. The Bell System technical journal . the frequency spectrum of thepulse is shifted by 180°. Therefore to compare the spectrum of a pulseposition modulated wave like that on Fig. 8 with the equivalent spectrum ofa phase modulated wave, what is needed is Fig. 9, showing the frequencyspectrum of a phase modulated wave of the form Cos{ct — k sin vt) as a func-tion of k for values of ^ up to -zr radians or 180°. The computation of thefrequency spectrum of such a phase modulated wave has been adequatelycovered elsewhere and all that is done here is to give the brief developmentshown in appendix B.
. The Bell System technical journal . the frequency spectrum of thepulse is shifted by 180°. Therefore to compare the spectrum of a pulseposition modulated wave like that on Fig. 8 with the equivalent spectrum ofa phase modulated wave, what is needed is Fig. 9, showing the frequencyspectrum of a phase modulated wave of the form Cos{ct — k sin vt) as a func-tion of k for values of ^ up to -zr radians or 180°. The computation of thefrequency spectrum of such a phase modulated wave has been adequatelycovered elsewhere and all that is done here is to give the brief developmentshown in appendix B. * Mr. W. R. Bennett has pointed out that this incommensurate case is the general requires a double Fourier series, which reduces to a single series when the signal andcarrier frequencies are commensurate. This analysis is based on the single Fourier series. SPECTRUM ANALYSIS OF WAVES 375 A comparison of the spectra on Figs. 8 and 9 shows that the sidebandshave the same general pattern. However comparative sidebands are not. 40 50 60 70 MODULATION IN PER CENT Fig. 8—Spectrum of pulse position modulated wave for case where the carrier frequencyC is 10 times the signal frequency v. quite equal in the two cases. In fact comparable upper and lower side-bands in the case of the pulse modulated wave shown on Fig. 8 are not 376 BELL SYSTEM TECHNICAL JOURNAL equal in absolute magnitude to each other. This lack of symmetry is dueto the fact that c is ()nl\ 10 limes v.
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