718 BELL SYSTEM TECHNICAL JOURNAL 



In the transmission of signals by frequency modulation, the instantaneous 

 radian frequency deviation, B'{t), is made to vary in proportion to the 

 signal amplitude, so that B{t) then varies in proportion to the time integral 

 of the signal amplitude. 



As a preUminary step to the discussion of the frequency detector itself, 

 we require a formula for the voltage drop across an impedance Z(/) when the 

 frequency-modulated current (1) flows through it. The point of view 

 usually adopted is to regard the impedance as a composite function of time, 

 viz., Z[/(/)], and to say that the voltage across it is 



v{i) = imifit)] = i{t)z [/o + ^ e'{i)^ . 



(3) 



This quasi-stationary viewpoint gives results that are nearly correct if the 

 rate of change, d"{t)/2Tr, of the variable frequency is not too large. The 

 magnitude of the error has been determined in a paper by Carson and Fry.^ 

 In the present paper, impedance is a function of frequency that is inde- 

 pendent of time, as in the classic theory of linear systems. The frequent 

 use of the term ''instantaneous frequency," as defined by (2), does not 

 imply a departure from this point of view. 



In the following, H{t) is, in general, the voltage response as a function 

 of time, of a network to a unit impulse of current applied at time / = 0. 

 In the case of a two-terminal impedance element, H{l) is the voltage drop 

 across the element when a unit impulse of current is sent through it. Then, 

 if the frequency modulated current (1) flow through the impedance, the 

 voltage drop is 



V{t) = h f cos Mt - r) + e{t - r)]HiT) dr (4) 



Jo 



where ojo = ^irfo . d{t) can have any form as a function of time. V{t) 

 can be written, 



V{t) =^/"°' [ e-''^''^''^'-'^ H{t) dr 

 2 Jq 



2 JQ 



In the frequency detector problem, the result finally desired is the envelope 

 of the voltage wave. It will clarify the discussion to explain first what is 

 meant by an envelope. If the voltage is of the form 



V{t) = c{l) cos M H- <^(/)] 



= a{l) cos Wo/ — b{l) sin coo/ 



= h[a{l) + ib{l)] i-"^' + i[a{t) - ib{t)] €-'"'' (6) 



' Item 1 in the bibliography. See formula 21 in that paper. 



