TRANSIENT RESPONSE OF AN FM RECEIVER 723 



where ^(/) is the F.T. of h exp [id{t)] and Z(/) is the impedance, (26), in this 

 case. The envelope function then is 



a{l) + ib(t) = r mm/ + Me'"' df 



J— 00 



= hke-^''^'-^"^' f e(-o+»r+v^(r) ^^ (28) 



J- 00 



J— 00 



This result is obtained by employing the convolution formula^, which states 

 that if Fi and F2 are F.T.'s of Gi and G2 , respectively, then the F.T. of F1F2 is 



r F,{f)F2{f)e''" df = f * G,{r)G2{t - r) dr. (29) 



J—ao J— 00 



(The upper limit of the integrals in (28) is / instead of go for the reason that 

 H{t) is zero for / < 0.) The result could have been obtained equally well 

 without using the Fourier transforms by substituting (27) in (8). When 

 6{t) is specified mathematically in the infinite interval (—00, «=) these 

 formulas give the resultant of the steady state and transient oscillations. 

 \'arious problems can be solved by specifying particular forms of variation 

 for d{t). Two examples follow: (1) where the instantaneous frequency, 

 6'{i), is an impulse and (2), where 0\i) is a sinusoidal wave, as for elemen- 

 tary signal transmission. 



Example 1 : Impulse Modulation 



Ignition interference comprises a sequence of sharp impulses, each of du- 

 ration very brief compared to the interval between them, so that the tran- 

 sient in the receiver produced by one impulse dies away before the next one 

 arrives. It is therefore sufficient to consider the disturbance caused by a 

 single impulse. 



If the receiver is perfectly tuned, an impulse produces, in the tuned cir- 

 cuits, a transient of the same nominal frequency^ as the signal carrier. 

 When superposed on the carrier, the interfering transient alters, or mod- 

 ulates, both the amplitude and phase of the carrier. The amplitude mod- 

 ulation is wiped out by the limiter, but the phase modulation remains to 

 produce noise in the output. The phase shift caused by the transient is a 

 random variable, because it depends upon the time of arrival of the impulse, 

 and this is entirely fortuitous. 



' See pair 202 of Item 10 in the bibliography. 



^ By "nominal frequency" is meant the frequency as determined by counting zeros of 

 the wave . The transient actually comprises a spectrum of frequencies spread over the 

 band of the tuned circuits, of course. 



