SPECTRUM ANALYSIS OF WAVES 377 



earlier. Since the frequency c— I'is 10% less thane, the unmodulated pattern 

 of the 10 subsidiary components, as shown on Fig. 7C, is the mirror image of 

 that for c + ^ in 7A, for the first component is now 360° less 10% or 324°, 

 and subsequent components are each retarded 36° with respect to the pre- 

 vious one. When the pulse train is modulated the effect is similar to the 

 case for c -\- v and, for the same per cent modulation, the Vector pattern 

 of Fig. 7D is formed. The resultant in this case differs from that of 7B 

 in sign as well as in magnitude. The difference in sign comes from the fact 

 that, since component 1 in 7A corresponds to component 9 in 7C and com- 

 ponent 2 in 7A to component 8 etc., the modulation in the case of c — t; rotates 

 these corresponding components in opposite directions. The difference in 

 magnitude is due to the fact that since c — v is an appreciably lower fre- 

 quency than c -\- v\\\ this case (approx. 20%), the phase shift corresponding 

 to a given shift in pulse position is proportionately less. Thus the corre- 

 sponding Vector components are not shifted the same number of degrees. 

 Thus the absolute magnitudes of c -f i' and c — v are not equal in this case. 



It is apparent that this difference in magnitudes oi c -\r v and c — v be- 

 comes smaller as the carrier frequency c becomes larger with respect to v. 

 In the limiting case of c very much greater than v, c -\- v and c — v would 

 each be shifted the same number of degrees as c itself. If this more or less 

 compromise shift of c is used to compute the c ± i', c ± 2v, and c db 3i; terms, 

 then the resulting frequency spectrum is that of the phase modulated carrier 

 on Fig. 9. 



The higher harmonics of c in the pulse position wave are similarly phase 

 modulated and the interesting point is that 2c is modulated through twice as 

 many degrees phase shift and 3c 3 times as many degrees, etc. Thus a 

 single pulse position modulator could be designed to produce a harmonic of 

 c with almost any desired degree of phase modulation. This is a useful 

 method for obtaining a phase modulated wave, or with a 6 db per octave 

 predistortion of the signal, a frequency modulated wave. 



Figure 8 also shows a term in v itself, which has been neglected so far in 

 the discussion. It is apparent that the components at v contributed by the 

 10 subsidiar}' unmodulated waves must form the same kind of vector pattern 

 as those oi c -\- v in Fig. 7. However, in this case c -\- v\% eleven times v in 

 frequency, so that the components of v are rotated only one eleventh as 

 much for a given pulse diplacement. Thus the magnitude of v at 100% 

 modulation is equal to that oi c -\- v at approximately 9% modulation. For 

 different frequency ratios of c to v the relationship of the v term io c -\- v will 

 vary, and it is apparent that for c very much greater than v, the v term will 

 vanish. The relationship is such that the amplitude of the v component out 

 of the modulator at a given per cent modulation is directly proportional to 

 its own frequency v for all frequencies less than approximately one quarter 



