SPECTRUM ANALYSIS OF WAVES 



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only multiply this wave by a pulse of the correct length and proper phase 

 with respect to the sinusoid to show only the desired piece of the wave. In 

 this simple case it is not difficult to derive the spectrum because what are 

 produced are the sum and the difference products of the modulating fre- 

 quency with the spectrum of the pulse. This gives two single pulse spectra 

 shifted up and down in frequency by the frequency of the modulation. An 

 example of this is shown in Fig. 4, where the spectrum of a single half c>cle 

 is determined. 



Pulse Position Modulation 



For the first example, a simple form of pulse position modulation will be 

 analyzed. The pulse train in this case is made up of pulses spaced T seconds 



U 0.2 



a -0.4 



I 2 3 4 5 , 



FREQUENCY, f, IN TERMS OF C (WHERE C = — ) 



Fig. 3 — Change in frequency spectrum with pulse length. 



apart and the width of each pulse is a very small part of the spacing T. 

 Such a pulse train is shown on Fig. 5. The pulse train is modulated by ad- 

 vancing or retarding the position (time of occurance) of the pulses by an 

 amount proportional to the instantaneous amplitude of the signal at sampled 

 instants T seconds apart. Figure 5 also shows the signal, in this case a sine 

 wave of frequency 1/lOr, and the resulting modulated pulse train. The 

 peak amplitude of the modulating sine wave is assumed to shift the position 

 of a pulse by 1 /-iT. The length and the amplitude of the pulses are the same 

 since neither is affected in this type of modulation. 



The first step in the analysis is to determine the spectrum of the pulse 

 train before modulation. Each pulse contributes a spectrum of the form 



