CARRIER NATURE OF SPEECH 515 



The effect of the selective transmission is allowed for by multiplying 

 Cv by the transmitting factor r(co, /), co indicating that the transmitting 

 factor is a function of frequency at any instant. Applying this factor 

 in equation (1) gives: 



n 



Transmitted C^ = E r(oj, t)Ak cos {kPt + dk). (4) 



The r factor is placed inside the summation to indicate that as k 

 changes the different frequencies have different values' of the multiply- 

 ing factor r. If a multiplicity of carrier waves is assumed, the trans- 

 mitting factor would be rk{t), individual to the ^th component. 



In normal voiced speech, Sv, these three modulations are all present 

 simultaneously, so that: 



Sv = s(t) J2 ^(w, t)Ak cos 



k=l 



kP f p{t)dt + 



'JO 



(5) 



Equation (5) shows how the message in the form of the 5, r, and p 

 functions has imprinted its characteristics on the original carrier C» 

 of equation (1). 



The derivation of (5) was for voiced speech. Unvoiced speech, 

 however, is also covered by (5) as a degenerate case. Nevertheless, 

 further information is presented by writing out the unvoiced carrier 

 separately. For unvoiced speech, the frequency P approaches zero 

 and the number of terms, w, approaches infinity, giving an integral 

 instead of a finite sum of components in equations (l)^and (5). The 

 unvoiced carrier C„ is then : 



Cu= I ' A(u)) cos [co/ + ^(co)]^a; (1') 



and the unvoiced speech: 



Su = s{t) I r(o}, t)A{co) cos lojt + d(co)^doj (5') 



with the continuously variable frequency cj (radians per second) vary- 

 ing over the audible range of energy contribution from coi to co2 and 

 the unvoiced carrier spectrum defined by amplitude A(oo) and phase 

 d(o}). The unvoiced speech has no inflecting factor but does have 

 switching and transmitting factors to make up the message impressed 

 on the carrier. 



