Rca1948FrequencyModu.. - The New Jersey Antique Radio Club

66 FREQUENCY MODULATION, Volume Icarrier by ± r^ ±s/x 2where r and s are positive integers or zero.Although, theoretically, an infinite number of side frequencies isproduced, in practice the ones separated from the carrier by a frequencygreater than the deviation decrease rapidly toward zero, so thebandwidth always exceeds the total frequency excursion, but neverthelessis limited. For large modulation indexes and a sinusoidal modulatingvoltage, the bandwidth approaches, and is only slightly greaterthan, the total frequency excursion.To show how the bandwidth changes with modulation index, exactmathematical expressions for the spectrum will now be obtained.The Spectrum of a Carrier Wave which is Frequency-Modulatedwith a Sinusoidal SignalWhen a carrier wave is frequency-modulated with a single audiotone, the equation for the voltage isD/\e = E sin I at -\— sin 2ir[it I(1)whereE = amplitude of the wave« = angular frequency of the carrier, radians per secondB = deviation, cycles per secondji= audio frequency, cycles per secondt = time in secondsD/jx = modulation index.This expression can be expanded in a spectrum consisting of acarrier and side frequencies, in accord with the result 3e = E ^ J n (D/fJ.) sin (£ + 2irn[it) (2)?l= COwhere J n (D/fx) is a Bessel function of the first kind of order n andargument D/y,.Graphs of the Bessel FunctionsTo plot the spectrum of a frequency-modulated wave for a givensA. Bloch, "Modulation Theory," Jour. I.E.E.,91, pp. 31-42; March, 1944.(London), Part III, vol.