366 BELL SYSTEM TECHNICAL JOURNAL 



break down the given complex wave into a series of single pulses. Next 

 the spectrum of each pulse is determined separately. Then the spectrum 

 of the complex wave is obtained by combining the spectra of the various 

 single pulses involved. One of the things to be demonstrated here is that it 

 is perfectly feasible in many cases to perform these summations graphically, 

 even tliough basically it does involve the handling of spectra each containing 

 an infinite number of frequency components. 



There are other wave forms that could be used as the fundamental build- 

 ing block instead of the single pulse. The unit step function is one possi- 

 bility, since it is used in transient analysis for a similar purpose. However, 

 the single pulse has obvious advantages when the complex wave to be ana- 

 lyzed is itself a series of pulses, as in pulse modulation. Again it would be 

 nice to be able to choose as the fundamental unit a wave that has a discrete 

 rather than a continuous band frequency spectrum, but it seems that any 

 wave flexible enough to make a satisfactory building unit is inherently non- 

 periodic and so has a continuous frequency spectrum. However the fact 

 that the fundamental units have continuous spectra does not of itself compli- 

 cate the results. If for example, the wave to be analyzed is periodic, the 

 sum of the spectra of the various pulses must reduce to a discrete frequency 

 spectrum. In the cases of interest here, when the pulse train under analysis 

 is repetitive, combinations of identical pulses will be found to occur with the 

 same fundamental period, and generally the first step in the summation of 

 such spectra is to group the series of pulses into periodic waves with discrete 

 spectra. 



Manipulations of Single Pulses 



In its use, the single pulse may be varied in amplitude, in length, and in 

 position with respect to time. These changes have independent efifects on 

 the frequency spectrum. A variation in the amplitude of a pulse does not 

 change its spectrum, except to increase proportionately the magnitudes of 

 all components. A change in position of a pulse with time does not change 

 the amplitude vs. frequency characteristic of the spectrum, but it does 

 shift the phase of each component by an amount proportional to the product 

 of the frequency and the time interval through which the pulse was shifted. 

 A change in the length of a pulse will change the shape of the amplitude vs. 

 frequency characteristic of the spectrum. Figure 3 shows this effect. How- 

 ever, if the center point of the pulse is not shifted in time, the relative phases 

 of the components are not afifected by such changes in length. 



The single pulse can also be modulated to aid in the resolution of more 

 complicated wave forms. This process is based on the use of the pulse as a 

 function having a value of unity over a chosen time interval and a value of 

 zero at all other times. Thus, to show a part of a sinusoidal wave, we need 



