ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR OVERMODULATION OF A CARRIER BY SINE WAVE AND GAUSSIAN NOISE Technical Report No. 75 Electronic Defense Group Department of Electrical Engineering By: D. R. Rothschild Approved by: A. B. Macnee 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 September 1957

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iii ABSTRACT iv ACKNOWLEDGMENTS v 1. INTRODUCTION 1 2. BASIC FORMULATION 1 2.1 Basic A-M Transmitter 1 2.2 Modulation Characteristic 2 3. INCREASE IN R-F POWER DUE TO MODULATION 4 3.1 Sinewave Modulation 4 3.2 Gaussian Noise Modulation 6 3.3 Square Wave Modulation8 4. CLIPPING OF THE MODULATING VOLTAGE 8 4.1 Clipped Sine Wave 9 4.2 Clipped Gaussian Noise 10 5. PEAK TO RMS VOLTAGE RATIOS OF THE MODULATED RF SIGNAL 12 5.1 Peak-to-RMS Ratio for Modulation by a Sine Wave 13 5.2 Peak-to-RMS Ratio for Modulation by Gaussian Noise 13 6. SUMMARY 15 APPENDIX 18 REFERENCES 23 DISTRIBUTION LIST 24 ii

LIST OF ILLUSTRATIONS Page Figure 1 Basic Model of an AM Transmitter 2 Figure 2 Modulation Waveforms 3 Figure 3 Increase in RF Power Due to Amplitude-Modulation for Sinewave, Gaussian Noise, and Square Wave Modulations 7 Figure 4 Reduction in RMS Modulation Voltaged Due to Clipping at + V VoltsSinewave and Gaussian Noise 11 Figure 5 Probability Density Function for Clipped Gaussian Noise 10 Figure 6 Peak to RMS Voltage Ratio of the Modulated RF Signal for Sinewave and Gaussian Noise Modulation 14 Figure 7 Test Set-Up for Measurement of Modulation Characteristics 18 Figure 8 Experimental Modulation Characteristic 20 Figure 9 Experimental Measurement of Increase in RF Power for Amplitude Modulation by Sinewave and Gaussian Noise 21 iii

ABSTRACT This paper discusses all degrees of amplitude modulation of a carrier by sine wave and Gaussian noise. The analysis is based on formulation of a basic model of an a-m transmitter and an idealized modulation characteristic. Appropriate parameters are defined to characterize the degree of modulation, including over-modulation. The increase in r-f power due to modulation is calculated and is shown to approach 3 decibels for complete overmodulation, for which modulation by a square wave is the limiting case. The effect of clipping of the modulating wave form is discussed. The peak-to-rms voltage ratio of the modulated r-f wave as a function of the degree of modulation is determined. It is shown that this ratio is not a satisfactory measure of the amount of modulation, and that the modulation parameters defined herein are a much better specification of the degree of modulation. iv

ACKNOWLEDGEMENTS The author wishes to acknowledge the assistance of Mr. T. G. Birdsall and the suggestions of Professors Gunnar Hok and Alan B. Macnee. v

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN OVERMODULATION OF A CARRIER BY SINE WAVE AND GAUSSIAN NOISE 1. INTRODUCTION An amplitude-modulated carrier is a signal commonly used in communication systems and for testing purposes. In some applications over-modulation occurs, either unavoidably or intentionally, and it is desirable to be able to describe in measurable terms the degree of modulation and to determine the increase in r-f power due to modulation. In this paper, equations and curves for these situations will be presented for two modulating waveforms: (1) a sine wave, and (2) Gaussian noise. Result of modulation by a square wave is given as a limiting case. However, no consideration is given to the spectral distribution of the energy of the modulated wave. A table of the principal mathematical results of this paper is presented in the concluding section. 2. BASIC FORMULATION 2.1 Basic A-M Transmitter A convenient basic block diagram model of an a-m transmitter is shown in Figure 1, with power amplifier stages omitted. The particular decomposition shown is convenient for analysis purposes. The unity gain amplifier havinginfinite 1. Some aspects of this problem have been treated previously by other authors. (See, for example. References 1, 2, and 3). However, much of their work has dealt with spectral considerations and some of the factors discussed herein have not been considered. The treatment of modulation in this paper is presented in convenient form and is slanted toward applicability in practical measurements.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN ZL-CI-6 V t S- iS-V Z9ZZ...- MODULATED' R.F. CARRIER MODULATOR MODULATED SOURCE S- TAGE R.F. OUTPUT L -------- I E at 11 ----- L- _ z TOTAL MODULATOR MODULATIO_ UNIYATED-_I VOLTAGE WIT DYNAMIC RAN FIG. 1. BASIC MODEL OF AN A.M. TRANSMITTER. dynamic range serves to isolate the modulation voltage source and the modulator; the modulator itself may or may not introduce clipping of the modulation voltage. This is shown by the external clipper and the shorting switch. 2.2 Modulation Characteristic In most modulators, the r-f signal amplitude cannot be more than doubled on positive modulation voltage peaks and its limiting value is zero on negative modulation peaks. Figure 2a shows an idealized modulation characteristic, which is a reasonably close approximation to characteristics obtained from good practical transmitters (See appendix). The abscissa is the modulating voltage and the ordinate is the instantaneous r-f signal amplitude normalized to the amplitude of the unmodulated carrier amplitude (equivalent to an unmodulated carrier amplitude of unity). Note that this plot is a dynamic characteristic, analogous to a load line on a set of vacuum tube characteristics. In Figures 2b, 2c, 2d are shown three modulation waveforms that will be considered below. ------------— L —........ —----------

LZ-1-O01 Y 9S-9 S-9 Z9ZZ f(V) NORMALIZED RF SIGNAL AMPLITUDE 2(a). MODULATION CHARACTERISTIC. RF SIGNAL AMPLITUDE AS A FUNCTION OF 20 MODULATION VOLTAGE. -Vo Vo 2(b). SINE WAVE MODULA- I MODULATING VOLTAGE v TING VOLTAGE. iT -vml -v -- o Vo Vm 2(c). RANDOM NOISE -V 3 v v MODULATING VOLTAGE. ~Vo -VP 0 VP vo v 2(d). SQUARE WAVE MODULATING VOLTAGE. 05 0.' 1.5 3

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The equation of the idealized modulator characteristic is (Vo> ) O, v < -V f(v) = 1 + V -V v + (1) 2.0, v + V, Note that +VO is that positive value of modulation voltage required to double the r-f signal amplitude, and that -V is that negative value of modulation voltage required to cause zero r-f signal amplitude. 3. INCREASE IN R-F POWER DUE TO MODULATION The modulating waveforms discussed in this section are assumed to originate from an ideal generator and would be measurable at point A in the model a-m transmitter of Figure 1. 3.1 Sinewave Modulation Let the sinewave of Figure 2b have the equation v(e) = Vm sin e, Vm> 0 (2) If PO is the unmodulated carrier power and P the total modulated signal power, the increase in power due to modulation is given by'P i 2 PO 2 ff [ (e) de (3) Po = it 2I The calculation must be divided into two cases: (1) for V < Vo, which means m 0 that -V0 - v(e) - +V, or that the modulation voltage is confined to the linear portion of the characteristic; and (2) for V > V, which means overmodulation m o occurs. ____________________^ 4 ___________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Let m be the modulation index, defined as, m Vm (4) V 0 For V/V = m B 1: Substituting Equations 1 and 2 into 3 yields L f - f (1 +m sin e)2 de (5) P0 2-A -r o - t2 2f1 (1 + m sin2 e) de (6) it 0 This is readily evaluated to give P l+m, m l (7) P o 2 where m = V/V m' o For Vm/V = m! 1: Define e0 such that V0 V =V sin e or sin = o 1 (8) 0 m 0 0 V m m For this case, substituting Eqs 1 and 2 into Eq 3 yields P _1 1+e 2 -e +e 2 i F 2g f 0(l+m sin e) de+ j 0 * d + f 0(l+m sine) de + f 0 4de + o -I( e//e-n -tn+e0 -e0 + 1i ( l+m sin e)2 de (9) it-eo e/it-e By changing variables in the first and last integrals as indicated below the integral, evaluating the second and fourth integrals, expanding integrals and eliminating symmetrical integrations of odd functions, Eq 9 can be simplified to ----- ------------------ ~ ~5 —--------— J

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN p- - 2e0) + 4 (1 + m2 sin2 e) de (10) P 0o which is readily evaluated to give 2P -2sin 2eo 2ek -sin 20 8 0 o = 2 - -1- sin2 m ( where m = V/V and sin 0 = l/m. Equations 7 and 11 can be combined into a single curve for all values of m > O. In Figure 3, P/Po, in decibels, is plotted versus m. 3.2 Gaussian Noise Modulation For Gaussian noise the probability density function is -v2/2No2 p(v) dv = 1 e dv (12) v/~ No where N is the rms noise voltage. For a random modulating waveform, Figure 2c, 0 the power increase is given by: 7P +00 PO I [f(v)] p(v) dv. (13) 0 -oo00 Substituting relations (1) and (12) into (13) yields: +V r v -v2/2N2 -V2/2N02 P 1 v 2 = 1 ~ + e dv + 4e dv(4) o iNo -V -vv - By changing variables, letting x = v/No, and noting that -x2/2 = f e dx =, Equation 14 can be simplified to l eo/ (15) ----------------— NOv2X ) 2-2

0 _ 00 l- -nFI. o 0 0 i') > 0 z o,~ z,, 0 __ (D Q~~~~~r'~^~~~~ c (n 0 Z ~ 0 a~os w w 0 0 E z I-t 0 " IL 001 4,i N > 6 N90 (Ni d \ 381030 NI l cNOILV-lO I O i I3a0

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN By integration by parts, Equation 15 can be evaluated in terms of tabulated quantities. The final result is FPlN o No ) P = 2 - - (16) P 22 N o V o where 1 t-x /2 ea(t) = e dx (16a) -t2/2 and _t'(t) = - e (16b) a(t) and a'(t) are conveniently tabulated on pages 209-213 of Reference 4. P/Po of Equation 16 is plotted in decibels versus No/V in Figure 3. 3.3 Square Wave Modulation For a square wave of amplitude V (Figure 2d) a similar but more elementary calculation gives the result: P v 2 V < P = F = i + 7 Vo <7 0 VO which is also plotted on Figure 3 in decibels. Note, that for large values of modulation voltage of both the sine wave and random noise, the -modulated r-f waveform approaches that of square wave modulation. Consequently, the power increase for all the waveforms approaches 3 db (or double the unmodulated carrier power) for arbitrarily large modulating voltages. 4. CLIPPING OF THE MODULATING VOLTAGE The modulator characteristic shown in Figure 2a can result from either the inherent behavior of the modulator itself or from clipping of the modulation at voltage levels of + V. It is possible that even though the modulator does 8

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN exhibit a characteristic like that of Figure 2a, the modulation may, in addition, be clipped at some voltage levels, + V (V >0), where usually Vc V0 In this section calculation of the ratio of the rms voltage after clipping to the rms voltage of the ideal waveform will be performed. Clipping will be assumed to occur symmetrically at the (modulation) input to the modulator, (as shown by the clipper of Figure l), so that the ideal modulation waveform would be measurable at point A and the clipped waveform at point B in the model transmitter. The purpose of this calculation is that, in some practical transmitters, it is not physically possible (particularly for noise modulation) to find a point where an ideal (or reasonably close to ideal) wavef-orm is obtained; it may only be possible to make a measurement of a clipped modulating voltage and yet it would be desirable to determine the degree of modulation with reference to an ideal waveform. 4.1 Clipped Sine Wave The equation of the sine wave (Figure 2b) is again v(0) = V sin e (2) Define 0 such that c Vc = Vm sin ec, or, sin ec = V/V (18) where V - V m c For purposes of analysis only, Vo and e0 of Figure 2b can be replaced by V and e0. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ c c Then, the mean-squared voltage can be written as: 2 2 -d+e 2 2 -e 2 +e 2 V2 = V2 f sin e2de +V2 C de + V 2 sin2 edie + rms 2v m c m L c m 0/- ~ -:n+e -e i-e C C 1. As long as Vc = Vo, the increase in power due to modulation, as calculated in Section 3, is valid. For Vc<Vo, a new calculation of power would have to be performed. 9

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN With the indicated change of variables the calculation is very similar to that of the second part of Section 3.1. The result is V 2e ( sin 20 1/2 rms = -... sin2 +- -e for V <- V (20a) V/%Fp- C C\ 2 c m V rms = 1.00 for V V (20b) Vm/%T2 c m where sin 8 = Vc/V (18) c c m and Vm//2 is the rms voltage of the ideal sine wave. Equation 20 is plotted versus Vc/Vm in Figure 4. 4.2 Clipped Gaussian Noise The probability density function for Gaussian noise (Figure 2c) is -v2/2N 2 again 1/ o p(v) dv = e dv (12) A8 (-Vc) A (VC), -6 ll — Z9AZ AREA A -P./AREA A -v0 o Vc V FIG. 5. PROBABILITY DENSITY FUNCTION FOR CLIPPED GAUSSIAN NOISE. The density function for clipped Gaussian noise is shown in Figure 5; at + VC are dirac-delta functions with area equal to the shaded area shown on the curve. The mean-squared voltage (for zero mean) is given by: ______________________________________l_ 10O

() LL I0 ci D1,+ U5 <> ~CU 0 8 | >oVE o > 0 (:::) L. L C) 1E >1~> 0 0> i 0 i\ ~ I \ (n \ OR d foCD~~~~~IC N -'- o 0o o 6 4 6 I6 ) o0 INIO_3-AVM 1V301 JO 3V.1IOA SNI W1tO3AVM 03dd13 _O 39V.LOA SWI 11

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 22 22 2 V+ 2 p1 d= L N V -v /2 -v 2No rms The evaluation of Equation 19 is similar to that in Section 3.2 and the resulting ratio of rms voltages after and before clipping is: V V2 V2 /V V V /2 rms q c +11 - a J- a (20) F2N2 N / \oI oN 0o 2N~o o xN v Equation 20 is also plotted in Figure. l is the peak-to-rms voltage ratio of the modulated r-f signal. The peak to rms rati e(t+ - 0 The increase in rms voltage, as a function of modulation voltage, is simply the a(t) = - ]' e dx (16a) and -t2/2 a'(t Peak e (16b) Equation 20 is also plotted in Figure 4.. PEAK TO RMS VOLTAGE RATIOS OF THE MODULATED RF SIGNAL A measure frequently used when a carrier is amplitude modulated by noise is the peak-to-rms voltage ratio of the modulated r-f signal. The peak to rms ratic for a sine wave-and a noise-modulated carrier can be readily computed from the results of Section 3a assuming no clipping of the modulation voltage below Vo The increase in rms voltage as a function of modulation voltage, is simply the square root of the increase in ower. For an unmodulated carrier the peak to rms voltage ratio is p2 or - For convenience in notation, let a be defined as rms modulated- r-f voltage and assume a carrier of unit amplitude. 1. The calculations of this section are still valid for clipping levels of the modulation greater than Vo, i.e., Vo < Vc. If Vc <Vo~ further computations are necessary, as is true in determining the increase in power due to modulation(See Footnote 1, page 9 ). ___________________ —---— 12 --------------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 5.1 Peak-to-RMS Ratio for Modulation by a Sine Wave For m - 1: The peak voltage is 1 + m, so that = J2 (1 + m) = 2 (1 + m) 1 (22a) 4Equation 7 + m/ For m = 1: The peak voltage is 2, so that a = 2 2, m = 1 (22b) svEquation 11 Equation 22 is plotted in Figure 6. Note the range of values of a for the following values of m. m: 0 0+ 1/2 1 2 o a: J2 42 2 4/(J3( 2.31) 2.12 2 5.2 Peak-to-RMS Ratio for Modulation by Gaussian Noise For any small amount of modulation the peaks of the noise will result in a peak r-f voltage of 2. Therefore, 2 _f2 ca = 2.2. 22 for N /V > 0 (23a) (/1/22) Equation 16 /'Equation 16 a = s2 for No/V =- 0 (23b) Equation 23 is plotted in Figure 6. Note the range of values of a for the following values of No/Vo No/V 0 0 + 1/2 1 2 a: 4-2 2T2(s 2.83) 2.555 2.30 2.145 2 ------------------------ 13

0 0 I^ -- LLU LL IZ 2 0 LL-... 0 - un - Q_ tI 0 zi "~LL~ ~ o0:: o 0 zi-o b -i.5 o o 0J5 /0 ~~~~~~~5:~~~~~~~I ffl LJ0J f __ 4 6\ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 IVNgIS d~ O2iV-fNOOIJ d o' OIIVy 39V'11OA si$Jiy-Oi->IV3d Ih~~~~~~

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN As seen by the range of values of a from the data above and from the curves of Figure 6 for sinewave and noise modulating voltages, the peak-to-rms voltage ratio, a, is multi-valued for sinewave, is not sensitive to changes in the modulation voltage, and, for noise, has a discontinuity at No/Vo = 0. Therefore, it is suggested that, for a conventional a-m transmitter having a modulator characteristic approximating that of Figure 2a, the peak-to-rms voltage ratio is not a satisfactory measure of the degree of modulation. Such a measure may only be appropriate to describe the behavior of a quasi-AM signal, such as obtained from a modulated oscillator, like a magnetron, where it is possible to much more than double the r-f signal amplitude and thus, in effect, to increase the average carrier power when modulating. 6. SUMMARY AND CONCLUSIONS i This paper has considered amplitude modulation of a carrier by a sinewave, Gaussian noise, and incidentally, a square wave for all possible values of modulation voltage from - oo to + oo. The degree of modulation (including overmodulation) has been characterized by an extended definition of the modulation factor, m = Vm/Vo, for a sinewave, and by an No/V ratio for Gaussian noise where: V = voltage determined from modulator characteristic (Figure 2a); V = amplitude of the sinewave modulation; m N = rms voltage of the noise. The effects of modulating and over-modulating have been analyzed, with the analysis applying to an idealized conventional a-m transmitter represented by the model of Figure 1 and having a modulator characteristic of Figure 2a. Three calculations have been performed for sinewave and Gaussian noise modulation. These are: 15

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN (1) increase in r-f signal power due to amplitude modulation, (2) reduction in rms modulation voltage due to clipping, and (3) peak-to-rms voltage ratio of the modulated r-f signal. The resulting mathematical expressions are summarized in Table 1. The power can increase up to 3 db for complete over-modulation, approaching that of a square wave. If the modulation waveform itself is clipped, the rms voltage after clipping is reduced. The peak-to-rms voltage ratio for the r-f signal was found not to be a critical function of the modulation voltage, and its use as a measure of the degree of modulation is not considered satisfactory. The modulation parameters discussed in this paper (m and No/Vo) are a much better measure of the amount of modulation. In order to experimentally demonstrate some of the results of this paper, some laboratory measurements on a commercial signal generator are included in the appendix. 16

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ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX In order to demonstrate the applicability of the theoretical analysis of this paper, some experimental measurements were made on a commercially available piece of equipment, a Hewlett-Packard 608D Signal Generator. A.1 Laboratory Test Set-Up The equipment employed in these measurements is shown in Figure 7. 22 2 A-S7-57 AT 10 — 57 BALLANTINE H.P. 430 B 320 RMS POWER METER G.R. UNIT METER a OSCILLATOR I KC. 477A BOLOMETE 1214-A G 7- H.P. 608D D M\ G.R. ATTEN. ISIGNAL b i|123-PA MOD. GENERATOR R. F. I The Hewlett-Packard signal generator was operated at 20 Mc. The carrier level was ket c t durig mula n by re g te OUT dulation Level control was kept in the full (clockwise) position. 5418 G.R. 1390-A XRANDOM NOISE GEN. 20 FIG. 7. TEST SET-UP FOR MEASUREMENT OF MODULATION CHARACTERISTICS. The Hewlett-Packard signal generator was operated at 20 Me. The carrier level was kept constant during modulation by- resetting the output level control so as to maintain a reading at "Set Level" on the Output Volts Meter (Reference 5). The Modulation Level control was kept in the full (clockwise) position. 18 _________ —---

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN A.2 Modulation Characteristics As a first step, it is desirable to obtain a modulation characteristic, similar to that of Figure 2a. A 1 kc/second voltage from the General Radio Unit Oscillator was applied to the modulation input of the signal generator. A trapezoidal modulation pattern was obtained on the oscilloscope by applying the 20 Me r-f modulated wave to the vertical plates and the 1 kc modulating voltage wave to the horizontal plates (Reference 5). The modulation factor, for m - 1, is normally obtained from a-b <(A.1) m - a' m < i (A.I) a+b m l where a and b are the maximum and minimum peak-to-peak scope deflections. The modulation characteristic, however, may also be obtained from this display. If the peak-to-peak scope deflection is unity with zero modulation input, then, with modulation, the maximum r-f signal amplitude is simply the deflection "a" and the minimum r-f signal amplitude is "b". The corresponding modulation voltages are + %$2 and - 42 respectively times the voltage measured at point A by the Ballantine RMS Meter. The experimental modulation characteristic obtained is shown in Figure 8. From an idealized approximation to this characteristic, a value of V = 0.215 volts is obtained. A.3 Increase in Power Due to Modulation The increase in r-f power due to modulation was measured with the Hewlett-Packard Power Meter and Bolometer. After completing the measurement with a sinewave modulation, a measurement of the power increase due to Gaussian noise modulation (from General Radio 1390A Random Noise Generator - 20 kc bandwidth) was obtained. The resulting curves are shown in Figure 9. These experimental curves agree quite favorably with the theoretical curves of Figure 3. The maximum discrepancy from the computed curve for either the sinewave or noise curve is 0.15 db and, for the most part, is considerably less. Certainly some discrepancy 19

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L2. — o01 S4d t —L-9 Z9gZZ 0 0 0 w 0 > 10 l S 0o o z I ~ 1 a: z wL. w z o Z > Z 0 0 LL _ (0 1 t 1 2 o wz ~ 0- 0 0 0O a-. 2 I Z ~ W 2 Z L.O 0 - Q 0 Li \ r) L ~ W: ) w 4 \ ~ 0* (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 S9381a3Q NI' d'NOlVgnoo0I 01 Oi 3 SV3rONI I3MOd d 21

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN can be attributed to the difference between the idealized and experimental modulation characteristic and the actual determination of VO from the experimental data. 22

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN REFERENCES 1. J. H. van Vieck, "The Spectrum of Clipped Noise," RRL Report No. 411-51, July 21, 1943. 2. D. Middleton, "The Distribution of Energy in Randomly Modulated Waves," Philosophical Magazine, Series 7, p. 689, July 1951. 3. D. Middleton, "On the Distribution of Energy in Noise and Signal Modulated Waves, Part I: Amplitude Modulation," Quarterly of Applied Mathematics, Vol. 9, January 1952. 4. Mathematical Tables from Handbook of Chemistry and Physics, Chemical Rubber Co., Cleveland, Ohio, Tenth Edition, December 1954, pgs. 209-213. 5. Operation and Maintenance Handbook for Model 608D VHF Signal Generator, Hewlett-Packard Company, Palo Alto, California, 1955. 6. Reference Data for Radio Engineers, International Telephone and Telegraph Corporation, New York, Fourth Edition, pgs. 529-530, 1956. 23

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65 Copies Transportation Officer, SCEL Evans Signal Laboratory Building No. 42, Belmar,. New Jersey FOR - SCEL Accountable Officer Inspect at Destination File No. 22824-PH-54-91(1701) 1 Copy J. A. Boyd Engineering Research Institute University of Michigan Ann Arbor, Michigan 1 Copy Document Room Willow Run Laboratories University of Michigan Willow Run, Michigan 12 Copies Electronic Defense Group Project File University of Michigan Ann Arbor, Michigan 1 Copy Engineering Research Institute Project File University of Michigan Ann Arbor, Michigan 25

UNIVERSITY OF MICHIGAN i39 1 3ll l ltllllll