SPECTRUM ANALYSIS OF WAVES 379 



special case where the phase relationship between spectra of adjacent pulses 

 does not change with modulation because the centers of the pulses are not 

 shifted by the modulation. The conventional form of pulse length modula- 

 tion, where one end of the pulse is fixed in position, combines both this 

 pulse width modulation and the pulse position modulation previously ana- 

 lyzed. The interest in this case of pulse width modulation arose in con- 

 nection with the analysis of ''hunting" ser\^omechanisms, and the analysis 

 provides a basis for a general solution of the response of a two-position 

 switch or ideal limiter to various forms of applied voltages. 



Since the unmodulated wave is a square wave with pulses of length 2L 

 recurring at intervals of T = 4L, it has the familiar square wave spectrum 

 including a d-c term, a fundamental term or carrier of frequency c = l/T, a 

 3rd harmonic with a negative ampUtude 1/3 that of the fundamental, etc. 

 Figure 10 shows clearly that this spectrum is the sum of single pulses of 

 width 2L spaced T = AL seconds apart. In the summation, all frequencies 

 cancel except harmonics of c and, since they all add directly in phase, the 

 component frequencies in the resultant spectrum have the same relative 

 amplitudes as they have in one single pulse. 



When this pulse train is modulated, the width of each pulse becomes 

 2{L-\- AL), where the magnitude of AL depends in some specified way on the 

 magnitude of thhe modulating function at the instant corresponding to the 

 center of the pulse. For simplicity, the case will be taken where AL is 

 proportional to the magnitude of the modulating function. For 100% 

 modulation, AL will be assumed to vary from — L to +L. Figure 3 shows 

 how the relative amplitude of the components of the frequency spectrum of 

 a pulse vary for 3 different values of AL , along with the equation that gov- 

 erns these amplitudes. 



If the modulating function has a periodicity v such that c = lOz', then 

 every lOth pulse, recurring at the same instant in each modulating cycle, 

 will be widened to the same extent and so can be formed into a subsidiary 

 unmodulated pulse train, as was done on Fig. 5 for the pulse position 

 modulated wave. 



Again vector diagrams like those in Fig. 7 may be formed showing the 

 contribution of each of these subsidiary pulse trains at various frequencies 

 such as c, r + v and c — v. ^^1len the waves are unmodulated, the vector 

 diagrams for the same frequencies will be the same as those for the pulse 

 position modulated case, except for the absolute amplitudes of the com- 

 ponents, as long as c = lOr in each case. When the pulse width system is 

 modulated, however, the modulation does not rotate the individual vector 

 components as in the pulse position case since the spacing between pulses is 

 not changed. What the pulse width modulation does is to change the 

 length of the individual component vectors exactly as it does in the case of 



