ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR A CONSTANT-AMPLITUDE RANDOM FUNCTION GENERATOR Technical Report No. 77 Electronic Defense Group Department of Electrical Engineering By: G. A. Hellwarth Approved by:' _ H. W. Farris Project 2262 TASK ORDER NO. EDG-7 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 194B November 1957

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iii ABSTRACT iv 1. INTRODUCTION 1 2. THE RANDOM SAWTOOTH GENERATOR 2 2.1 General Description of Operation 2 2.2 Integrator Operation 4 2.3 Discussion of Laboratory Model 6 2.4 Limitations of the Sawtooth Waveshape 7 3. THE RANDOM TRIANGLE, COSINE, AND SQUARE WAVE GENERATOR 7 3.1 Design Considerations 7 3.2 Analysis of Generator Operation 8 3.2.1 Relation Between Integrator Input and Output Side Time 8 3.2.2 Integrator Input Requirements 11 3.2.3 Effect of Shaping on the Noise Signal Amplitude Probability Density 13 3.2.4 Relation Between Noise Signal Amplitude Probability Density and The Side Time Choice Probability Density 14 3.2.5 Output Wave Side Time Probability Density 16 3.2.6 Side Time Probability Density Using Gaussian Noise 18 3.3 Experimental Determination of the Output Power Spectra 20 4. GENERAL CIRCUIT DISCUSSION 22 4.1 Comments 22 4.2 Circuit Diagram 27 4.3 Oscillographs of Output Wave 27 APPENDIX A DETAILED CIRCUIT OPERATION OF THE FUNCTION GENERATOR 33 APPENDIX B CALIBRATION AND ADJUSTMENT 36 APPENDIX C SUGGESTIONS FOR A TRANSISTORIZED GENERATOR 38 DISTRIBUTION LIST 41 ii

LIST OF ILLUSTRATIONS Page Figure 1 Constant-Amplitude Random Functions 3 Figure 2 Block Diagram of Random Sawtooth Generator 5 Figure 3 Simplified Equivalent Circuit for Generating a Side 5 Figure 4 Block Diagram of Constant-Amplitude Random Function Generator 9 Figure 5 Operation of Integrating Circuit 10 Figure 6 Dual Noise Shaper Characteristics 12 Figure 7 Effect of Shaping Circuit on Rectangular Amplitude Probability Density Function 12 Figure 8 Relation Between Noise Voltage, eS, and Generated Side Time by a Sample of e 15 Figure 9 Graph of Output Side Time Probability Density for Noise with a Rectangular Amplitude Probability Density 17 Figure 10 Output Wave Side Time Probability Density, r(t), for eS Being Gaussian Noise 19 s Figure 11 Cosine Wave Power Spectra as Noise Voltage, e, Is Varied 21 Figure 12 Cosine Wave Power Spectra Showing Effect of Changing Shaper Gain Ratio for Large Side Times 23 Figure 13 Power Spectra for the Three Constant-Amplitude Random Functions 24 Figure 14 Cosine Wave Spectra for Triangle and Sine Wave Noise Signals 25 Figure 15 Photograph of the Laboratory Generator 28 Figure 16 Three Output Modes at Steady State 28 Figure 17 Random Triangle and Cosine Wave Oscillograph 29 Figure 18 Random Sawtooth Generator Circuit 30 Figure 19 Constant Amplitude Random Function Generator Circuit 31 Figure 20 Generator Cosine Shaper and Power Supply 32 Figure 21 Transistorized Integrator 39 iii

ABSTRACT The equipment design discussed in this report resulted from the requirement for a generator capable of producing an audio frequency waveform with a constant peak-to-peak amplitude but randomly chosen paths between peaks. A mathematical analysis is given for various aspects of the several types of waveforms treated: sawtooth, triangle, and cosine. Probability density functions and power spectra describing the outputs of the generator are shown. The equipment is described in detail and results obtained from it are shown graphically. iv

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN A CONSTANT-AMPLITUDE RANDOM FUNCTION GENERATOR 1. INTRODUCTION The constant-amplitude random function generator described in this report resulted from the need for an electronic generator which would combine some of the properties of random noise waveshapes with the known characteristics of periodic, constant amplitude waves. In this instance, a waveshape was desired which had as a main feature a constant peak-to-peak amplitude, but whose path between peaks was in some way randomly chosen. At first inspection the simplest way to generate such a function would be that of amplitude clipping any noise-like waveshape. However, this was thought to be unsatisfactory because the wave spends too much time at the peak value, or clipping level, rather than traversing between peaks. The next criterion on the waveshape, then, was that it reverse its direction immediately after reaching each peak. There appeared to be several common waveshapes that would accomplish this task. The most obvious were: 1. A sawtooth wave with the time length of its tooth randomly determined 2. A triangular wave with the time lengths of both the positive and negative slopes randomly chosen. 3. A cosine wave with its half-cycle time length randomly chosen. 4. A wave with a random path between peaks, each path maintaining a slope of constant sign.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN At this point, it becomes necessary to define some standard terms for discussion of the proposed waveshape. Given a constant amplitude waveshape, a single excursion of the wave between two peaks shall be called a "side". The time derivative or slope of a side is a constant for a sawtooth or triangle wave. A side of a cosine wave shall be the value of the cosine function between 0 and 7t, or jt and 2rt. The side timelength, or "side time", is defined as the time duration of a side's excursion from one peak to the next. The side time of a triangle or sawtooth wave is inversely proportional to the slope of the side. A constant side time triangle or cosine wave (not random) has a fundamental frequency which is the reciprocal of twice the side time. The fundamental frequency or repetition rate of a sawtooth wave is the reciprocal of the side time, assuming a zero flyback time (see Figure 1). Perhaps the most evident reason for the introduction of this terminology is that the functions will actually be generated by randomly choosing each side time-length, resulting in the description of the wave being a probability distribution of side times. In practice, however, it was found to be much easier to measure the power spectrum than the side time distribution, so both frames of reference will necessarily be used. 2. THE RANDOM SAWTOOTH GENERATOR 2.1 General Description of Operation The first type of random waveshape generated was the sawtooth, primarily because this type of generator was thought to be the simplest to design and build. The generator was to operate at audio frequencies and was to have as much flexibility as possible in controlling its side time-length distribution. The random - I —--— i-,i ——, —,-,,, 2 -,,-,_, —--------------

SIDE ------ FLYBACK SAWTOOTH - SIDE TIME - SIDE PEAK TO PEAK AMPLITUDE TRIANGLE -^ SIDE TIME - SIDE COSINE S /M SIDE TIME NEGATIVE SLOPE ~ ^~ \~~ 4 / -POSITIVE NOISE-LIKE SLOPE -A SIDE TIME 4 SQUARE ____l -- SIDE TIME 4 DIAGRAMS ARE INSTANTANEOUS AMPUTUDE VS. TIME FIG.I CONSTANT - AMPLITUDE RANDOM FUNCTIONS 3

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN sawtooth block diagram is shown in Fig. 2, and a schematic circuit diagram is shown in Fig. 18. Essentially, the generator consists of a Gaussian noise source being sampled at the time of the sawtooth flyback, the noise sample being held and integrated until the output reaches some constant value. At this point the integrator is reset (causing the sawtooth flyback) and a new noise sample is taken and integrated, etc. The integrator is a constant-current pentode discharging a capacitor, resetting being due to a thyratron recharging the capacitor. The noise samples are capacitor-coupled into the pentode grid, a controllable grid bias determining the center side time (the center side time is that side time resulting from a zero noise sample, see Equation 4b). 2.2 Integrator Operation A simplified integrator circuit as shown in Figure 3, is helpful in observing the generator operation during the formation of a single side. The current sources in Figure 3 represent a pentode whose only plate load is a capacitor, C. Thus, during the length of time required to generate one side, a constant current is being fed into the capacitor to produce the ramp function of voltage across the capacitor. If g is the transconductance of the pentode; I~ = gm E is the pentode current resulting from the grid bias voltage, E; and I = gm E is the pentode current resulting from the noise sample voltage, o n m n E, then the capacitor voltage, E(t), will be: Io+In gm(Eo+E ) E(t) _1 m n t 0 < T (la) C C And since (Eo+ En) is constant during a side, the time length of the side is that time, T, required for the capacitor voltage to change by E volts, the peak-topeak voltage of the sawtooth wave. Therefore, 4~

SAWTOOTH NOISE SAMPLING OUTPUT SOURCE AND. INTEGRATOR HOLD CIRCUIT TRIGGER DIFFERENTIATOR RESET FIG. 2 BLOCK DIAGRAM OF RANDOM SAWTOOTH GENERATOR I0=9gEn I_=gm E c E(t) FIG.3 SIMPLIFIED EQUIVALENT CIRCUIT FOR GENERATING A SIDE 5 2262 A-S7-60 PS 2- 6-58(10- 23-57 SA)

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN CE CE m m T =_ m _m (lb) T gm(E+En) +In Thus, Equation lb gives the relation between the sample of noise, En, and the side time, T, produced by such a noise sample. The effect of varying the pentode grid bias, Eo, can also be seen. A quantity to note in Equation lb is the transconductance, gm, which in practical cases is never a constant, but is a function of E. By choosing differing types of pentodes in the generator circuit, one is able to obtain different relations between E and T according to the particular way a pentode transconductance varies with its input. Another tool available for controlling the side time, T, produced by a noise sample, En, is the nonlinear amplitude attenuating filter, which can be inserted between the noise source and the sampling circuit. This type of network will effectively introduce to Eq. lb a nonlinear factor, a monotonic function of Eo, which multiplies E. The circuit described in Figure 18 contains a sharp cut-off pentode and no amplitude attenuating network. Amplitude attenuating filters are discussed in greater detail in Section 3.2.2. 2.3 Discussion of Laboratory Model The random sawtooth circuit is relatively stable and reliable. The most critical element appears to be the pentode, since its DC characteristics are used to determine the center side time. However, the output power spectrum is not seriously affected by slight changes in the pentode characteristics. A limit can be set on the maximum time length of any side by simply paralleling the integrating capacitor with a suitable resistance so that if the pentode were cutoff by a large noise sample, the resistor would discharge the capacitor at a predetermined exponential decay rate. _______________________ 6 ______1_________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The center side time is controllable between 0.2 and 10 milliseconds by varying the pentode grid bias. Maximum deviations of side time from the center depend on the pentode grid bias setting, but 200:1 maximum to minimum side time ratios are achievable with the pentode grid bias near the middle of its range. A much more satisfactory way of changing the center side time would be to vary the size of the integrating capacitor, always leaving the grid bias in the middle of its range. This would yield a side time distribution which does not vary with the center side time setting. 2.4 Limitations of the Sawtooth Waveshape The output power spectrum of the random sawtooth is very wide and contains a considerable band above audio frequencies. (There are no spectrum or side-time curves in the appendix for the sawtooth, since it became only a forerunner of the more significant triangle and cosine generator.) The higher frequency spectral power, caused chiefly by the rapid flyback of the sawtooth, requires a correspondingly wide bandwidth for any associated circuit to pass the wave and maintain its constant-amplitude character. It was then decided to try to generate one or more of the other waves described in Figure 1, waves which would not have the large amount of high frequency power resulting from discontinuities. In the triangle and cosine waves, the power spectrum was expected to be much more concentrated in the range corresponding to the side time distribution, resulting also in increased ability to control the output power spectrum. 3. THE RANDOM TRIANGLE, COSINE, AND SQUARE WAVE GENERATOR 3.1 Design Considerations In planning the design of a random triangle generator, there were several problems of circuit choice and complexity that were not encountered in the sawtooth ------------------------- 7 —-----------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN generator. The basic block diagramof the circuit, as shown in Figure 4, is similar to the diagram of the sawtooth, but the contents of the blocks are quite different. The first design consideration resulted in choosing an operationalamplifier integrator circuit which could integrate positive and negative inputs. The output of the integration operates between constant peak-to-peak limits by the action of the amplitude discriminator circuit. When the integrator output reaches a certain magnitude (i.e., the pre-set peak value) the bistable discriminator circuit is "flipped", resulting in a trigger pulse to the sampler and a reversal of polarity of the integrator input signal. This action takes place at both positive and negative peaks, thus confining the integrator output between the peak values. The input to the integrator is the sum of two signals: (1) the constant-amplitude square wave output of the amplitude discriminator, E, and (2) the random amplitude square wave output of the noise sampling and holding circuit, En. The operation of the circuit is wholly dependent upon the successn ful generation of the integrator input signal. 3.2 Analysis of Generator Operation 3.2.1 Relation Between Integrator Input and Output Side Time. Given an integrator circuit as shown in Figure 5a, an equation of operation similar to Eq. 1 can be written. The time, T, required for E to change by Em volts is given by Eq. 4a, since (Eo + En) is constant throughout the interval T. E T = RC m (4a) E +E o n The "center" side time, To, is again defined as that side time generated when the noise voltage sample, En, is zero. _____________________________ ~8 ______,,,____

In 0 an n E~n+ SOURCE DISCRIMINATINGOR HOLDSHAPE SHAPER _ SAMPLER ---- CIRCUIT En - ^ ADDER INTEGRATOR WAVE WAVE WAVE e3 TRIGGER NOISE AMPLITUDE COSINE SOURCE DISCRIMINATOR SHAPER SQUARE COSINE TRIANGLE WAVE WAVE WAVE FIG.4 BLOCK DIAGRAM QE CONSTANT-AMPLITUDE RANDOM FUNCTION GENERATOR 9

OPERATIONAL AMPLIFIER En * vA/- E R E C dE En+Eo R C (a) INTEGRATOR CIRCUIT E - Em zo (b) GENERATING A SIDE < o'7 i 1T' 0 ~-,, CE R I- o T o/ 2 -- TIME 2 To lOT. I J —.O-Eo.IEo EE 1OE0 EI+E0 (c) INTEGRATOR CHARACTERISTIC FIG. 5 OPERATION OF INTEGRATING CIRCUIT 10

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN That is: E T RC m * (4b) 0 Figure 5b shows the expected ramp function of voltage vs. time obtained at the integrator output, when a constant is integrated. The differing times required for the output to change by Em volts when the input is varied is indicated by the three sides pictured. Figure 5c is a quasi log-log plot of Eq. 4a. 3.2.2 Integrator Input Requirements. The first requirement to be noted is that at every proposed peak of the integrator output, the polarity of the integrator input signal (Eo+E ) must be reversed. The signal Eo, which is a positive constant, is intended to produce the necessary changes of polarity at the integrator input by alternately appearing as +E or -E. However, in order that the total input signal change polarity as dictated by the sign of E, then Eq. 5 must hold: En> - E when +E is integrated (5) En < Eo when -E is integrated If side times shorter than - T are to be desired, then there must be two separate 2 o sources of noise signal, En, from which the integrator receives its input. One source shall be defined as En+, and it will provide the input noise signal in conjunction with +Eo; the second source, En will provide the input with -E. 0 n- 0 The dual noise shaper takes a single noise voltage input, es, and feeds this signal through two parallel shaping amplifiers (or attenuators)whose character istics are opposite in polarity. These two shaped signals are then alternately sampled and held yielding En and E, respectively. The output vs. input Ia + I n - amplitude characteristic of the shaper circuit is arbitrary, as long as the clipping levels maintain the limits on en as indicated in Figure 6. A relatively simple shaper characteristic was chosen for this generator: a fixed gain for noise signals of the same polarity as E, and a smaller, variable gain for signals ~11

en e,,FOR DIFFERENT GAIN RATIOS,R, WITH A'R=4' _ I 2 II R=41 PLOT OF OCTPUT eP,VS. INPUT VOLTAGE,eL, FOR DIFFERENT GAIN RATIOS,R, WITH A FIXED CLIPPING LEVELEc FIG.6 DUAL NOISE SHAPER CHARACTERISTICS (xc) g(x) R f y) Yb-Yo Yb Yo Yb- Yo X X ------— () —--------— N,,y, — -x Yo Yb y xb Yb (a) BEFORE SHAPING (b) AFTER SHAPING FIG.7 EFFECT OF SHAPING CIRCUIT QON RECTANGULAR AMPLITUDE PROBABILITY DENSITY FUNCTION 12

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN opposite in polarity to E; the ratio of these two gains is designated as the shaper gain ratio, R. The amplitude characteristics of the two channels of the shaper circuit are alike except for polarity, as shown in Figure 6. 3.2.3 Effect of Shaping on the Noise Signal Amplitude Probability Density. The amplitude probability of the noise voltage is affected by the shaping circuit as follows. If f(y) is the amplitude probability density of the noise voltage, if the clipping level is x, and if the shaper gain ratio is R, then the amplitude probability density g(x) after shaping is given by g(x) =, x < X (6a) g(x) = 6(xc), x = xc,where (6b) x +e Rx C c lim f g(x)dx = f f(y)dy = area of 6(x ), e-0 x -E g(x) = Rf(y/R), x< x<0 (6c) g(x) = f(y), x > 0 (6d) As an example, Figure 7 and Equation 7 show the shaper effect on a noise voltage with a rectangular amplitude probability density function: g(x) = O, x < x (7a) g(x) = 5(xc), x = Xc, where (7b) x +e Rx Rx -y lim g(x)dx = c dy = = area of 5(x) e-~O xc - Yb -Ya b-Ya g(x) = R x< x < 0 (7c) Yb- Ya ~ g(x)= 1 O~x <yb (7d) g(x) =, < x < (7d) 13

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Figure 6 and Equation 4 can now be combined to give the overall relationship between the initial noise signal, es, and the output side time, T. Included in the es vs T curves shown in Figure 8 is the shaper gain ratio as a parameter, assuming a fixed gain for the signals opposite in polarity to E, An arbitrary clipping level, Ec, is indicated: E T T = 0E e < E > -E (8a) E +E es c oc o o c ET T = 00 E e <(8) E +e' s o s E T T = E e > 0 (8c) E +Re s 0 S The shaper gain ratio, R, can either indicate attenuation of noise voltages of one polarity, or amplification of noise voltages of the other polarity. However, the two actions are equivalent if the overall gain is properly adjusted, as shall be assumed in this discussion. 3.2.4 Relation Between Noise Signal Amplitude Probability Density and the Side Time Choice Probability Density. If a given noise voltage, e, has an amplitude probability density, p(es), then from Eq. 8 and 9 the probability density, q(T), of choices of side time can be calculated: de q(T) = p(es) s (9) where de E d /To E T =T -= _-o -o which is (10) dT R dT T R T2 valid for E < e < 0 and 0 < e Ec s s ___________________________ 14

.OITo, (~~~R=E~ ~I. CY I I I I! y^ |^' ^^-^^ 1GAIN RATIO CY REM _____ Eo T- CREm E+Re 0.... r —---------- ---- lOT. FIG.8 RELATION BETWEEN NOISE VOLTAGE,s, 8 GENERATED SIE TIME _BXA SAMPLE es _ IO0T. _. -.~. -. -E. -5E. 0 5E E 15IE. 2E. 2.5E. NOISE VOLTAGE, e 15

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN At e = 0, the gain ratio, R is not defined and at the clipping level, s E C de /dT is an impulse function whose area is given by Equation 6b. Thus, the probability density of choices of side time, T, is known. 3.2.5 Output Wave Side Time Probability Density. To find the probability density, r(T), that the output of the integrator is generating a side time T, where T < T< (T1 + dT1), one weights the q(T) density with the factor, T, since each choice, T, of a side time lasts T seconds, and, therefore, occupies a portion of time proportional to T. r(T) = r(T1< T < T1 + dT1) - ~(T) (11) a where Ta = T q(T) dT maintains unity area under the density curve. An example of interest is one where p(es) is a rectangular distribution as defined in Figure 9a. Equation 12 gives the general relation for R(T). Excluding discontinuities; de E T r(T) = T/T ~ q(T) = T/Ta p(e,) dT T 1 T (12) a- 0 T R Therefore, R(T) is found to be for this case: T R(T) =0, T< (13a) +RB T T < T< T 2R T T R(13b) a 2TaT, To < T < 10 T (13c) a = (10 T), T = 10 To, area =1/20 for Ec = -.9E (13d) =0, T > 10 T (13e) Equation 13 is plotted in Figure 9b. 16 ____

:in I N 0 TI I I I I eL I I I II I i, -Eo Eo.5To To 2To lOTo (a) (b) NOISE SIGNAL AMPLITUDE OUTPUT WAVE SIDE TIME PROBABILITY PROBABILITY DENSITY, p(es) DENSITY, R(T) FIG.9 GRAPH OF OUTPUT SIDE TIME PROBABILITY DENSITY FOR NOISE WITH A RECTANGULAR AMPLITUDE PROBABILITY DENSITY 17

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 3.2.6 Side Time Probability Density Using Gaussian Noise. A second example of even more interest is the case where es is Gaussian noise, the amplitude probability density being 1 -(e) p(e) - exp - s 2 422 aa 2c where a, the standard deviation, will be some constant times E. Let E = 0.9 E de E T Then r(T) = T q(T) T p(e) s = ( (la) a a 2 E2 ex j -) ox 2R2 J x 0 < T < T (14b) 2f aT RT/T o a 2 2 1 E exp[ -1' E= --— x...-'- -, T < T < 10 T (14c) 4' at Ta T/TO -.9E0 e 2 = 5(10T0), area =. f exp - -2 de, T = 10 T (14d) -oo 2a 0, T >10 To (14e) Figure 10 shows a plot of r(T) for four values of a. For a particular shaper gain ratio, R, appropriate values of a on either side of T = T are chosen. The value of the standard deviation, a, for Gaussian noise is just the rms value of the noise, so it would be relatively easy to determine a/E0 in the generator circuit by measuring the unshaped noise voltage and calculating the effective gain of the following circuitry. In the laboratory model constructed, however, there are several variable gain controls affecting the noise sample amplitude with respect to E, so the value of o/Eo is not so easily determined. From inspection of Figure 10 and Eq. 14, it is seen that as the noise voltage, es, is increased, r(T) approaches a 1/T curve, since the Gaussian density function is being clipped for noise voltages opposite in polarity to EB, and the 18

2 I 2 3 4 5 6 7 8 910 2 3 4 5 6 7 8 9 0 ^T, - I I T, I - I - - -" 3 - ~ ~ ~ ~/r -~ \ E. /2 Vf2 2 L- i; /11L 1 11! ~rsto9 i I I C-_ \ 1 =_ 6 / 8 - - _ — ^10i -\\10 \1 (OTo) r(t) /Ir=2EoI t 3 2l111 I I-1 I --— 1 -1~ I 1 1.02 i /T7 9 I o \_ 7 1 1 I FIG. K OUTPUT W0E SIDE 2 r ----- - ----- - _ - - _ - - I T IM E P R O B A B IL IT Y - - - DENSITY,r(t) FOR es.002 i ______ To.i.IT. 2. 3.4 5.6 7.8.9 IT. 2 3 4 5 6789 IOT. 19 SIOE TIME,T

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN noise causing these large side times approaches a rectangular amplitude probability density, as analyzed for Figure 9- More important, however, is that the area represented by the impulse at the shaper clipping level, E c, becomes larger and larger as the noise voltage, es, is increased. This means that the output triangular wave will have more and more side times equal to the maximum side time, which might be undesirable from the standpoint of the output power spectrum. One could arbitrarily set the noise voltage such that, say, 1/20 of the total area under the side time probability density curve, r(T), is contained in the impulse. This would be when a = Eo/1.645, which would also appear to be a good value from the standpoint of obtaining a large side time density for large side times. One could construct a shaper circuit which has a more complex characteristic to achieve different side time densities. This would, of course, depend on the application of the random function generator. For the purpose of this report previous discussion is considered to be sufficient with respect to calculating the output wave side time probability density, given an input noise voltage and shaper gain characteristic. 3.3 Experimental Determination of the Output Power Spectra It would be desirable to know the relationship between the side time probability density, r(T) (previously determined), and the output power spectrum. Much time was spent in an attempt to relate these two functions, but without success. However, in spite of the advantages of having the mathematical equation, some practical insight into this relationship can be derived experimentally. When the noise signal, e, is Gaussian noise, the measured power spectrum for a cosine output (as shown in Figure 11) consists primarily of two relatively straight rolloffs from the center frequency, fo = 1/2To. It would then seem necessary only to relate these roll-offs with the amplitude of the side time density on either side 20

e 0 3 IAJV ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~V I \I I r II COSINE WAVE POWER SPECTRA AS NOISE \ VOLTAGE,es,IS VARIED R= I Ec 2 -.9Eo To — 1/2 m sec es=GAUSSIAN NOISE 10 20 50 100 200 500 1000 2000 5000 10,000 FREQUENCY, CYCLES PER SECOND

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN of T. Unfortunately, the constructed laboratory generator was not built to provide easy measurement of the noise rms voltage, a, the clipping level, Ec, and the shaper gain ratio, R. Also, the accuracies of the shaping, sampling, and hold circuits are not sufficient to guarantee a high degree of reproducibility of settings. Approximate values are stated, however, on the power spectrum measurements shown. Figure 12 shows the effect on the output power spectrum when the shaper gain ratio is varied for large side times. It is noted that in varying the amount of long side times, the high side of the power spectrum is unaffected. Conversely, it can be experimentally shown that the high frequency side of the power spectrum only slightly affects the lower frequencies for the cosine wave output. Figure 13 shows the power spectra for the three constant-amplitude random functions which the generator produces (i.e., the square wave, the triangle wave, and the cosine wave). Very little difference in the power spectra is noted between the triangle and cosine waves. Figure 14 is included to show the power spectrum obtained from a random cosine output when a triangle wave is used for the noise signal, es, the triangle having a rectangular amplitude probability density such as described in Figure 9. This curve is not easily related to the side time density, differing from the case using Guassian noise. Also included in Figure 14 is the power spectrum of a random cosine wave using a sine wave as the noise voltage. 4. GENERAL CIRCUIT DISCUSSION 4.1 Comments The laboratory generator was designed and constructed as an experimental piece of equipment. Consequently, there are many potentiometers built in for 22

L G-'-01 Sd t1l-L 9- Z9ZZ 0 _ 0. -10 ___ - - I / I A I o o w R= i -30 R A ^ y FIG.12 COSINE WAVE POWER A"[,,, SPECTRA SHCWING |i I j 5 EFFECT OF CHANGING \ ~-40 - ~ - - _-R SHAPER GAIN RATIO FOR LARGE SIDE TIMES A To= 1/2 m sec Ec- -.9Eo -50 ------ --- -- _ - ----- es GAUSSIAN NOISE -- --- a- -1/2 Eo fo= To 2To 50 100 20a 500 K0 2000 5000 10,000 FREQUENCY, CYCLES PER SECOND

NORMALIZED TO Odb X I_ ___ I -0 | / FIG.13 PWER SPECTRA FOR THE A NS. o |/ THREE CONSTANT AMPLITUDE | I ~ A I _10,, ____ _ 10 50 100 500 1000 5000 10,000 50,000 FREQUENCY, CYCLES PER SECOND FREQUENCY, CYCLES PER SECOND

L2g- -01 Sd 9bl-9-V-L_ Z9ZZ w ~2-30 -.2To.S %92Efop| to pk I~~./'| TRIA E ANFREU ENCY CY SCLES PER SECOND | TRIANGL E AND SINE WAVECOND FREQUENCY, CYCLES PER SECOND

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN adjusting parameters in the various parts of the circuit, most of the controls being either for gain or for DC bias. Adjustment and calibration procedures are given in Appendix B. Perhaps the major problem encountered in the design and construction of the constant amplitude random function generator was insuring that the input to the integrator reversed polarity at the end of each side. In the circuit developed, this was done, as mentioned in Section 3.2.2, in the noise shaping circuit by the clipping diodes. Since the noise waveform in each channel of the shaping circuit is unbalanced, the shaping, sampling and hold circuitry must be direct-coupled in order to preserve the DC voltage level corresponding to the average value of the noise voltage before shaping. This requires that the two channels hold their DC potentials, with respect to each other, or the clipping levels, Ec, will vary; the tolerances on E are fairly small if side times as large or greater than 10 T are desired. Another problem in feeding the generator input properly is whether or not to capacitor couple the output of the hold circuit to the integrator input. A large capacitor was used in the circuit, but it is suspected that the small currents flowing through this coupling capacitor are sufficient to disturb the capacitor potential, since these currents are random. A recommendation would be that the clipping level, E, be accomplished at the integrator input (see Appendix C),so that the circuit could more effectively guarantee reversal of polarity after each side. If the application of this function generator does not demand that the clipping level, Ec, be more nearly equal in amplitude to Eo than about -0.8 Eo0 then the present circuit will operate reliably; in other words, this generator will easily produce side times five times as long as To (the center side time), but the probability that the integrator input polarity will not reverse after every side increases as one attempts to set the clipping level closer to -E and obtain longer side times 26 0

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN in the output. When the input polarity does not reverse after a side, the output will pass out of the constant peak-to-peak region and take about 0.1 second before the circuit resumes proper operation. 4.2 Circuit Diagram Figure 19 shows the circuit of the laboratory model of the constantamplitude random function generator. Figure 20 shows the cosine shaper and power supply. A general discussion of the circuit operation is given in more detail in Appendix A, while a calibration procedure is outlined in Appendix B. 4.3 Oscillographs of Output Wave Several photographs of an oscilloscope displaying the output waveshape vs. time are included, each one a time exposure of many sweeps of the oscilloscope.with each sweep beginning at a peak of the wave. Figure 16 shows the three output modes running at steady state; that is, without any noise voltage. Figure 17a shows three views of the triangle wave using different sweep rates on the oscilloscope; Figure 17 b shows three similar views of the cosine wave. On these photographs the side time distributions can be observed with respect to statistical variations in the choices of side times, the wide range of chosen side times, and the apparently continuous representation of side times when a large number of oscilloscope traces are recorded. Figure 17c and 17d differ from the above in that fewer traces are pictured in order to show statistical variations; all of these samples are at the same oscilloscope sweep speed. Figure 15 shows the laboratory model with self-contained, regulated power supply. 27

FIG. 15 PHOTOGRAPH OF LABORATORY GENERATOR (a) TRIANGLE (b) COSINE (c) SQUARE FIG. 16 THREE OUTPUT MODES AT STEADY STATE 28

(a) (b) TRIANGLE COSINE (c) (d) TRIANGLE COSINE FIG. 17 RANDOM TRIANGLE AND COSINE OSCILLOGRAPHS 29

1 —--------------------------------- +300 VDC 6S K 270K I M 28K IM 28 K < 820 K 10O K 2W I <4W 2 K 4W 50 pf - f 39 K50a pf r 1T /2 12AT7 12X7 ~~~~~~~~~~~.1 V5 r 1 A 2. D7,10IOOK 1|/212AoT7- 220 K.I.f 5600K V7 0K 5 I __7 X V 0 ^-1/2 12n7 M\ -.OOK K2 12A4 5,'__ 7 I >/2 6AS > PO L/ 6A95 | - V 39K T 1 - 2 6AL5. 05005 568 FIG.- -- 470 K 47 K 75pt R O AG150 K.002ELf RA - g,'i2W 560KP 2W 2 6D2 +300VD C47K IK 3 K IW FIG. 18 RANDOM SAWTOOTH GENERATOR V3 JAILf POT GENERAOTOR I ~ IW J30Q

g-l-8 93 ~1-N30-G 29ZZ FIG.19 CONSTANT AMPLITUDE RANDOM FUNCTION GENERATOR 6.8 K -150V 2W 180K 180K < II 470K f I IN39A -33K POTAt I PoA IsI 2 P 4 ~ *-W NOISE 6.8 K 450VS V A V4A. - r 39K'~f " ~,o,~,::~-*,. ______ ], |.~~, _______ ~; 1-3939K OUT 2 56K 8.2 K 4500^ l 2K 9 27 30p "0 Kt'~' I I rNPOT A STDUAL V2 GU8S O POT S I 5502 R ~~~~~~~~~~~~POT OLN39GA p IN54A.004F. f - io,,uj l-OOK:470K 56K —:- ^ 15 0V 10 0 K - n IOK; 120K 20,uP ~^ 20 2 o IM 27K ]i o. - 450V 450V II50 FILAMENT VOLTAGE 400pt 1N54A IOUTWTI 481( 474!o"TPuT!,,6 390K -_22K 2w ^, 7 \ -/2 12AU7? GUS 4v7K 10500*>( Pt^z _ fI f 6A9 5 V9A- 260K Vsp r0f 10 220K W fflS,_2'vlOB V^ / 1/2 12ArU7 1^ IOK.;____________| CNTE250K 4: 590V |2FT [OUTPUF GAiN I / — oT-o —- NOTE: I. ALL RESISTORS I/2 WATT UNLESS 2262 D-CM-13 REG 8-1-57

56Q POWER SUPPLY 6 6AS7 iOQ V13 3 4300V-DC REG. 5A. FUSE 5U4 AT 0250MA. ~.O 6.3 V —AT -450V 450V 220K- 6BY5-G V 12 56 6^'2W - 375v?4 6AH6 ~~~~RED 7Vlz3~M 20/525K POT 2 1OK1 IOK 10W -3005.0V AT 3 A.5W 3.5K 5W 3 YELLOW _ OA v15 6.3V AT 5A FILAMENT FIG. GENEATOR COSUPPLY ~~~~GREEN~ ~2262 C-CM-12 REG 8-1-57 0== 37 ~=P825f 25V ^, /-j-^OA2 P l 7 /I450V 600V.~ ~ ~ ~ ~ ~~~IoKlf 20K 25K V17 \, /1 I I I, y' — ^ 81\. 5.0 K 25pf 25V |80 K -300V-DC REG 3.5K 5W 3.5K 5W -300V 0-150V-OC REG ^^ 7e.AT7 330fl N IN54A IK VIOA' 25/.1 + 7KS -25^^ f"S 4-25Vr 25V T 54.7K 2V 2-.N34A COSINE WAVE SHAPER FIG.20. GENERATOR COSINE SHAPER AND POWER SUPPLY 2262 C-CM-12 REG 8-1-57

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX A DETAILED CIRCUIT OPERATION OF THE FUNCTION GENERATOR The noise source, e, is a 6D4 gas triode (V1) operating as a diode in s a magnetic field (see circuit of Figures 19 and 20). In order that the instantaneous noise voltage remain relatively constant during the 10 microsecond sampling gate, the noise is bandwidth-limited by a single RC lowpass filter with a time constant of about 40 microseconds. That is, 106 BW = 2i-x4 = 4000 cps The filtered noise is then fed through a gain control (P-9) and amplified (V2) before being put into the dual shaping circuit. The dual shaping circuit consists of four back-biasea diodes. Two diodes (lN39A) effect the hard clipping of the noise, one dioue in each channel, while a potentiometer (P-2) controls the bias on these diodes and therefore, the clipping level, E. A second pair of diodes (1N34A), each in series with a variable resistance, attenuates the instantaneous signals of the same polarity, in each channel, as those being clipped. A dual potentiometer (P-l) controls the amount of attenuation, thus controlling the shaper gain ratio. The dual sampling circuit is made as follows: Two parallel triodes (V3) are operating with a common plate resistor in a normally cut-off state. The shaped noise voltage is direct-coupled to the grids of the triodes, but the grid bias is controlled by two cathode followers (V4). When a sample is taken, a 10 microsecond pulse is applied to the grid of one of the cathode followers, thus driving the corresponding grid of a sampling triode upward by a fixed amount plus an amount determined by the noise signal. With this arrangement, then, the two 33

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN shaped noise signals are alternately sampled and fed out at the same point. When the grid biases are properly adjusted (P-3, P-4) on the sampling triodes, the plate (TP-3) will have 100-volt, 10 microsecond negative pulses appearing. The voltage on the negative peaks of the sample is fed to the holding capacitor (200 pf) through a diode (V8), while another diode and pulse generator (V7, V8) sweep the holding capacitor voltage up to meet the level of the new sample, if the previously held sample was below the new sample. The trigger pulses for both the sampling tubes and the holding capacitor sweeping circuit are obtained by differentiating the two square wave outputs of the amplitude discriminator. The amplitude discriminator circuit (v6) is a bi-stable multivibrator with a large amount of hysteresis, such that if the input is direct-coupled the multivibrator will flip into one state when the input rises to some specified value, and flop into the other state when the input decays to a second specified value. In this manner, the output of the integrator is contained within constant peak-to-peak limits, since it is the square wave output, E, of the amplitude discriminator circuit which is being integrated. (Of the two out-of-phase square waves available, only one is of the proper phase to be fed to the integrator. Similarly, proper arrangement of the pulses triggering the dual sampler is necessary). A potentiometer (P-5) controls the effective amplitude of the sampled and held noise compared to the constant amplitude square wave, Eo, as the two signals are added and power amplified by a cathode follower (V1OB). The center side time, To, is controlled by varying the resistor (P-10) at the integrator input. If a large change in To is desired, the feedback capacitor (500 pf) should be changed. The minimum side time this circuit will generate is about 50 x 10 seconds, it being limited by responses of the amplifiers. 34

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The integrator operational amplifier (V-5) contains an internal positive feedback loop adjusted by a potentiometer (P-12) until a minimum voltage appears at the input grid of the integrator. Two front panel switches are provided. The first (S-l) is a function selector, giving either the triangle, cosine, or square wave mode at the output terminals. The second switch (S-2) is an output attenuator, the upper position giving a maximum output of about 100 volts, peakto-peak, while the lower switch position yields about 4 volts. A potentiometer (P-ll) continuously controls the output level. The cosine shaping circuit consists of biased diodes arranged to attenuate gradually the peaks of a fixed amplitude triangle wave. Adjustment of the circuit is made by a potentiometer (P-8) controlling the input triangle amplitude and a potentiometer (P-7) adjusting the level of hard clipping at the waveshape peaks. 35

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX B CALIBRATION AND ADJUSTMENT The following procedure is suggested for making necessary potentiometer adjustments on the laboratory generator while setting desired parameters of operation. 1. Allow a 10 minute warm-up period. Rotate P-5 full counter clockwise to reduce the gain of the random signal, En, at the integrator input. Adjust noise gain control, P-9, to maximum and observe the noise voltage on an oscilloscope at TP-1 (test point). 2. With P-l, the shaper gain ratio control, and P-2, the clipping level control, set such that no shaping of the noise voltage is present, measure the rms voltage of the noise at TP-1. Adjust P-l until the pattern on the oscilloscope shows the desired gain ratio. This is done because P-l attenuates those noise voltages, in each channel, which produce long side times; when P-l is used to change the shaper gain ratio, then P-2 must be adjusted also to maintain the same clipping level relative to the long-side-time producing voltages (see Figure 8). Check TP-2 on the oscilloscope to insure that the pattern is symmetrical with respect to that seen at TP-1. The shaper gain ratio and the clipping level relative to the noise voltage is now known. 3. Reduce P-9 to zero, and observing TP-3 adjust the DC biases for the sampling circuit, P-3 and P-4, such that the negative peaks of the samples seen at TP-3 are approximately +200 volts. 4. Adjust P-3 until no AC output is observed at the output of the hold circuit, TP-5. This completes the balancing of the dual sampling circuit. 36'

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 5.Adjust the integrator input bias, P-6 such that the output waveform negative and positive going side times are equal. 6. Adjust P-10 to the desired center frequency, fo = 1/2To. 7. Set the noise gain control, P-9, to maximum, and rotate P-5 clockwise until the gain of the random signal at the integrator input is such to cause deterioration of the output signal due to failure of the total integrator input to reverse polarity after each side. Turn P-5 slightly counter-clockwise until proper circuit operation is obtained. The clipping level is now approximately determined to be -.8 to -.9 times the constant amplitude square wave, Eo, and thus the amplitude relation between the different signals in the generator are known, and the triangle and square wave outputs are operating. 8. To adjust the triangle to cosine shaper, reduce P-9 to zero and adjust P-7 for symmetry and P-8 for general cosine similarity while observing the cosine wave output on an oscilloscope. 37

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX C SUGGESTIONS FOR A TRANSISTORIZED GENERATOR Since the construction of the random function generator, the author has observed some advantages that transistors offer over vacuum-tubes in implementing the circuit ideas of this generator. Some qualitative suggestions for transistorized circuitry will briefly be shown in this Appendix, along with the operational advantages which can be achieved. In general, the use of complementary symmetry with transistors enables the design of circuits balanced about ground potential in a push-pull, class B fashion. An integrator is easily made by connecting a capacitor to the collectors of two transistors in parallel, one a pnp and the other an npn (as shown in Figure 21). In this configuration, one of the high-output impedance transistors will feed a current into the capacitor for integration of signals of one polarity, while the other transistor will be used to draw a current from the capacitor for integration of signals of the opposite polarity. The most attractive feature of this integrator for use in the function generator is obtained from the nature of the transistors and the integrator input arrangement, which is such that the input polarity can be reversed after each side with certainty, no matter what amplitude of noise signal is being integrated (see Section 4.1). The input to the emitters is the square wave output of an amplitude discriminator sensing the integrator output, E(t) (see Figure 4). When the square wave is positive, the noise sample E will determine the constant current flowing in the capacitor, while En should be equal to +Eo during this time in order for 38

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN En+ pnp ~E, R OUTPUT o0 ---- -...... ------- V ------ i — E (t) SQUARE WAV+E I ) INPUT EnFIG. 21 TRANSISTORIZED INTEGRATOR the current through the npn transistor to be negligible and to have the same polarity as the current through the pnp transistor. Should the amplifier providing the noise sample En+ present a sample more positive than Eo the transistor becomes a perfect clipper in that such a noise sample cannot possibly deliver a current of the wrong polarity to the integrating capacitor. Conversely, during integration of -Eo and its corresponding noise sample En _ En+ should always be equal to -Eo. The resistor R determines the longest side time that can be produced. Note that the output peak amplitude will necessarily be less than Eo. In view of the previous discussion, the noise sampling circuit need only to provide a shaped noise signal with no clipping, thus eliminating any exacting requirements on the accuracy of the sampling circuit and its amplifiers. It is suggested that use of complementary symmetry again be employed in the construction of two separate sampling circuits, one providing E alternately with the second circuit which supplies En. Since the useful ranges of output voltage of these 39

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN two noise samplers differ only in polarity (see Figure 6), the two sampling circuits would especially be suited by complementary design; that is, the npn and pnp transistors in one are replaced by pnp and npn transistors in the other, respectively, and the power polarities are opposite. The amplitude discriminator, shaping circuit, noise source, and other parts of the circuit can be transistorized without any special changes from the vacuum-tube design. Thus it is believed that a transistorized constant-amplitude random function generator could be built with a much higher degree of reliability than was accomplished in the vacuum-tube circuit of this report. 40

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