https://ntrs.nasa.gov/search.jsp?R=19740002895 2020-03-23T13:44:34+00:00Z I m I s-2 ' r C RELATIVE SIDEBAND AMPLITUDES V) f ld 0 a 4- :a- urn VS. MODULATION , * C 2 3N a r- INDEX FOR COMMON FUNCTIONS USING m o'a FREQUENCY AND PHASE MODULATION , .4 2 -..* . 0 CY + 4U 0 OW 0 mu 4 cabm. 4' z* ; k am FRANK STOCKLIN T.--* - k- p14hU .6 - QSUZ HWZ V) an .i - % 3 woom mHWt 5 HWCG p elat4 44 A 4304 wL3zm .. - i*OH-14 mn= 3 AV) l-3'4Z rV)zo InffiUH ornun CI*C(IIO I xV)==I IM3Q EQk0 $33 E 8 Ha I' 4now - 6000ARO SPACE FLHtCENTER CMEIWELT, MARYUIID RELATIVE SIDEBAND XMPLJTUDES VS. MODULATION INDEX FOR SOME COMMOPj FUNCTIONS USING FREQUENCY AND PHASE MODULATION Frank Stocklin November 1973 Goddard Space Flight Center Greenbelt, Maryland PRECEDING PAGE BLANg NOT FILMED RELATIVE SIDEBAND AMPLITUDES VS. MODULATION INDEX FOR SOME COMMON FUNCTIONS USING FREQUENCY AND PHASE MODULATION ABSTRACT The equations defining the amplitude of sidebands resulting from either frequency modulation or phase modulation by either square-wave, sine-wave, sawtooth or triangular modulating functions are presented. Spectral photographs and computer-generated tables of modulation index vs. relative sideband ampli- tudes are also included. iii PRECEDING PAGE BLANK NOT FILMED CONTENTS -Page INTRODUCTION ....................................... 1 SQUARE-WAVE FREQUENCY MODULATION (FM). .............. 2 SQUARE-WAVE PHASE MODULATION (PM). .................. 10 SINE-WAVE FREQUENCY MODULATION AND PHASE MODULATION . 13 ACKNOWLEDGMENTS .................................. 25 REFERENCES ........................................ 25 APPENDIX A - ANALYSIS OF SQUARE-WAVE PHASE MODULATION AND FREQUENCY MODULATION. .... 26 APPENDIX B - ANALYSIS OF SAWOOTH-WAVE AND TRIANGULAR- WAVE PHASE MODULATION .................. 35 ILLUSTRATIONS -Page 1 Basic Test Configuration Used to Obtain Spectra. ........... 6 2 Frequency Modulation Spectra (Square-Wave and Pseudo-Random) ................................. 7 3 Phase Modulation Spectra (Square-Wave and Pseu 10-Random) ... 12 4 Frequency and Phase Modulation Spectra (Sine-Wave). ........ 24 A1 Square-Wave Phase Modulating Function ................. 27 A? Square-Wave Frequency Modulating Function .............. 31 B1 Sawtooth-Wave Phase Modulating Function ................ 35 B2 Phn se f.!odulation Spectra (Sawtooth-Wave) ................ 40 B3 Trianguiar-Wave Phase Modulating Function. .............. 41 B4 Phase Modulation Spectra (Triangular-Wave) .............. 49 RELATIVE SIDEBAND AMPUTUDES VS. MODULATION INDEX FOR SOME COM.LION FUNCTIONS USING FREQUENCY AND I~HASE MODULATION INTRODUCTION An important consideration in the. design and testing of a communications system is the relative amplitude of sirleband components resulting from modu- lation of a radio frequency (RF) carx5er. This spectral information is useful in compatibility testing to determine modulation parameters and can also be used to precisely select optimum receiver bandwidths. This document is intended to be used as a handbook for engineers involved in such activities. The majority of the information is presented in tabular form (i.e. modulation index vs. relative sideband amplitudes). Spectrum photographs are included for illustrative purposes. Theoretical background material is given for completeness and convenience. Appendix A contains the square-wave FM and PM analysis and develops the equatioas for the sideband amplitudes. Appendix B extends the PM analysis for the sawtooth and triangular modu- lating function. This illustrates the generality of the analysis technique. The general expression (derived in Appendix A) for the amplitude of the nth sideband resulting from square-wave frequency modulation of a unity-amplitude RF carrier is Taking the limit as ,b - n RF carrier deviation bf) where ,B modulittion index in radians = modulating frequency (fm) ' n = sideband order (0, 1, 2, . ), JC/ = absolute value of nth order sideband. The expression for the residual carrier amplitude is obtained by setting n = 0 in Equation (1): Note that the right-hand side of Equation (3) has the form of sin x/x, which has a limit value of 1 asx - 0; for the no-modulation case, the carrier ampli- tude is equal to 1. The expression for the first sideband amplitude term (n = 1) is A convenient ratio for measurement purposes is the ratio the residual carrier amplitude to the firs' ier sideband amplitude. This ratio is given by This result can bc used to determine the relative difference between the residual carrier level and the first sideband level on a spectrum analyzer. Expressions for the higher-order sidebands are given in Appendix A. Table 1 contains the solution of Equation (1)for the first five sidebands for modulation indices up to 10; all sideband amplitudes are nor~,~:dizedto unity as follows: Relative sideband amplitude = 20 log ,, (1/ c,I ) = -20 log,, / C,i . Table 2 illustrates the use of Table 1. Unrnodulated carrier power level: 1 watt (30 dbm) ; Fhl mod index = 1. The following salient properties df square-wave frequency modulation should be noted: (a) The residual carrier is zero for all even-integer modulation indices; @) When 3 is an odd integer, the odd-ordered sidebands equal zero (except the 3 = n sideband); (c) When 3 is an even integer, the even-order sidebands equal zero (except the 9 = n sidedand); (d) When ,b = n, one-half the spectral energy resides in the ij: = n sideband; the remaining spectral energy is contained in the residual carrier and other sidebands ; (e) Except for the special cases noted previou=ly, all sidebands are present in the FM spectrum although only odd harmonics exist in the modulating signal. Figure 1 illustrates the basic test setup used to obtain spectrum photographs, These photographs are shown in Figure 2. The modulation indices were chosen to demonstrate the spectrum characteristics discussed previously. A decibel scale (phot~g~aph(e)) is provided for measuring amplitudes from the pictu~s. This scale is valid for all RF spectrum photographs in Figures 2, 3, 4, B1 and B4. For each modulation index, the corresponding spectrum resulting from the equivalcct pseudo-random (PR) modulating signal is shown.* This is done both because the PR spectrum is a good approximation to the actual spacecraft data spectrum, and also to illustrate the problem in the setting of modulation index for complex signals. * Equivalent PR sign01 is defined os follows: Assilme tie square wove to be gerleroted by usi~g on alternating 1,0 NRZ-L pottern. The PR code is then clocked at the same square-wove rate, but with its own unique 1,0 pattern. 3 Table 1 Square-Wave Frequency Modulation: Modulation Index VS Sideband Amplitude BETA c0 c 1 co/c1 c 2 c3 c 4 ( RAD 1 (DR) (DR) (nn) (OR) (n~) (DR) 0.10 -0.04 -23.3 4 23.31 -52.06 -43.11 -64.11 0.20 -0.14 -17.98 17.34 -40.06 -37.38 -52.16 0.30 -0.32 -14.56 14.24 -33.08 -34.38 -45.27 0.40 -0.58 -12.21 11.63 -28.13 -32.65 -40.49 0.50 -0.91 -10.45 9.54 -24.43 -31.79-36.90 0.60 -1.33 -9.10 7.77 -21.42 -31.71 -34.08 0.70 -1.83 -8.03 6.20 -18.93 -32.48 -31.84 0.80 -2.42 -7.19 4.77 -16.82 -34.51 -30.02 0.) 0 -3.11 -6.53 3.41 -15.02 -39.22 -28.58 1.00 -3.1 2 46.02 2.10 -13.46 -27.44 1.10 -4.86 -5.65 0.79 -12.11 -37.04 -26.60 1.20 -5.94 -5.41 -0.53 -10.94 -30.11 -26.04 1.30 -7.20 -5.28 -1.32 -9.12 -25.78 -25.76 1.40 -8.69 -5.26 -3.42 -9.03 -22.57 -25.79 -10.45 -5.35 -5.11 -8.27 -20.00 -26.18 -12.62 -5.54 -7.08 -7.62 -17.86 -27.02 -15.39 -5.85 -9.55 -7.08 -16.04 -28.52 -19.23 -6.26 -12.97 -6.63 -14.46 -31.13 -25.61 -6.79 -18.82 -6.28 -13. 09 -36.32 -7.44 -6.02 -11 88 -26.48 -8.24 -18.24 -5.85 -10.82 -34.87 -20.97 -9.20 -11.77 -5.76 -9.89 -28.23 -18.02 -10.34 -7.68 -5.76 -9.08 -24.14 -16.14 -11.71 -4.43 -5.84 -8.37 -21.14 Table 1 (continued) Square-W ave Frequency Modulation : Modulation Index VS Sideband Amplitude BETA CO C 1 CO/Cl c2 C 3 C4 C5 (RAD) (DB) (DB) (DB) (DB) (DB) (DB) (DB) Table 2 Sample Sideband Energies Siqle Total db Value dbm Sidebond Sidebnd Spectrum Element From Power Pcwer Power Table 1 Lwel (watts) (watts) Residual carrier Iwel - 3.92 26.08 0A055 0.a055 Fint sideband lwel' - 6.02 23.98 0.250 0.500 Residual carrier/Fint sideband 2.10 - - - Second sidebad level -1 3.46 1 6.54 0.045 0.090 Third sideband levelb Not Resent - 0 .OW 0 .OOO Fourth ridebond lwel -27.44 2.56 ' 0.0018 0.0036 Fifth sidebnd levelb Not Present Q 0 .OOO 0 .000 TOTAL POMR 0.9991' '~hmnB = n, half of the totol ~p.ctrulew b~np =odd integer, oll odd sibbond m raidrr in tM pair of si&M (1/4 in ~h 1.m except the B = n ribbmd. ridoband). Th. mini*) power midm in the :,igh.r- abr *:.&on&. RANDOM GENERATOR pc-1 1 WAVE TRANSMITTER SPECTRUM ANALYZER TOOTH i TEKTRONIX - 1 L20 SQUARE PM PLUG IN WAVC TRANSMITTER ,GEP!ERATOW TRIANGULAR GENERATOR Figure 1. Basic Test Configuration Used to Obtain Spectra Figure 2, Frequency Modulation Spectto (Square-Wove ond P seudo- Pondom) 7 (h) 5q~0rn-Wme Wobldlon (i) Pseudo-Ron& Modulation IR a 91 (P = 2) (1) Squore-Wave bduldion (m) Re"&-Ron&m F-rbdtllotion rn = 71 (P = 3) Figure 2(continued).