374 BELL SYSTEM TECHNICAL JOURNAL 



The above solution assumed a special case where c was an exact multiple 

 of V. The purpose of this assumption was to simplify the problem to the 

 extent that the periodicity of the modulated wave would be the same as 

 that of the modulating function. There are two other possible cases. For 

 one, the ratio of c to v could be such that a pulse would occur at the same 

 instant of the modulating period only once every so many periods. The 

 actual periodicity of the modulated pulse wave would be reduced accordingly 

 because it would make the same number of periods of the modulating func- 

 tion before the modulated pulse train is repeated. This is a result of the 

 fact that pulse modulation provides for a discrete sampling rather than a 

 continuous measure of the modulating wave. The technique of spectrum 

 analysis demonstrated above is just as applicable to this case as it was to the 

 simpler one. However, there will be comparatively more terms to be 

 handled. The other possible case is the one where c and v are incommen- 

 surate.^ In this case, the resulting modulated wave is non-periodic. How- 

 ever, on the basis that the spectrum is practically always a continuous 

 function of the signal frequency, this case has received no special attention 

 here. 



At frequencies for which c is very much greater than v, so that the number 

 of component pulse trains becomes too numerous to handle conveniently in 

 the above fashion, the sidebands about each carrier or harmonic of the 

 switching frequency can be computed by the standard methods for phase 

 modulation, as the next section will demonstrate. This result follows 

 directly from the theorem that as the carrier frequency c becomes large with 

 respect to v, pulse position modulation merges into a linear phase modulation 

 of each of the carriers. 



Pulse Position Modulation vs Phase Modulation 



When a pulse, in a pulse position modulated wave, is shifted by 1/2 the 

 spacing between pulses (100% modulation) it is apparent from the previous 

 discussion that the component of the carrier in the frequency spectrum of the 

 pulse is shifted by 180°. Therefore to compare the spectrum of a pulse 

 position modulated wave like that on Fig. 8 with the equivalent spectrum of 

 a phase modulated wave, what is needed is Fig. 9, showing the frequency 

 spectrum of a phase modulated wave of the form Cos{ct — k sin vt) as a func- 

 tion of k for values of ^ up to -zr radians or 180°. The computation of the 

 frequency spectrum of such a phase modulated wave has been adequately 

 covered elsewhere and all that is done here is to give the brief development 

 shown in appendix B. 



* Mr. W. R. Bennett has pointed out that this incommensurate case is the general one. 

 It requires a double Fourier series, which reduces to a single series when the signal and 

 carrier frequencies are commensurate. This analysis is based on the single Fourier series. 



