two highest frequencies and 0.0042 v for the others) in an attempt to keep the detection constant for all signals. All runs were done in a background of white Gaussian noise of 20 KC bandwidth at an amplitude of 0.06 volts. Conditions were counterbalanced and so presented as to include at least one run of every ensemble size in every experimental session, Thus, in any one session those four frequencies which comprised the foursignal ensemble of the experimental session were presented also individually, making four runs of ensembles-of-one, and in pairs making two 3

ensembles-of-two. For example, if the ensemble-of-four which was to be presented on a particular day consisted of 1007, 1157, 1207, and 1357 cps, detection indices were obtained for that ensemble; for each of these frequencies presented individually; and also for the pairs 1007 or 1357, and 1157 or 1207 cps. Approximately sixteen runs were obtained for each condition. The data thus obtained were then compared with predictions made on the basis of the hypotheses, or models, as described in Sec. II - above. Table I gives the obtained data for the ensemble-of-one condition both in percent correct and in d' for each of four observers. In Tables IIA, IIB, IIC, and IID data relating to the multiple ensemble conditions are presented. Here predictions are listed in percentcorrect for the first three hypotheses discussed above. The deviations of these predictions from the obtained data, and the obtained data are also listed. 5.1 Calculation of Predictions for Multiple Ensemble Performance 3.1.1 By Narrow-Band Scanning Model. The predicted percent correct is determined in the following manner: since the observer can attend to only one stimulus at a time, that stimulus in the ensemble for which detection is best (i.e., percent correct is greatest) will be the stimulus the observer will listen for. He will, therefore, have the same percent correct for that stimulus as he did in the one-of-one condition through all ensemble sizes. When the stimulus to which he is attending does not occur, the observer will not have heard a signal in either presentation interval and his response, therefore, will be based upon a guess. In a two-alternative forced-choice type of experiment such as this is, the

TABLE I Results Obtained From Four Observers on the Detection of Single Signals in White Gaussian Noise Data are presented as percent correct and their respective d's. 0:SS 0:SR O:GP 0: SD Signal Frequency P(C) dC)' c) d' P(C) d' P(C) d' 1007 89.13 1.74 79.50 1.16 71.00 0.78 85.70 1.51 1057 91.28 1.92 74.17 0.91 70.11 0.75 88.28 1.68 1107 91.22 1.92 75.25 0.96 67.71 0.65 84.28 1.425 1157 88.06 1.66 74.38 0.92 66.57 0.61 82.08 1.29 1207 893.8 1.76 77.89 1.08 67.86 0.66 82.92 1.35 1257 87.61 1.63 70.92 0.78 68.71 0.68 87.00 1.59 1307 91.38 1.93 77.17 1.04 76.38 1.02 89.15 1.73 1357 91.00 1.90 80.25 1.20 72.50 0.85 89.60 1.77 5

probability of a guess being correct is 0.5..1.2 By Multiple Band Model. More recently, Green3 has obtained data which seem to support a multiple band observer model, rather than the single narrow-band observer described above. In the multiple band observer model, d' is a distance in sigma units on a decision axis between the means of two Gaussian distributions of equal variance, noise (N) and signal-plusnoise (S+N). The mean of N is, for simplicity, set at zero, variance at 1; the S+N distribution for a set of signals S1 (all signals within the set being within the same critical band) has, then, mean d' and variance 1. Now consider two different sets of signals-in-noise, S1 and S2, such that d'l = d' If these signals are presented randomly one at a time and — 1 -2' the observer is listening for either of two, the noise distributions of the two critical bantrs are added (the mean remains zero, variance is multiplied d'1l by 2) and the predicted values for S1 and S2 in sigma units are N and _'2 N a, respectively. The equivalent percentages of these new d' values, weighted according to the relative frequency with which they occurred during the trial, constitute the predicted percent-correct for that condition. The same logic is applied to l-of-X signal sets, variance always being multiplied by the number of critical bands in operation, hence, the original d's are always divided by the square root of the number of critical bands in operation. 3.1.3 By the Null Hypothesis. Performance on the component signals is averaged to predict performance on the multi-ensemble conditions. 3.2 Comparison of the Obtained Data with Three Hypotheses. The deviations of the predicted P(C) from the obtained P(C) by each of our three models discussed above are presented in Tables IIA through IID 6

TABLE IIA A Comparison of the Obtained Data and the Probabilities of Correct Detections as Predicted by the Three Models O:SS Narrow-Band Model Multiple-Band Model Null Hypothesis Predicted Predicted Predicted Signal Predicted Minus Predicted Minus Predicted Minus P(C) Ensemble(cps) P(C) Obtained P(C) Obtained P(C) Obtained Obtained 1157 or 1207 60.58* -27.82 88.72 0.32 88.7 0.30 88.4 1107 or 1257 70.45 -12.50 85.2 2.25 89.5 6.55 82.95 1057 or 1307 71.42 -11.39 86.72 3.91 91.3 8.49 82.81 1007 or 1357 71.81 - 9.44 81.9 0.65 90.0 8.75 81.25 1107, 1157, 86.15** 1207 or 1357 60.17 -23.83 82.8<>89.5 2.15 89.6 5.6 84.0 1057, 1107, 1257 or 1307 59.48 -22.12 85.96 4.36 90.3 8.7 81.6 1007, 1157, 1207 or 1357 61.23 -21.27 76.4 -6.1 89.9 7.40 82.5 1007, 1057, 1307 or 1557 61.14 -19.39 84.31 3.78 90.7 10.17 80.53 TABLE IIB A Comparison of the Obtained Data and the Probabilities of Correct Detections as Predicted by the Three Models 0:SR Narrow-Band Model Multiple-Band Model Null Hypothesis Predicted Predicted Predicted Signal Predicted Minus Predicted Minus Predicted Minus P(C) Ensemble(cps) P(C) Obtained P(C) Obtained P(C) Obtained Obtained 1157 or 1207 64.76* -10.49 76.14 0.89 76.1 0.85 75.25 1107 or 1257 63.22 - 7.78 66.95 -4.05 73.1 2.10 71.00 1057 or 1307 62.96 -10.44 68.6 -4.8 75.7 2.30 73.40 1007 or 1357 66.44 -10.16 72.2 -4.4 79.9 5.30 76.6 1107, 1157, 71.32** 1207 or 1257 57.26 -15.24 68.05<>74.6 0.82 74.6 4.10 70.50 1057, 1107, 1257 or 1307 56.22 -11.08 67.78 0.48 74.4 7.10 67.30 1007, 1157, 1207 or 1357 57.98 -14.77 67.22 -5.53 78.0 5.25 72.75 1007, 1057, 1307 or 1357 58.36 -15.78 70.40 -3.74 77.8 3.66 74.14 * If one were to consider this ensemble to be within the same narrow-band, the predicted P(C) would be the same as the null, *M The midpoint of this interval, used in calculating predicted minus obtained. 7

TABLE IIC A Comparison of the Obtained Data and the Probabilities of Correct Detections as Predicted by the Three Models O:GP Narrow-Band Model Multiple-Band Model Null Hypothesis Predicted Predicted Predicte Signal Predicted Minus Predicted Minus Predicted Minus P(C) Ensemble(cps) P(C) Obtained P(C) Obtained P(C) Obtained O btained 1157 or 1207 60.0 * - 6.92 67.22 0.30 67.2 0.28 66.92 1107 or 1257 60.32 - 7.09 62.95 -4.46 68.2 0.79 67.41 1057 or 1507 64.67 - 7.53 67.15 -4.85 73.2 1.20 72.00 1007 or 1357 62.90 - 9.54 65.95 -6.49 71.8 -0.64 72.44 1107, 1157, 65.2 ** 1207 or 1257 54.56 -12.44 62.7<>67.7 -1.8 67.7 0.70 67.00 1057, 1107, 1257 or 1307 55.5 -15.27 65.00 -5.77 70.8 0.03 70.77 1107, 1157, 1207 or 1357 56.7 -10.3 61.58 -5.42 69.5 2.50 67.00 1007, 1057, 1307 or 1357 57.81 -10.06 66.5 -1.37 72.5 4.63 67.87 TABLE IID A Comparison of the Obtained Data and the Probabilities of Correct Detections as Predicted by the Three Models 0:SD Narrow-B.nd Model Multiple-Band Model Null Hypothesis Predicted Predicted Predicte Signal Predicted Minus Predicted Minus Predicted Minus P(C) Ensemble(cps) P(C) Obtained P(C) Obtained P(C) Obtained Obtained 1157 or 1207 1107 or 1257 1057 or 1307 1007 or 1357 1107, 1157, 1207 or 1257 1057, 1107, 1257 or 1307 1007, 1157, 1207 or 1357 1007, 1057, 1307 or 1357 66.85 * 67.4 70.79 69.75 60.17 59.00 59.91 59.58 -15.06 -13.47 -13.21 -11. 35 -19.03 -21.85 -20.69 -19.78 82.50 77.55 80.3 79.50 80.0 ** 75.9<>84.1 78.82 72.65 79.90 0.59 -3.54 -3.7 -1.60 0.80 -2.03 -7.95 0.54 82.5 85.7 88.5 87.7 84.1 87.5 85.1 88.5 0.59 4.83 4.5 6.60 4.90 6.65 4.50 9.14 81.91 80.87 84.00 81.10 79.2 80.85 80.60 79.36 same narrow-band, the predicte * If one were to consider this ensemble to be within the P(C) would be the same as the null. ** The midpoint of this interval, used in calculating predicted minus obtained. 8

and shown graphically in Figure 1. An examination of these data reveals a consistent underestimation of observers' performance by the narrow-band scanning model, and with one exception a consistent, though small, overestimation of observers' performance by the null hypothesis. Using nonparametric statistics, predictions by these two models are found to differ significantly from the data, predictions by the multiple band model, however, do not. The obtained data fell shorter of the null predictions in the oneof-four ensembles than in the one-of-two ensembles. This is a reflection of a greater decrement in performance with greater complexity of stimulus field. As regards effect of pattern of performance, there is no indication in the present study that this is a relevant variable. A Chi-Square test on the distributions oi P(C) for the two ensembles-of-four of equal range, showed no significant differences for all four observers. 3.3 Calculation of the Mathematically Optimum Detector Finally, in Table III the obtained data in d' are listed along with the d' expected of the mathematically optimum detector. For a signal ensemble of one, d" pi =pN where E is signal energy and N is optimum 0 O noise power per unit bandwidth. For a signal ensemble of M orthogonal f/E\ 2E 1 N0 N - - M optimum 1 M signals - M optimum -l(l-1 + M e ),and since 1-e is very much greater than 1 -, we can for simplicity eliminate 1 - from the equation. Our final estimate is d'n (see footM No note 4;) A comparison of the obtained with the mathematical optimum is expressed in terms of efficiency5 or i, where a = (d' bbt./d' opt.)2. The efficiencies thus calculated are presented graphically for the four observers in Figs. 2A, 2B, 2C and 2D, respectively. 9

FIG.I IL.l 0 LJ 0z - (p z 0 0c Q. +20 +10 O:SS 0 -10 -20 -30 +20 +10 O:SR 0 -10 -20 +20 +10 O:GP 0 -10 -20 +20 +10 O:SD 0 NARROW-BAND MODEL 00 0 0 ~* * * - * * L 000.0.00 MULTIPLE BAND MODEL. 0 0..0 * 0. 0 0 * 1 NULL HYPOTHESIS *~ * ~ ~ 0 In 00 * 0 ** * * *a0 0*00 SIGNAL FREQUENCIES CONSTITUTING THE MULTIPLE ENSEMBLE -10

TABLE III A Comparison of d' for the lMathematically Optimum Detector with d' Data for Three Observers, and Efficiencies (l's) Calculated for All Experimental Conditions Signal Signal ____ | O:S]R O:GP 0:'SD Frequency (cps) d' opt. d' obt. ) ct' obt. Tl d' obt. T d' obt. Tr 1007 5.66 1.92 0.114 1.16 0.042 0.94 0.027 1.66 0.086 1057 5.66 2.04 0.150 1.06 0.035 0.92 0.026 1.80 0.101 1107 5.66 1.82 0.103 1.02 0.032 0.74 0.017 1.51 0.071 1157 5.66 1.69 0.089 1.00 0.031 0.73 0.017 1.39 0.060 1207 5.66 1.90 0.112 1.16 0.042 0.72 0.016 1.42 0.063 1257 5.66 1.72 0.092 0.79 0.019 0.73 0.017 1.52 0.072 1507 6.02 1.81 0.090 0.96 0.025 1.12 0.034 1.56 0.067 1357 6.02 2.06 0.117 1.28 0.046 0.96 0.025 1.76 0.086 1157 or 1207 5.59 1.73 0.096 0.95 0.029 0.66 0.014 1.531 0.055 1107 or 1257 5.59 1.43 0.065 0.84 0.023 0.71 0.016 1.28 0.052 1057 or 1307 5.78 1.29 0.050 0.83 0.021 0.87 0.023 1.42 0.060 1107 or 1357 5.78 1.28 0.049 1.08 0.035 0.89 0.024 1.33 0.053 1107, 1157, 1207 or 1257 5.54 1.45 0.068 0.83 0.022 0.69 0.016 1.19 0.046 1057, 1107, 1257 or 1307 5.63 1.36 0.058 0.76 0.018 0.92 0.027 1.30 0.053 1007, 1157, 1207 or 13557 5.63 1.539 0.061 0.87 0.024 0.71 0.016 1.28 0.052 1007, 1057, 1307 or 1557 5.72 1.20 0.044 0.95 0.028 0.75 0.017 1.22 0.046....1 5............ _......................,....rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr oll.6 TTTTTTTTTTTTTTT1NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN.41o~4l.27.6mmmmmmm1.8 11

7 =(d' obt/d' opt)2 0 gn 0 n1' Uf~ 1007 1057 1107 1157 1207 1257 1307 1357 1007 1057 1107 1157 1207 1257 1307 1357 1007 1057 1107 1157 1207 1257 1307 1357 I U b o b b b~ b o 6 11obb,b -- W' - fm 4, -, -I i-'1I I i I I I 5 ro A 4 a) -4 i i I I I I I I I I I 1.... _ S * * * *.. I, I I If In. Vt'91E

~1 = (d'obt/d' opt)2 o O 0n 0 G) 0r I I I I I I I I I * I I 1007 1057 1107 1157 1207 1257 1307 1357 I I I I I I I I I I 0.. S... 0. C) z r 0 C I z - 0n 1007 1(07 1057 1107 1157 1207 1257 1307 1357 i4 r 1007 1057 1107 1157 1207 1257 1307 1357 I I I l I I I 8Z91IJ

3.4 Comparison of the Obtained Data with Mathematically ptinmum Detector 3.4.1 i as a Function of Ensemble Size. If mean I's for each ensemble size are considered, it is found that efficiencies for all observers are greatest for the smallest ensemble sizeo Three of four observers (O:SS, O:SR, and O:SD) show a further though smaller decrease in efficiency as the ensemble size increases from two to four. One observer (O:GP) shows no difference between the multiple ensembles of different size. If we compare efficiencies on each of the four-signal ensembles with the two-signal ensembles of equal frequency range (Table IV), we find that 0:SS shows greater efficiercies, three of four times, in the larger ensembles than in the smaller ensembles of comparable range. O:SR and O:SD and, with a single exception, O:GP, show lower efficiencies in the larger ensemble conditions of comparable frequency range~ 3.4.2 Tr as a Function of Ensemble Frequency Range.. comparison of efficiencies within the ensemble-of-four condition, taken observer by observer, shows no systematic change as a function of frequency range. Furthermore, for O:SR and O:SD, no systematic change is noted in the ensemble-of-two. As has been noted, these results are predictable by the multiple band model. Upon examination of the data of O:SS and O:GP, however, an interesting pattern emerges. O:SS, who was the most efficient observer through all the experiments reported herein, shows a decrease of efficiency as the two signal frequencies become more widely separated. O:GP,who was the least efficient observer, shows an increase in efficiency as a function of increasing frequency separation. If we speculate that there is a narrow-band 14

TABLE IV A Comparison of Efficiencies, (j' s), Obtained for Two Ensemble Sizes and Three Frequency Ranges Ensemble Ensemble Frequency Range Observer Size 150 cps 250 cps 350 cps 2 0.065 0.050 0.049 Ss 4 0.068 o.o58 0.061, 0.044 2 0.023 0.021 0.055 SR 4 0.022 0.018 0.024, 0.028 2 0.016 0.023 0.024 GP 4 0.016 0.027 0.016, 0.017 2 0.052 0.060 0.053 SD 4 0.046 0.053 0.046 0.05 0.052, 0.046 15

observer, we should expect that observer to behave un(ier these conditions of increasing frequency range as O:SS. If, on the other hand, there were a broad-band observer, we should expect him to behave as O:GP. It is interesting to note at this point, that in an experiment that may be characterized as a wide-band detection problem —one calling for the detection of noise in noise, these two observers reversed themselves in their relative abilities to detect. In that experiment, O:GP's efficiencies were indeed higher than 0: 3Ss. These differences in the observers, while only fragmentary, indicate a possibility that there is a population of observers whose distribution ranges from narrow-band observers to broad band observers with the greatest number of observers perhaps, being those who can shift their operation from narrow- to multiple- to broad-band listening as the task requires. 4. THE SECOND EXPERIMENT 4.1 Introduction The purpose of the second experiment was to explore more extensively the applicability the narrow-band and the multiple-band observer models. The original experimental design and equipment were retained but modifications in the experimental variables were as follows: (1) Greater frequency separations were introduced. The eight frequencies dealt with singly were: 507, 707, 907, 1107, 1307, 1507, 1707 and 1907 cps. The pairs used to make up ensembles-of-two were: 1107 or 1307, 907 or 1507, 707 or 1707, and 507 or 1907 cps, giving range differences of 200, 600, 1000 and 1400 cps, respectively. Two different ensembles-of-four included 707, 1107, 1307, and 1707 cps, giving a range of 16

1000 cycles; and 507, 907, 1507, and 1907 cps, giving a range of 1400 cps. With these ensembles the number of critical bands involved in calculating predictions by the multiple-band model is the same for everj ensemble of equal size. Thus, in every ensemble-of-two, the number of critical bands is two; in every ensemble-of-four the number of critical bands is four. (2) An ensemble of eight equally-likely occurring signals was included. This ensemble included all of the individual frequencies listed above. (3) An increase in voltage (to 0.067 v) in the background noise was introduced with no relative increase in the signal voltages. Detection was more difficult for comparable ensembles in this experiment than in the first experiment (see section III). (4) Only one of our former observers, 0:SD, remained with us for this experiment. In addition, two new observers were employed and trained, and data for three are presented. 4.2 Results 4.2.1 Detection of a Single Signal in Noise The results in terms of percent correct and d' for the detection of a single signal in noise are presented for all three observers in Table V. With the exception of O:SD in detecting the 1507 cps signal, the performance of each observer is uniform through all frequencies. 4.2.2 Detection of Multiple Ensembles Predictions of performance on the multiple-ensemble conditions are based again upon (1) performance in detecting a single signal and (2) the mathematics.described in section 3.1. The predictions and the o bt'ined data are presented in Tables VIA, VIB, and VIC. It will be noted- that the 3-7

TABLE V Results Obtained From Three Observers on the Detection of Single Signals in White Gaussian Noise Data are presented as percent correct and their respective d's. l____ l 0:BH O0:CR_ O:SD Signal Frequency P(C) d' P(C) d' P(C) d' (cP S-) - - -__ 507 75.12 0.96 76.12 1.00 85.50 1.50 707 71.38 0.80 73.88 0.90 74.12 0.92 907 78.00 1.09 77.25 1.05 79.50 1.16 1107 72.50 0.84 77.00 1.04 77.00 1.04 1307 74.00 0.91 75.58 0.89 78.25 1.10 1507 74.37 0.92 79.50 1.16 57.00 0.25 1707 75.50 0.98 76.38 1.02 76.12 1.00 1907 74.87 0.94 75.57 0.97 80.25 1.20 18

TABLE VIA A Comparison of the Obtained Data and the Probabilities of Correct Detections as Predicted by Three Models O:BH Narrow Band Model Multiple Band Model Null Hypothesis Predicted Predicted Predicted Signal Predicted Minus Predicted Minus Predicted Minus P(C) Ensemble (cps).P(C) Obtained P(C) Obtained P(C) Obtained Obtained 1107 or 1307 62.00 -4.62 66.97 0.35 73.25 6.63 66.62 907 or 1507 64.00 -3.00 69.25 2.25 76.18 9.18 67.00 707 or 1707 62.75 0.50 67.20 4~95 73.44 11.19 62.25 507 or 1907 62.56 -3.19 67.75 2.00 75.00 9.25 65.75 707, 1107, 1307 or 1707 56.38 -4.37 62.21 1.46 73.34 12.59 60.75 507, 907, 1507 or 1907 57.00 -3.50 63.29 2.79 75.59 15.09 60.50 507, 707, 907, 1107, 1307, 1507, 1707 or 1907 53.50 -8.06 59.08 -2.48 74.47 12.91 61.56 TABLE VIB A Comparison of the Obtained Data and the Probabilities of Correct Detections as Predicted by Three Modeis 0:CR Narrow-Band Model'Multiple Band Model Null Hypothesis Predicted Predicted Predicted Signal Predicted Minus Predicted Minus Predicted Minus P(C) Ensemble (cps) P(C) Obtained PC) Obtained PC Obtained Obtained 1107 or 1307 63.50 -1.38 68.50 3.62 75.19 10.31 64.88 907 or 1507 64.75 2.38 71.17 8.80 78.37 16.00 62.37 707 or 1707 63.19 -1.06 68.15 3.90 75.13 10.88 64.25 507 or 1907 63.06 2.19 68.90 8.03 75.74 14.87 60.87 707, 1107, 1307 or 1707 56.75 2.63 63.19 9.07 75.16 21.04 54.12 507, 907, 1507 or 1907 57.38 0.51 64.39 7.52 77.06 20.19 56.87 507, 707, 907, 1107, 1307, 1507, 1707 or 1907 53.69 -3.12 59.84 3.03 76.11 19.30 56.81 ~~~~~,,,..,,.................. _ 19

TABLE VIC A Comparison of the Obtained Data and the Probabilities of Correct Detections as Predicted by Three Models O:SD Narrow-Band Model Multiple Band Model Null Hypothesis _ Predicted Ped ed redictededte Signal Predicted Minus Predicted Minus Predicted Minus P(C) Ensemble (cps) P(C) Obtained P(C) Obtained P(C) ObObtained Obtained 1107 or 1307 64.12 -6.00 70.40 0.28 77.62 7.50 70.12 907 or 1507 64.75 2.00 63.65 0.90 68.25 5.50 62.75 707 or 1707 63.06 0.56 68.42 5.92 75.12 12.62 62.50 507 or 1907 67.52 2.15 75.05 9.68 82.88 17.51 65.37 707, 1107, 1307 or 1707 57.06 -5.69 63.94 1.19 76.37 15.62 62.75 507, 907, 1507 or 1907 58.88 -4.62 64.06 0.56 75.56 12.06 63.50 507, 707, 907, 1107, 1307, 1507, 1707 or 1907 54.44 -6.00 62.65 2.21 75.97 15.53 60.44 20

difference between the predictions made by the narrow-band model and those made by the multiple-band model are smaller in this experiment than in the original experiment. (For direct comparison, see both sets of predictions for O:SD.) It is generally true, that the differences in the predictions diminish as the original percent correct, upon which the predictions are based, decreases. The deviations of the obtained data from the predictions of the three models, for all observers, are presented in Figure 5. Performance:n all multiple ensembles for all observers falls short of the predictions by the null hypothesis. Which, then of the remaining hypotheses best fits the data? Figure 4 offers a direct comparison of the narrow-band and multiple-band models for each observer. For each multiple-ensemble condition a line is drawn from the (less deviant) better prediction to the zero line. Thus, we can see that the multiple band model more closely resembles the obtained data 6 of 7 times for O:BH, only 1 of 7 times for O:CR, and 5 of 7 times for O:SD. In the first experiement, however, the multiple-band model approached the data more closely than the narrow-band model in all conditions for all observers. Whether this shift toward the narrow-band model is related to the greater frequency separations, to the smaller E/N ratios, to individual differences, or to the interaction of these variables is not determinable in the present experiment. The present research indicates only that both models may be descriptive of certain detection phenomena, though neither is sufficient to handle all the data. 4.2.3 Efficiencies Efficiencies, or _'s, which represent the comparison of the obtained d's with d's expected of the mathematically optimum observer are presented for all conditions and all observers in Table VII and are shown graph 21

Figure 3 NARROW-BAND MODEL MULTIPLE- BAND MODEL NULL HYPOTHESIS +20 +10 0:BH 0 0* * * * -I0 -20 0 ICL LI 0 C-) Z LU (I CD,Li 0 +30 +20 +10 O:CR 0 ~, -10 -20 +20 +10 O:SD 0 0 -10 -20 I I I I I I i.I I * * * a * * *0 _ _ i.* * *0.00 0 * 0 0 0 ~~~~~~~~~~~~~~~~~~~~~~~~ ~* 0 0 0 0. -AA, A A,v O Aq' \ 0) 49' (3 I9 -A N & &N & & -Jo4 O& A, A A0 A $'V to'' 4' SIGNAL FREQUENCIES CONSTITUTING THE MULTIPLE ENSEMBLE 22

+ NARROW-BAND MODEL * MULTIPLE BAND MODEL +8 + E w IlJ 0 z O L0 Q O w z 0 (I) m z 0 C) LU a +-4 +; C O: BH 3 + * - + 2 + + L ~ ~+ >_. I p p 3- ~ I I I i,, I I a_.I O:CR +.I 4. ~, ~ ~! i! + O: SD + t i t + | t + * 0 1 I T a 1 I I - 2. rf wj -4 -6 -E A A A A & & & 04 ^ & fA A _ A0' A \0 ~\O' V A ~F,-?^na 30 ^~Cgg0 P S>1 /~p:' 6b-.s,: A \N s A,\ A. ^'V 6' -o I"I o' SIGNAL FREQUENCIES CONSTITUTING THE MULTIPLE ENSEMBLE

TABLE VII A Comparison of d' for the Mathematically Optimum Detector with d' Data for Three Observers, and Efficiencies (s's) Calculated for all Experimental Conditions Signal 0:BH 0: CR O0:SD in cps' d' opt. d obt. d' obt. d' obt.. 507 4.008 0.955 0.057 1.005 0.063 1.50 0.140 707 4.008 0.795 0.039 0.898 0.050 0.915 0.052 907 4.816 1.09 0.051 1.052 0.048 1.165 0.059 1107 5.050 0.845 0.028 1.04 0.043 1.04 0.043 1307 5.199 0.91 0.051 0.886 0.029 1.102 0.045 1507 5.199 0.925 0.032 1.165 0.050 0.25 0. 002 1707 5 ~440 0.975 0.032 1.016 0.035 1.005 0.034 1907 5.804 0.94 0.026 0.97 0.028 1.202 0.043 1107 or 1307 5.05 0.607 0.014 0.542 0.011 0.745 0.022 907 or 1507 4.95 0.62 0;016 0.443 0.008 0.46 0.009 707 or 1707 4.83 0.44 0.008 0.52 0.012 0.45 0.009 507 or 1907 4.97 0.572 0.013 0.427 0.007 0.56 0.013 707, 1107, 1307 or 1707 4.94 0.39 0.006 0.143 0.0002 0.46 0.009 507, 907, 1507 or 1907 4.87 0.38 0.006 0.246 0.002 0.49 0.10 507, 707, 907, 1107, 1307, 1507, 1707 or 1907 4.98 0.416 0.007 0.242 0.002 0.378 0.006........... ~..... [~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 24

(dobt 2o ~ =- ( d'obt /d opt ) 0 (i) gr 0 o 6 0 0 N LJ wP U 0 8 QOQ~82OQo R 0 Ox I I I I I I * I I I I I I S 507 707 907 1107 1307 1507 1707 1907.. S S S.. * S. S S S S...... G) C) z r-n 0 O c z rr Z c) 0 u~ 507 707 907 1107 1307 1507 1707 1907 507 707 907 1107 1307 1507 1707 1907 -t I I I 507 707 907 1107 1307 1507 1707 1907 - 4 I - -I I -4< -4( - I -4 S a9jn9T

ically in Figure 5. 4.2.4 ri and P(C) as a Function of Ensemble Size and Ensemble Frequency Range If mean I's for each ensemble size are considered, it is found that for all observers I's (1) are greatest for the detection of the single signal; (2) decrease considerably (from 50 to 75 percent) for the ensemble-of-two; (5) decrease additionally for the ensemble-of-four. 1 continues to decrease for one of the observers (O:SD) as the ensemble is increased to eight. The other two observers, however, improve somewhat under this largest-ensemble condition. The results are similar even if viewed in terms of percent correct. Although it has been shown that all performance on multiple ensemble falls short of null predictions, predictions by the null hypothesis have been based throughout upon percent correct detection of the single signals. If, however, predictions for the four-signal-ensemble are based on detection of the two-signal-ensemble; and if predictions for the eight-signal-ensemble are based upon performance in the four-signal-ensemble condition, we find the following: (1) Percent correct detections obtained for the two-signal-ensembles are less than predicted from single signal performance, (2) Percent correct detections obtained for the four-signal-ensembles are less than predicted from performance on two-signal-ensembles. (3) Percent correct detections obtained for the eight-signal-ensembles are less than predicted from performance on four-signal-ensembles for only one observer, 0:SD, For the other two observers, O:BH and 0:CR, performance is better than predicted. 26

A check experiment was performed to determine whether this reversal, found both in rk and in P(C) as described above, was attributable to the fact that the observers were operating at efficiencies so close to zero in the four-signal-ensemble that any decrement virtually would be unmeasurable. Wide-band noise was dropped from 0.067 v to 0.040 v and data for four and eight-signal-ensembles were obtained at the new E/N levels for two observers, O:BH and O:CR. The data, based upon four runs of each of the four-signalensembles and eight runs of the eight-signal-ensemble are presented in Table VIII. At these new energy levels, we obtain (1) a loss of efficiency as the ensemble increases from four to eight, and (2) percent correct detections on the eight-signal-ensemble that are smaller than average performance on the four-signal-ensemble. TABLE VIII __: BH 0: CR Signal Ensemble (cps) d' opt. d' obt., P(C) d' obt. P(C) 707, 1107, 1307 or 1707 7.550 2.40 0.106 95.50 2.29 0.097 94.75 507, 907, 1507 or 1907 7.452 2.40 0.104 95.50 2.37 0.101 95.25 507, 707, 907, 1107, 1507, 1507, 1707 or 1907 7.354 2.13 0.084 93.57 2.09 0.081 95.00 If we compare efficiencies on each of the four-signal-ensembles with the two-signal ensembles of equal frequency range (Table IX) we find lower efficiencies for the larger ensembles of comparable range five of six times. For the frequency range of 1000 cps, however, O:SD shows no change in efficiency with increase in ensemble size. In comparing the 27

TABLE IX A Comparison of Efficiencies, (a's), Obtained for Three Ensemble Sizes and Two Frequency Ranges Observer Ensemble Ensemble frequencj Range Size 1000 cps 1400 cps 2 0.008 0.013 BH 4 0.006 0.006 8 0.007 2 0.012 0.007 CR 4 0.0002 0.002 8 0.002 2 o.009 o.013 so 4 0.009 0.010 8 o.oo6 28

ensemble-of-eight with the ensemble-of-four of equal range, it is seen that the efficiency of O:BH increases, of O:CR remains unchanged and of O:SD decreases. A comparison of efficiencies within ensemble size shows no systematic change as a function of frequency range for any of the three observers. 5. CONCLUSION'Within the experimental conditions described in the present paper, it has been found: (1) that for 6 of 7 observers, performance in detection decreased as a function of increasing ensemble size, (provided that the level of performance from which decrement was measured was not so low as to make decrement measures unfeasible), (2) that no conclusive statement can be made about performance as a function of signal frequency range of the ensemble, (3) that the multiple band model was consistently better in predicting the results of the first experiment, but that (4) in the second experiment (A) the narrow-band model more often predicted tir etvi-ier. the obtained data for at least one observer, whereas for the other two observers(6 of 7 and 5 of 7)of the multiple ensembles were more closely predicted by the multiple-band model, (B) On the other hand, a comparison of the average deviation of the narrow-band model predictions from the obtained with the av6 deviation of the multiple-band model predictions from the obtained dataa shows that: 29

(i) the multiple-band predictions have the smaller average deviation for one observer (O:BH), (ii) the narrow-band predictions have the smaller average deviation for one observer (O:CR), (iii) the average deviations of the two sets of predictions from the obtained are virtually equal for one observer (O:SD). Finally, we conclude that both the narrow-band scanning model and the multiple-band observer model may be descriptive of certain detection phenomena, though neither is sufficient to handle all the data. 3o

REFERENCES 1. Tanner, W. P,, Jr., Swets, J. A. and Green, D. M., "Some General Properties of the Hearing Mechanism," Electronic Defense Group, University of Michigan, Technical Report No. 30, March 1956. 2. d' as a detection index in a two-alternative forced-choice test is simply a normal transform of the probability of correct detection. For a more detailed treatment of d', see Tanner, W. P,, Jr.,, and Birdsallj T. G., "Definition of d' and a as Psychophysical Measures," Electronic Defense Group, University of Michigan, Technical Report No. 80, February 1958. 3. Green, D. M., "Detection of Signals in Noise and the Critical Band Concept," Electronic Defense Group, University of Michigan, Technical Report No. 82, April 1958. 4. For a more detailed treatment, see Peterson, W. W. and Birdsall, T. G., "Theory of Signal Detectability," Parts I and II, Electronic Defense Group, University of Michigan, Technical Report No. 13 (1955), and Tanner, W. P., Jr. and Birdsall, T. G., op. cit. The author wishes to express her thanks to T. G. Birdsall for his guidance in the mathematical treatment of the data. 5. Tanner, W. P., Jr. and Birdsall, T. G., "Definition of d' and a as Psychophysical Measures," Electronic Defense Group, University of Michigan, Technical Report No. 80, February 1958. 6. Unpublished study conducted in the Electronic Defense Group laboratories, University of Michigan, by Marius Smith. 31

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