522 BELL SYSTEM TECHNICAL JOURNAL 



and by substitution in (34) we get 



I = E&^p(i r ^{T)dT + id'\ 



X I ( 1 + ^s(n~) I Y(ico.)\ + E^CnCO 



, (35) 



which corresponds precisely with (29) except that it is expressed in 

 terms of the retarded time t'. If the transducer introduces a large 

 phase delay, (35) may be much more rapidly convergent than (29) 

 and should be employed in preference thereto. 

 Corresponding to (30) we may write 



F(ico, + ic.)6-^-^(") = U^^^\\ Y(iu>,) I + R, 



which defines the remainder. Then 

 I = Eexpli r ndr + id' 





where 



+ E exp (ico/ + id')D{t'), (36) 

 D{t') ^ r R(coc,co)-F(icc)e-''^''do^. (37) 



iJ — CO 



Formulas (36) and (37) correspond precisely with (31) and (32) and 

 the same remarks apply. 



II 



The foregoing will now be applied to the Theory of Frequency 

 Modulation. A pure frequency modulated wave may be defined as a 

 high frequency wave of constant amplitude, the "instantaneous" 

 frequency of which is varied in accordance with a low frequency signal 

 wave. Thus 



PF = exp q coc/ + X s(t)dt 



I 



Jo 



(38) 



is a pure frequency modulated wave. Here coc is the constant carrier 

 frequency and s{t) is the low frequency signal which it is desired to 

 transmit. X is a real parameter which will be termed the modulation 

 index. The "instantaneous" frequency is then defined as 



It is convenient to suppose that s(t) varies between ± 1; in this case 



