SPECTRUM ANALYSIS OF WAVES 



369 



apart, the contributions from all pulses add in phase. These are the fre- 

 quencies nc, where n = 1,2,3 and c "^ Tj.- It is also apparent that at fre- 

 quencies for which the phase differences between the components are not an 

 exact multiple of 2ir radians apart, the contributions from enough pulses 

 must be spread in phase over an effective range of to 2x radians in such a 

 way as to cancel one another. For example, take the particular frequency 

 for which the difference in phase between pulses is 361° instead of 360°. 



-^- 



-^■ 



UNMODULATED 



PULSE TRAIN 



(PERIOD T) 



MODULATING 



FUNCTIOM 



OR SIGNAL 



(PERI0D=10T) 



POSITION 



MODULATED 



PULSE TRAIN 



(AT, ~e|,ETC) 



(REFERENCE) 



(AT,orO) 

 TIME.t — »■ 



Fig. 5 — Formation of pulse position modulated pulse train and its resolution into subsidiary 



unmodulated pulse trains. 



The contribution from each preceding pulse will be effectively advanced in 

 phase 1° with respect to its successor, so that the contributions from pulses 

 180 periods apart will be exactly 180° out of phase. Therefore over a 

 sufBcient number of pulses, the net contribution is zero. 



The spectrum of the unmodulated pulse train is thus made up of a do 

 term plus harmonics of the frequency C = \/T. The dc term is the average, 

 and therefore is equal to £ X 2L/T, where E is the magnitude of the pulse. 

 All of theother components have the same relative magnitudes that they have 



