THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR A RE -EVALUATION OF WEBER'S LAW AS APPLIED TO PURE TONES TECHNICAL REPORT NO. 47 Electronic Defense Group Department of Electrical Engineering By: W. P. Tanner, Jr. Approved by: A. B. Macnee Project 2262 TASK ORDER NO. EDG-3 CONTRACT NO. DA-36-039 sc-63203 SIGNAL CORPS, DEPARMENT OF THE ARMY DIPARTMENT OF ARMY PROJECT NO. 3-99-04-042 SIGNAL CORPS PROJECT NO. 194B August, 1958

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iv ABSTRACT v I. INTRODUCTION 1 II. EXPERIMENTAL PROCEDURE 7 III. ANALYSIS OF DATA 9 IV. DISCUSSION OF RESULTS 10 V. CONCLUSION 23 REFERENCES 24 DISTRIBUTION LIST 25 iii

LIST OF ILLUSTRATIONS Page Figure 1 Illustration of Amplitude Difference as Basis for Discrimination in the Case of Pure Tones. (A) Simple Detection Situation, (B) An Amplitude Discrimination 2 Figure 2 d' Versus AV for Observer 1, NG = 0 11 Figure 3 P(c) Versus V x 105 for Observer 1 12 0 Figure 4 P(c) Versus V x 10 for Observer 1 12 o Figure 5 P(c) Versus V x 103 for Observer 1 13 0 Figure 6 P(c) Versus V x 103 for Observer 1 13 0 Figure 7 d' Versus AV for Observer 2, NG = 0 14 Figure 8 P(c) Versus V x 105 for Observer 2 15 0 Figure 9 P(c) Versus V x 10 for Observer 2 15 0 Figure 10 P(c) Versus V0 x 103 for Observer 2 16 Figure 1 P(c) Versus V x 103 for Observer 2 16 0 Figure 12 d' Versus AV for Observer 3, NG = 0 17 Figure 13 P(c) Versus V x 105 for Observer 3 18 0 Figure 14 P(c) Versus V x 10 for Observer 3 18 Figure 15 P(c) Versus V x lO3 for Observer 3 19 0 Figure 16 P(c) Versus V x 103 for Observer 3 19 Figure 17 Data for Observer 1 for a Signal Which is a Sample of White Gaussian Noise 20 Figure 18 Data for Observer 2 for a Signal Which is a Sample of White Gaussian Noise 21 Figure 19 Data for Observer 3 for a Signal Which is a Sample of White Gaussian Noise 22 iv

ABSTRACT The fact that Weber's law appears to apply in the same way both to intensity discrimination for pure tones and to intensity discrimination for white noise poses a theoretical paradox: in the case of pure tones, the human observer becomes less efficient as the intensity of the tone is increased, while in the case of white noise he exhibits a constant efficiency independent of intensity, An inventory of the various possible noise sources which may exist is made, and the way in which these may be expected to effect the detectability of a signal leads to the equation E (d/)2 rl NG ANE+ki where nr is the individual observers efficiency, NG is the noise introduced by the experimenter, NE is the uncontrolled noise present in the experimental situation, and k is a constant indicating that small amplitude variation in the oscillator constitute a noise source proportional to the power of the lower of two signals to be discriminated, Data for three observers over four noise levels is described by this equation sufficiently well to suggest that the hythesis that Weber's law is merely a reflection of the oscillator noise (kV<) is plausible. v

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN A RE-EVALUATION OF WEBER'S LAW AS APPLIED TO PURE TONES I. INTRODUCTION This study is conducted within the framework of the theory of signal detectability. It is recognized that a signal in noise has a capacity to lead to detection. This capacity is utilized as a standard against which to compare the performance of an observer in a psycho-acoustical experiment. The subject matter of this study is the application of Weber's Law to two types of signals: pure tones, and samples of white Gaussian noise. That Weber's law should apply to intensity discrimination behavior for both types of signal is somewhat of a paradox within the framework of the theory of signal detectability, because an observer whose data follows Weber's law is rated as becoming less efficient as the intensity of a pure tone is increased, but is rated as having a constant efficiency at all intensities for a white noise signal. For the case in which two bursts of pure tones are presented to the observer, the tones being identical in every respect except amplitude, the capacity of these tones to lead to discrimination is expressed in terms of the energy of the difference signal, as illustrated in Figure 1. The uppermost of the three graphs is a voltage waveform as a function of time Vo(t). The middle graph illustrates a voltage waveform Vl(t). These are the two signals upon which the discrimination must be based. The bottom graph is the waveform of the difference signal, AV(t), obtained by subtracting point for point in time the voltage V~ from the voltage V1~

Vo V VI O > CASE A Av o T - - Vo'!~il~ A~l~~~ A/~ A A DCASE B VI o Av o T = FIG. I. ILLUSTRATION OF AMPLITUDE DIFFERENCE AS BASIS FOR DISCRIMINATION IN THE CASE OF PURE TONES. (A) SIMPLE DETECTION SITUATION. (B) AN AMPLITUDE DISCRIMINATION.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The energy of the difference signal AV(t) is given by: E = f LV2(t) dt, (1) t=O where the observation interval begins at t = 0 and ends at t = T. The capacity of these two signals to lead to discrimination can be expressed in terms of 2 2E (d'), described in a previous paper (Ref. 1). 2 NA (d' opt) = (2) 0 where N is the noise power per unit bandwidth. o Since the two waveforms are assumed perfectly correlated, the following voltage relation holds: AV = - V, (3) where all of the voltages are expressed as r.m.s. voltages. Ignoring corrections introduced to handle irregularities for those signals near the absolute limits of hearing, Weber's law states that, to lead to a constant level of performance, the following relation holds, = constant, (4) "0 I3

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN where I0 is the intensity of the lower of the two signals, and LI[ is the difference in the intensity of the two signals. In terms of voltages, used above 10 [V02 I aC V 2 2 I1 - I = ccV1 V 0-v since V1 V0 + AV then AI c V 2 + 2V0 AV + V2 - V02' and nr AV AAV2 C2- +2 I V0 2 Thus, if equation (4) holds then for a constant level of performance, it is also true that ~2 = constant = C. (5) V0 1 In the previous paper, efficiency is defined by the following relation (d)2 =, A NO (6) Further, let the constant level of performance usually defined as "threshold" in the two alternative forced-choice advocated by Harris (Ref. 2) and Blackwell (Ref. 3) be the case where d' = 1. Incorporating the relation AV2 = C V0 from equation (5) into equation (6), the statement of Weber's law is given by: ___________________________ 4

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN cvO t N = -, (7) 2 0 and. is found to vary as (V~). q thus decreases as the reciprocal of I0. For the case of a signal which is a sample of Gaussian noise, the capacity to lead to discrimination is expressed as: o o o d'opt = - S 2 S opt Ne2_N_ Nwhere N is power measurement linearly related to I, and N + S bears the same linear relation to I1. For Weber's law to apply [equation (4), then S N = constant. (9) 0 It is the relative efficiency which is of interest here. By the same argument above, it can be observed that, if, in the relation d' -1 t N S 2 S (10) o 1 o)+ + 0 0 S is constant, then r is also constant. For the case of signals which are o samples of white Gaussian noise, the Weber's law observer exhibits an efficiency which is independent of I or N in the equation. o-5

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN A possible solution to the paradox can be reached by considering a number of sources of noise which combine to make up the effective value of N employed in the experiment. One such source is that noise generated by a noise generator and introduced by the experimenter, designated NG. A second source is the residual noise of the experiment designated NE. A third source can arise from amplitude and frequency variations in the output of the audio oscillator employed. While noise of this type may not be large, it is narrow band around the frequency of the signal, and is all relevant to the experiment. Since in the experiments reported below the oscillator always has the same output and 2 the voltages are obtained by attenuation, this noise is designated by kV. Since 0 the three sources of noise are assumed to be independent, and thus additive, equation (6) becomes 2E(d')2 2EA (11) NG +E 0+ k o Since previous experiments supporting the Weber's law application to pure tones were carried out in silence (NG = 0), one would expect Weber's law to apply as soon as k V >> NE. If the experimenter introduces noise, o E 2 then Weber's law should apply only after k V >> G + NE. It is obvious that there must be noise of the type introduced by the audio oscillator. The only question is one of magnitude. Is it sufficiently great to account for the experimental results which appear to support Weber's law? The problem of determining the magnitude of this noise, particularly in term of an N type of measure, is severe from the standpoint of physical mea____o0 6 __ __ _ __ _ __ _ __ _ __ _ __ _ __ _ __ _ 6 __ _ __ _ __ _ __ _.... _ _ __ _ __ _

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN surement. McPherson has observed amplitude variation in the output of the Hewlett-Packard type audio oscillator used in the experiments reported below. The major source of this noise is a tube. While both the amplitude and spectrum of this noise can vary widely from tube to tube, quantitative measurement is difficult since it is so low relative to the over-all output level of the oscillator. It is further very difficult to measure since the noise is a cycle to cycle variation, and any measure depending on time averages is not sensitive to the variation. Since direct physical measurement of NE and k are difficult, these constants will be estimated by fitting equation (11) to the data. Since the constants NE and k are conceived to characterize the equipment, the attempt is to achieve this fit with a single value for these constants for the three observers, while a value of r will be assigned to each observer. II. EXPERIMENTAL PROCEDURE In the experiments conducted in this study of Weber's law, the two alternative forced-choice technique was employed throughout. In this technique the two signals are presented in random order and the observer is asked to state which of the two is of the greater amplitude, the first or the second. The three observers, University of Michigan undergraduates, listened through PDR-8 earphones connected in parallel. In each of their booths there was a series of lights which informed them of the progress of the experiment. The flash of the first light warns the observers to ready themselves for the obser* Personal communication i! 7. i i~~~~~~

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN vation. A second light flashes twice marking the position in time of the two signals. A third light flashes indicating that it is time to designate their choice by pressing one of two buttons. On the flash of a fourth light they are presented with the correct answer information. At this point, the data for this trial are recorded on an IBM card. The apparatus then recycles for the next trial. The entire trial occupies approximately 3 1/2 seconds. The random ordering is achieved by a radiation programmed random number generator. A digital counter is driven by a 2500 cycle oscillator. A decade counter counts the output of a Gieger tube on which the average time between counts is approximately 0.1 seconds. When the decade counter completes its cycle, the digital counter is stopped indicating the interval in which the signal of large amplitude is presented. Both signals are generated by the same audio oscillator fed through separate attenuators and gates. The voltages at the top of each attenuator are adjusted to be equal when read across the terminals of the earphones. One of the attenuators is then set to yield the value V, and this is gated on its first positive zero crossing in each of the observation intervals. The second attenuator is set to yield the value AV and this gated on its first positive zero crossing during the observation interval determined by the random number generator. Each gate passes the same integral number of cycles. In the observation interval in which both signals are gated they are combined in a linear adding network. Two hundred trials constitute a datum point. Four different levels of NG were investigated. Except for the case where NG = 0, a single value of AV was maintained throughout that noise level. When N = 0, two different levels of AV were studied for low values of V, and G G three different levels for the higher values of V. In those experiments in.___________________ —- 8 _____.__________________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN which a measureable value of N was introduced by the experimenter, the nth value of V was the (n - l)st value of Vo plus AV. III. ANALYSIS OF DATA First, the data for NG = 0 were analyzed. The first step is to fit a curve of d' as a function of AV for each value of V. It is assumed that this 0 curve should be a straight line intersecting the origin (d' = 0 when AV = 0). From these curves, an estimate of that value of AV necessary to lead to a d' = 1.00 is made. This value of AV is then plotted as a function of V. Since by the 0 relation from equation (11) QAV2t = NE + k V 2 (12) the intercept on the ordinate is an indication of the residual noise of the equipment and the slope is an indication of k. These plots were fit satisfactorily by straight lines. The intercepts and slopes were different for each of the observers. ThLen the efficiencies of the three observers were determined from the experiment with noise. This permitted an estimate of a single set of parameters for the equation with only the efficiencies varying from observer to observer. Finally all of the data were compared to predictions so determined, and a single correction was made by eye. This lead to the equation. 2E (d' )2 = (13) -12 -- 2 NG + 5.6 x 10 + 2 x10 V L_________________ — 9 ---------------------— *

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The data for the several experiments are illustrated in Figures 2-15. The curves are those calculated using Equation (13) with values of 2 =.28,.21, and.17 for observers 1. 2, and 3 respectively. S In the white noise experiments in which - was maintained roughly cono stant, q was calculated from equation (10). In Figures 16-17-18, 1 is plotted as a function of the logarithm of N. For two of the observers 1 decreases slightly as N is increased, while for the third it appears to be more or less constant. IV. DISCUSSION OF RESULTS Equation (13) agrees reasonably well with the obtained results. While certain departures are apparent (it should be noted that all of the data has. been presented in a way which emphasizes these departures), none of these departures is great. In fact, had the data been treated in terms of db scales, the departure would have been hidden completely. If equation (13) is acceptable, then one has a satisfactory explanation for departure from Weber's law for low intensities. The explanation is that there is noise present even if the experimenter doesn't put it there. It is difficult to claim at present that the factor dependent on the level of V is entirely due to the audio oscillator variations. From the data, o it appears that the third observer might be better fit with a somewhat greater constant. Even here, considering the method of fit, the departure is not so great that the hypothesis should be rejected. A large quantity of data, over four different noise levels and three different observers is reasonably well satisfied by this single equation (13). The equation is based on a logical 1LO

099 THE LINES ARE CALCULATED 3 3 FROM EQUATION (13), AND THUS ARE NOT PURPORTED / / 2 093 2 TO BE BEST FITS FOR THE / INDIVIDUAL GRAPHS. ALL d I / NUMBERS SHOWN ON SCALES - SHOULD BE MULTIPLIED BY 10-5. 4 o0 I t I I I l 0 I 2 3 0 2 3 Av Av ------ 3, —------- 3 099 3 2 - 0 2 2 d'~ d't - d 0 0 o i I - I I I o I I I I I I I I o I 2 3 0 1 2 3 0 2 3 FAv -. Av -. Av ---- 3d t 3 3 97 96 2 - 2 - 2 92 dl dl 082 *86 df 0 V0=12 V0= 14 V0= 16 0, I I I I I I I I 0 1 2 3 0 1 2 3 0 2 3 Av -* Av - v AFIG. 2. d' VS. Av FOR OBSERVER NO.1, NG 0. -I I -

.0x.9 - - x -X-.8 x P(c).7 FIG. 3..6 NGO 0.5' I I I l I 0 2 4 6 8 10 12 14 16 18 20 Vo x 105 — ~.0 iii —-----------------— V0 -K 1.0.9 x x x.6- NG = 6.22 x 10-9 5 1 1 1 1 1_ I I I 1, O 10 20 30 40 50 60 70 80 90 100 Vo x 104 xOBSERVER NO. 1. THE LINE IS CALCULATED FROM EQUATION (13). P(c) IS THE PERCENTAGE OF CORRECT RESPONSES. 12

I.8o P(c).7 FIG. 5. *6 NG = 9.73 x 10-9 * 2 4 6 8 10 12 14 16 18 20 Vo x 103 P(c) FIG. 6..6 ~ NG - 4.92 x 10-9.5 I I I T I N 1 0 4 8 12 16 20 24 28 32 36 40 Vo x 103OBSERVER NO. 1. THE LINE IS CALCULATED FROM EQUATION (13). 13

THE LINES ARE CALCULATED 3 / 3 FROM EQUATION (13), AND V 4 THUS ARE NOT PURPORTED / V2 2 2 TO BE BEST FITS FOR THE INDIVIDUAL GRAPHS. ALL d' 0 NUMBERS SHOWN ON SCALES I SHOULD BE MULTIPLIED BY 10-5. o' / I I I I I o l I I I I 0o 2 3 0 i 2 3 3 3 3 Vo 6 Vo=8 - Vo10 94 2 0 2 2 / 87 087 It~ /~ i i~T O ~ T d8 078 0 j-0 0 0o I I I I I0 o I I I I I I I - 0 I 2 3 0 I 2 3 0 I 2 3 Av -; Av -; Av -; 3 3 3 V Vo=12 95 Vo=14- Vo=16 2 2 294 92 dl 87 dl- 85 80 dl 82 078 080 076 0 0 064 067 Y I I0 I 0 I I I I I I 0 I I I I I 0 2 3 0 2 3 0 2 3 Av -s Av -g Av - FIG. 7. d' VS. Av FOR OBSERVER NO.2, N,=O. -14

1.0I.9 x ~.8 P(c).7 x FIG. 8. x.6. x NG =.5 II I I I I I II 0 2 4 6 8 10 12 14 16 18 20 Vo x 105 1.0.9 T7x x.8 x P(c) x FIG. 9. X NG =6.22 x 0-9.5 I I I I I I I I 0 10 20 30 40 50 60 70 80 90 100 Vo x 104 - OBSERVER NO. 2. THE LINE IS CALCULATED FROM EQUATION (13). P(c) IS THE PERCENTAGE OF CORRECT RESPONSES. 15

1.0 x X.9 P(c) IX x.7F x X X FIG. 10..6 N NG = 9.73 x 10-9 5 I I I I I I I I i 0 2 4 6 8 10 12 14 16 18 20 Vo x 1031.0 X l x9 x x I.8 7 X FIG. II..6 NG = 4.92 x 10-9.5 I I I I I I I.0 4 8 12 16 20 24 28 32 36 40 Vo x 103-3 OBSERVER NO. 2. THE LINE IS CALCULATED FROM EQUATION (13). 16

THE LINES ARE CALCULATED 3 3 FROM EQUATION 13, AND Vo=2 095 Vo=4 THUS ARE NOT PURPORTED 0 2 92 2 TO BE BEST FITS FOR THEt / INDIVIDUAL GRAPHS. ALL 8 85 NUMBERS SHOWN ON SCALES 76 6 SHOULD BE MULTIPLIED BY 105. V o=6 Vo =8 \- =10 2 - 2 2 dL - d t O d - 00o 0 D k'I I I I I I O I I I I I I oI 0 1 2 3 2 3 0 2 3 AV -~ Av g- Av - 3 3 3 Vo =12 Vo 14 Vo=16 2 - 93 2 93 2,::rT / t~~~~~~~~~~A /t90 dl 087 I 076 072 OBSERVER NO. 3, N =0. -17d~ ~ ~ ~ OSRE T03 - 82 9 I~~~~-7

I.0 49 1.8 Fx P(c) 07l FIG. 13..6 NG =0.5 2 4 6 08 1O 12 -1-4 1 1 20 l.0 F I G4..9x NG 6.22. L0 " p(c) \ x x' r \x x x K K.6'.L -^^ -L" o 10 20 30 40 X50 60780 80 90 100 OBSERVER NO. 3. E LINE IS CACULATED FROM EQUATION (13. P() THE LINE IS CALCU E CORRECT RESPONSES. IS THE PERCENTAGE OF CORREC 318

1.0.9 x FIG. 15. t*8'^^^^ ^ ~NG = 9.73 x 10-9.8 x x P(c) x.7x x 0O 2 4 6 8 10 12 14 16 18 20 Vo X 103 1.0 X.9 Lx FIG. 16. i. 8 X NG = 4.92X I0"-9 P(c).7 - X X.tx~ x 09 4 8 12 16 20 24 28 32 36 40 Vo x 103 -- OBSERVER NO. 3. THE LINE IS CALCULATED FROM EQUATION (13). 19

1.0.9 oo FIG.17.8 - OBSERVER NO. I..7 5 O.2.1 0 Log N DATA FOR OBSERVER I FOR A SIGNAL WHICH IS A SAMPLE OF WHITE GAUSSIAN NOISE

.0.9 - FIG.18.8 OBSERVER NO. 2..7 0 Q.6 _ 0.775.4 (3.0 0.3.2 _ oI I I I I 1111 I 11111 Log N DATA FOR OBSERVER 2 FOR A SIGNAL WHICH IS A SAMPLE OF WHITE GAUSSIAN NOISE

1.0 19 FIG. 19.8 OBSERVER NO. 3..7.6 77 O.5.3 ( 0 o 00.2 0 0 0L I S I III 1111111 1 I 111111 Log N DATA FOR OBSERVER 3 FOR A SIGNAL WHICH IS A SAMPLE OF WHITE GAUSSIAN NOISE

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN argument which suggests that the parameters might not assume the same values in different laboratories, or for that matter with different oscillators. Our study of Weber's law for signals which are a sample of white Gaussian noise is in essential agreement with the results reported by Miller (Ref. 4). When r is plotted as a function of the logarithm of N for each of our observers (Figures 16-17-18), there appears to be a slight fall-off in efficiency as the intensity of the signals is increased but this drop is not great. It certainly would not be noticed had db scales been used. V. CONCLUSION The hypothesis that Weber's law as applied to intensity discrimination of pure tones reflects a condition of the environment rather than the hearing mechanism is found plausible. There are controlled sources of noise in experiments. Even experiments performed in "silence" involve noise, and this noise may be the limiting factor of hearing. 23

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN References 1. Tanner, W. P. Jr. and Birdsall, T. G., "Definitions of d' and T as Psychophysical Measures," Tech. Rept. No. 80, Eletronic Defense Group, University of Michigan, Ann Arbor, 1958. 2. Harris, J. D., "Remarks on the Determination of a Differential Threshold by the So-Called ABX Technique," JASA, 1952, 20, 160. 3. Blackwell, H. R., Psychophysical Thresholds: Experimental Studies of Methods of Measurement. Ann Arbor, University of Michigan Press, 1953 (Eng. Res. Inst. Bulletin No. 36). 4. Miller, G. A., "Sensitivity to Changes in the Intensity of White Noise and its Relation to Masking and Loudness," JASA, 1947, 19, 609. 24

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