Spectrum Analysis of Pulse Modulated Waves 



By J. C. LOZIER 



The problem here is to find the frequency spectrum produced by the simul- 

 taneous application of a number of frequencies to various forms of amplitude 

 limiters or switches. The method of solution presented here is to first resolve the 

 output wave into a series of rectangular waves or pulses and then to combine the 

 spectrum of the individual pulses by vectorial means to find the spectrum of the 

 output. The rectangular wave shape was chosen here as the basic unit in order to 

 make the method easy to apply to pulse modulators. 



Introduction 



The rapidly expanding use of pulse modulation^ in its various forms is 

 bound to make the frequency spectrum of pulse modulated waves a subject 

 of increasing practical importance. The purpose of this paper is to show 

 how to determine the frequency spectrum of these waves by methods based 

 as far as possible on physical rather than mathematical considerations. The 

 physical approach is used in an attempt to maintain throughout the analysis 

 a picture of the way in which the various factors contribute to a given result. 

 To further this objective the fundamentals involved are reviewed from the 

 same point of view. 



The method is used here to analyze two distinct types of pulse modulation, 

 namely, pulse position and pulse width modulation.^ These two cases are 

 especially important for illustrative purposes because their spectra can be 

 tied back to more familiar methods of modulation. Thus it will be shown 

 that, as the ratio of the pulse rate to the signal frequency becomes large, 

 pulse position modulation becomes a phase modulation of the various carrier 

 frequencies that form the frequency spectrum of the unmodulated pulse 

 wave, and pulse width modulation becomes a form of amplitude modulation 

 of its equivalent carriers. The analysis also shows certain interesting input- 

 output relationships that may be obtained from such modulators, treating 

 them as straight transmission elements at the signal frequency. 



These relationships are of more than theoretical interest. The pulse 

 position modulator has already been used as phase or frequency modulator 

 to good advantage.^ The use of a pulse width modulator as an amplifier is 



' E. M. Deloraine and E. Labin, "Pulse Time Modulation", Electrical Communications , 

 Vol. 22, No. 2, pp. 91-98, Dec. 1944; H. S. Black "AN-TRC-6 A Microwave Relay Sys- 

 tem", Bell Labs. Record, V. 33, pp. 445-463, Dec. 1945. 



2 By pulse position modulation is meant that form of pulse modulation in which the 

 length of each pulse is kept fixed but its position in time is shifted by the modulation, and 

 by pulse width modulation that form in which the length of each pulse varies with the 

 modulation but the center of each pulse is not shifted in position. 



' L. R. Wrathall, "Frequency Modulation by Non-linear Coils", Bell Labs. Record, 

 Vol. 23, pp. 445-463, Dec. 1945. 



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