TRA NSIENT RESPONSE OF AN FM RECEI VER 7 1 9 



the complex function, 



»>(0 



[Vm = c(l) e"'" = a(t) + ib(t) (7) 



is here called the "envelope function" of the voltage with respect to the 

 radian frequency coo , c{t) being a real amplitude modulation factor, which is 

 the envelope* itself, as usually conceived, and exp [i(f>{t)] a complex fre- 

 quency modulation factor, in which <f>'(t) is the instantaneous deviation of 

 the radian frequency from the reference value, coo . If such a modulated 

 voltage wave is applied to an ideal linear detector, the output voltage across 

 the load circuit of the latter is the real envelope, c(t) = [a'^(t) + b^{t)f ^. 

 This concept of an envelope function provides a convenient generalization 

 of modulation ideas. Both amplitude modulation and frequency modula- 

 tion vary the envelope function, but in different ways. In amplitude mod- 

 ulation, the real magnitude, c(/), is varied while the angle </> is constant, 

 whereas, in frequency modulation, c is constant and it is the angle, </>(/)) that 

 is varied. 



It will be seen that (5) is in precisely the same form as (6), so that we can 

 write the envelope function of V{t) immediately, as follows: 



[V(t)] = ait) + ib{t) = h f g--o^+»<'('-) h{t) dr. (8) 



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The conjugate envelope function then is 



[Vii)] = a{t) - ib{t) = h f «'"»'-'•"-" ff(r) dr. (9) 



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The spectrum of the envelope function is also of interest. To obtain the 

 spectrum, which we shall call, Fo(/), we find the Fourier transform (hereafter 

 abbreviated, F.T.) of both sides of (8), viz.: 



Fo(/) = r [F(/)]e-*"' dt = h r e-''-' f e-'''''-'''''-''-H(T) dr dt. (10) 



JLoo J- 00 Jo 



It is permissible to reverse the order of integration of r and /, obtaining 

 Fo(/) =hf e-'^-'^Hir) f " e--'+«('-> dt dr. (11) 



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The F.T. of h exp [ie{t)] will be designated, ^(/), i.e. 



^(/) = h r e'''''-"^' dt. (12) 



J— 00 



* The "envelope", so defined, is an engineering concept and is not quite the same thing 

 as the envelope of mathematics, which is always tangent to a curve or set of curves. 



