SPECTRUM ANALYSIS OF WAVES 373 



cellation of the 10 components, a net component of this frequency is pro- 

 duced in the frequency spectrum of the pulse wave. Taking the example 

 shown in Fig. 5, the 10 components in Fig. 7A would be shifted to the posi- 

 tions shown in Fig. 7B. These shifts in relative phase are determined in the 

 following way. Figure 5 shows that the number 1 pulse is retarded an 

 amount AT^i equal to 15% of T, the normal spacing between pulses. Thus 

 at the carrier frequency c, the phase shift between the component from tkis 

 retarded pulse and the reference pulse is 15% more than 360° or 414°. 

 Thus the component at the carrier frequency c from the first subsidiary 

 pulse train is shifted 54° from its unmodulated position. 



. At c -f- V, since the frequency is 10% higher, the net shift is 10% more than 

 at c or 59.5°. Thus the number 1 component on the vector diagram of 

 Fig. 7B is rotated 59.5° clockwise from its unmodulated position shown on 

 Fig. 7A. 



Similarly pulses 2 and 3 are each shifted in position by equal amounts, 

 AT2 and AT3 . These shifts in position give 85° phase shift at the carrier 

 frequency. Hence components 2 and 3 Sit c -\-v are each rotated 10% more 

 or 93.5° from their respective unmodulated reference positions shown on 

 Fig. 12 A. Component number 4 is shifted 59.5° clockwise just as number 1 . 

 Component 6 and 9 are also shifted 59.5° each, but in this case the modulat- 

 ing function has the reverse polarity so that the components are rotated 

 counterclockwise. Similarly components 7 and 8 are rotated 93.5° 

 counterclockwise. 



The sum of these components in the vector diagram of Fig. 7B gives a 

 resultant that is negative with respect to the reference direction and the 

 magnitude that is 58% of the reference magnitude, where the reference mag- 

 nitude and direction are those for the carrier c with no modulation. 



This gives the relative magnitude and phase of the c-\-v term produced by 

 pulse position modulation for the case where the modulating function is a 

 sine wave of frequency v — c/10 with a peak amplitude just large enough to 

 shift a pulse by 1/4 of T, where T is the spacing between unmodulated pulses. 

 A shift of this magnitude will be defined here as 50% modulation on the 

 basis that 100% modulation should be 1/2 T, the maximum displacement 

 that can be used without possible interference between pulses. 



In the same way the other component frequencies in the spectrum such as 

 c,c — v,c±2v,etc., have been computed for the above case of 50% modulation, 

 and for other peak ampUtudes of the modulating sine wave giving 25%, 

 70% and 100% modulation. In all cases the frequency of the modulating 

 function was held at z; = c/10. This information is plotted on Fig. 8, show- 

 ing V, c and the various components of the frequency spectrum that represent 

 the sidebands about the carrier frequency c, as a function of the peak % 

 modulation. 



