. The Bell System technical journal . l be 1-kJT radians apart. It is appar-ent that frequencies for which lirjT is 2x or some multiple of 27r radians 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 ^ It is also apparent that at fre-quencies for which the phase differences between the components are not anexact multiple of 2ir radians apart, the contributions from enough pulsesmust be spread in phase over an effective range of 0 to 2x radians in such away as to cancel one another. For example, take
. The Bell System technical journal . l be 1-kJT radians apart. It is appar-ent that frequencies for which lirjT is 2x or some multiple of 27r radians 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 ^ It is also apparent that at fre-quencies for which the phase differences between the components are not anexact multiple of 2ir radians apart, the contributions from enough pulsesmust be spread in phase over an effective range of 0 to 2x radians in such away as to cancel one another. For example, take the particular frequencyfor which the difference in phase between pulses is 361° instead of 360°. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <u —»■ u I inT o D1- p.— ^\ 62 °^:v H ^_^^ a< ^^\ \ \ ^^ ^^\ 1 i :i :i 1 TIME,t—»• 1 i 1 1 1 1 i 1 1 ;AT| -►1 U- -J U-AT2 —»• -^- -^■ UNMODULATED PULSE TRAIN (PERIOD T) MODULATING FUNCTIOM OR SIGNAL (PERI0D=10T) POSITION MODULATED PULSE TRAIN (AT, ~e|,ETC) (REFERENCE). (AT,orO)—»■ 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 inphase 1° with respect to its successor, so that the contributions from pulses180 periods apart will be exactly 180° out of phase. Therefore over asufBcient number of pulses, the net contribution is zero. The spectrum of the unmodulated pulse train is thus made up of a doterm 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 of theother components have the same relative magnitudes that they have 370 BELL SYSTEM TECHNICAL JOURNAL in the single pulse spectrum. This gives a spectrum like that shown onFig. 6. Figure 6 also shows for comparative purposes the spectrum of thesubsidiary pulse wave consisting of every 6th pulse. Thus in the unmodu
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