SPECTRUM ANALYSIS OF WAVES 371 



pulses at the carrier frequency c and its harmonics will no longer add directly, 

 because of the phase shifts that accompany the change in position. This 

 phase shift is equal to AT, the shift in position, times the radian frequency 

 of the component in question. 



However, when the signal function is periodic, each pulse will have the 

 same shift in position as any other pulse that occurs at the same relative 

 instant in a later modulating cycle. Furthermore, when the carrier fre- 

 quency is an exact multiple of the signal frequency i.e., c = nv, there will 

 be a pulse recurring at the same relative instant in each cycle of v. Under 

 these conditions, the pulse position modulated wave can be broken down into 

 a group of unmodulated waves, each being made up of that series of pulses 

 that recur at a given part of each modulating cycle, as shown in Fig. 5. 

 These subsidiary waves are eflfectively unmodulated because, as each pulse 

 recurs at the same instant in the modulating cycle, they are shifted to the 

 same extent and hence will be uniformly spaced. This uniform spacing 

 between pulses in a given wave is equal by definition to the period of the 

 modulating function, and there will be as many of these unmodulated pulse 

 trains as there are pulses in a single cycle. Thus, if c = nv, there will be n 

 such pulse trains. 



The reason for grouping the pulses into these unmodulated pulse tarns is 

 that unmodulated periodic trains have spectra of discrete frequencies. Since 

 the pulse widths are all equal, and since the spacing between pulses is the 

 same for each wave, the spectra of these unmodulated waves will all be 

 identical. Furthermore, these spectra will be the same as that of the 

 original carrier wave of pulses before modulation, except for two factors. 

 First, the fundamental frequency is now i', corresponding to the modulating 

 period, so that there are n times as many components as before. Secondly 



the amplitudes are reduced by the factor - because there is only one pulse 



in these new waves to every n pulses in the original wave. Thus, instead 

 of having a spectrum made up of the carrier frequency and its harmonics, 

 we now have one made up of harmonics of v. Since c = nv, such frequencies 

 as c, c, ± t, c ± 2v, etc., are included. An example of the spectra of both 

 the subsidiary and original pulse waves is shown on Fig. 6, for the case 

 where n = 6. 



Thus the problem of finding the spectrum of such a pulse position modu- 

 lated wave is reduced by this procedure to adding up the ;/ equal components 

 at each of the frequencies of interest, such as c and c dz v, allowing for the 

 phase difference between components corresponding to the position of one 

 pulse with respect to that of the other n-l pulses in one modulating cycle. 

 As an example, suppose n = 10 and the frequency to be computed is c + ^• 

 Now <- + I) is 10% higher in frequency than c. Thus in the unmodulated 



