378 BELL SYSTEM TECHNICAL JOURNAL 



of c, and the phase is 90° with respect to the input. Thus the modulator 

 puts out a signal component that is the derivative of the input signal. 



To summarize the case of pulse position modulation, the frequency spec- 

 trum may be determined by the methods based on subdividing the modu- 

 lated pulse train into a series of unmodulated ones when the ratio oi c ta v 

 is small, and by treating each harmonic of the carrier as a phase modulated 

 wave of the form Cos n (ct -\- 6), where 6 is the modulating function, when the 

 ratio of c to D is large. In the case treated here, the modulating function was 

 a simple sinusoidal wave. Of course the analysis holds for more complicated 

 wave shapes having frequency spectra of their own. In this event however 

 the restriction on the relative magnitudes of the frequencies v and c should 

 be taken as one on c and the highest frequency in the modulating spectrum. 

 The complexity of the modulating function does not affect the analysis when 

 it is done by this technique of subdividing the pulse train, since all that need 

 be known is how much each pulse is shifted, and this can be done graphically. 

 The analysis given here has neglected the length of the individual pulses. 

 This was done when it was assumed that the individual contributions from 

 the various pulse trains had the same amplitude at all frequencies. For any 

 finite pulse width, the relative magnitudes of the various components must 



silt X 

 be modified by the factor of the single pulse, as shown on Fig. 6. 



As mentioned in the introduction, a complex wave could be analyzed by 

 multiplying its magnitude-time characteristic by unit sinusoids at each 

 frequency in question, sampling the product at a sufficient number of points 

 uniformly spaced over a cycle of the envelope of the complex wave, and then 

 averaging the values of the product thus obtained. This technique is par- 

 ticularly applicable to the analysis of pulse position modulated waves since, 

 by taking the centers of the pulses of the modulated wave as the sampling 

 instants, it is possible, with a finite number of samples (same as the number of 

 pulses) to get the same results as though a very much greater number of 

 uniformly spaced samples were taken. The interesting thing to note here 

 is that the actual computations that would be involved in applying this 

 sampling method of analysis to a pulse position modulated wave are almost 

 identically the same calculations as required by the technique of resolving 

 the pulse train into unmodulated subsidiary pulse trains used here. 



Pulse Width Modulation 



Pulse Width Modulation as defined here could also be termed "pure" 

 pulse length modulation. The pulse train in the reference or unmodulated 

 condition is a recurrent square wave, and the lengths of the pulses will be 

 varied by the modulation without changing the position of the centers of 

 the pulses. The term "pure" pulse length modulation is appHcable to this 



