514 BELL SYSTEM TECHNICAL JOURNAL 



by nature, the carrier C„ may be written as a function of the time / thus: 



n 



a = T. Ai, cos {kPt + dk). (1) 



Here Cv is composed of n audible harmonics of relatively high fre- 

 quencies with the ^th of amplitude -4 a;, frequency kP radians per second, 

 and phase dk- The choice of fundamental frequency P is somewhat 

 arbitrary but may well represent the average of the talker over the 

 period of interest. 



By modulation processes, there is molded on to this carrier the total 

 message information at the relatively low syllabic frequencies. The 

 message is divided into three parts: (a) the starting and stopping of 

 the carrier; (6) the instantaneous fundamental frequency; and (c) the 

 selective transmission through the resonant vocal tract.^^ These three 

 message functions as they manifest themselves in varying the carrier 

 will be represented by s, p, and r, respectively. Equation (1) will be 

 modified to indicate the effect on the carrier of each of these modula- 

 tions separately, after which the equation will be rewritten to show the 

 effect of all three acting simultaneously. 



The effect of starting and stopping the carrier is described mathe- 

 matically as a function of time by multiplying C„ by the switching 

 function s{t), giving: 



n 



Switched C„ = ^(0 L ^t cos (kPt -f 9k). (2) 



fc=i 



For simple on-off switching, s(t) alternately equals zero and unity, 

 although it may in general represent more gradual changes or even any 

 variations of intensity over the frequency range. 



The instantaneous fundamental frequency is obtained by multiply- 

 ing P by the inflecting factor p{t). The effect of the frequency modu- 

 lation ^^ is represented by substituting for Pt the integrated quantity 



r Pp(t)dt = P \ p{t)dt. 

 Jo Jo 



Writing this value for Pt in equation (1) gives the inflected carrier wave: 



Inflected Cv = Jl Ak cos 



k=l 



kP f prnt + eA, (3) 



" As in the body of the paper, the effect of phase modulation is neglected here. 



13 "Variable Frequency Electric Circuit Theory with Application to the Theory 

 of Frequency Modulation," J. R. Carson and T. C. Fry, Bell Sys. Tech. Jour., Vol. 16, 

 p. 513 (1937). 



