Rca1948FrequencyModu.. - The New Jersey Antique Radio Club

422 FREQUENCY MODULATION, Volume Imodulated transmission problem. The method is rigorous and general,but not practical. Calculating the response pattern of (a) in Figure 2by the method just outlined, for example, would require the labor ofa staff of clerks for several years, because it would be necessary to calculateabout 100,000 points for a smooth curve of the radio-frequencyvoltage oscillations during one modulation cycle, each point being thesum of the instantaneous values of more than 500 side-frequencyvoltages.It is fortunate that the operations outlined by steps 2, S, 4, and 5can be done analytically, initiating a practical equation for the outputor response-voltage envelope in terms of the network parameters andthe frequency-modulated input-wave parameters. This equation will bederived with time as the independent variable and transformed to afunction of instantaneous input frequency to correspond with theabscissae of Figure 2. The method used in this report for bridgingsteps 2, 3, 4, and 5 is very similar to the pattern followed by Cherryand Eivlin. 1 The problem has also been considered by Kulp, who developeda solution which is practical when the modulation index of thefrequency-modulated input wave is small. 2First a unit frequency-modulated wave is defined. The equationsderived in this paper are based upon unit magnitude of the inputsignal.o) = 2tt X the center frequency (1)K = 2tt X the frequency deviation (2)v = 2tt X the modulating frequency (3)Km f= — = the modulation index (4)vS, = wq + K cos vt = 2-7T X the instantaneous input frequency. (5)It has been shown that 3e ln= sin (a t + m t sin vt)1E. C. Cherry and R. S. Rivlin, "Nonlinear Distortion, with ParticularReference to the Theory of Frequency-modulated Waves," Part II, Phil.Mag., vol. 33, pp. 272-293 ; April, 1942.2M. Kulp, "Spektra und Klirrfaktoren Frequenz- und AmplitudenmodulierterSchwingungen," Part II, Elek. Nach. Tech., vol. 19, pp. 96-109; June,1942.8John R. Carson, "Notes on the Theory of Modulation," Proc. I.R.E.,vol. 10, pp. 57-64; February, 1922.