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BELL SYSTEM TECHNICAL JOURNAL 



and take the real part, we get 



/(/) = 2Mexp(-ai) 



[(cosh 8ai sin 5r)" + (sinh dai cos 5t)']' 



COs(a;ir - (0i - coi^J) + /3 - 7)- (5) 



Let us consider first the extreme case where the spectrum of the signal is 



Fig. 1 — Elementary disturbance corresponding to a narrow segment of ttie spectrum 

 uniform in amplitude in the neighborhood of coi, so that ai is zero. Then 



sin 5r 



fit) = 28M exp (-ai) -- — cos (wit - (dx - wi^i)). 



8t 



(6) 



Here the amplitude includes a constant factor which is proportional to 

 the bandwidth, 25, and to the magnitude, M exp( — aO, at the frequency, 

 oji, and a function of time, a plot of which is shown in Fig. 1. This function 

 consists of a sinusoidal wave of frequency, coi , the amplitude of which varies 

 with time, the envelope being symmetrical about the instant, Te =^ 61 , 



