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THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 



H'^inin be greater than W„,^-^ . The difference M = W^m — W.^^x , 

 which represents the margin for distinction between pulse amphtudes, 

 becomes 



g - 1 



P(0) - (A„ax - An.in) Z [P^ M 

 n = l 



+ p-{nr) + P^i-nr) + P'i-nr)], 



(13.04) 



or: 



Al — (.-Amax ^minj 



Lq 



P(0) _ y^ 



— 1 n=l 



P(nr) I + I Pi-nr) I 



, (13.05: 



where ] P(± nr) \ designates the absolute values of the impulse charac- 

 teristic. 



Equation (13.05) shows that for a given value of q the margin depends 

 on the maximum pulse excursion ^max — Amin and is thus the same with 

 Aniax = 1 and ^min = as with Amax = 0.5 and Amin = —0.5. As an 

 example, equation (13.05) shows that with two pulse amplitudes, g = 2, 

 it is possible to distinguish between pulses and spaces, or between posi- 

 tive and negative pulses, if the sum of the absolute values of the impulse 

 characteristic at all the sampling points, excluding 0, is less than the 

 amplitude P(0) of the impulse at sampling point 0. 



The maximum margin against errors is obtained without pulse over- 

 laps, i.e. when the summation term in (13.05) is zero, and is 



Mu 



\-Amax Aminj 



P(0) 



1 



(13.06) 



The ratio of the margin M as given by (13.05) to the maximum margin 

 becomes : 



M/Mrn.. = 1 - ^^ E I Pinr) I + |P(-nr) 



P(0) t^i 



(13.07) 



This equation may be employed to determine the maximum possible 

 pulsing rate for a given impulse characteristic and number of pulse 

 amplitudes, obtained when M/M^ax = 0, or to determine the margin 

 for a given pulse transmission rate. An example of the latter application 

 is illustrated in Fig. 43, which shows the margin M/Mmax in per cent, 

 obtained when (13.07) with g = 2 is applied to the curves shown in 

 Fig. 23 for various degrees of delay distortion. The pulse interval is 

 taken as r = l/2/i = l//max , where/i is the frequency at the 6 db down 



