230 BELL SYSTEM TECHNICAL JOURNAL 



If it were greater we should have 



k = 271 ±1. 



It is then obvious that the time of maximum absolute value, Ta , is given 

 by the slope of the line OE. It differs from Te by the difference in slope of 

 the lines OE and GE. 



We have then deduced from the spectrum of the disturbance its time of 

 occurrence in terms of two definitions of the latter. The next step is to 

 compare these times for the input and output and determine the corre- 

 sponding delays. Let us consider first the case where the attenuation is 

 independent of frequency, so that ai is zero in the output signal also. We 

 may then confine our attention to the phase, 6. Let us represent its value 

 at the input by b, and the phase shift of the system by B. Then at the 

 output 6 will be equal to b -{- B. If we take the time of occurrence as 

 determined by the maximum envelope, these times at the input and output 

 are 



Te, = ^'l' + B[ . 



The delay is then 



De — Tei — Tgfj — Bi , 



which is by definition the envelope delay of the system. 



If we take the time of occurrence based on the maximum absolute value, 

 we have, at the input, 



_ ^1 + hir 

 where 



At the output, 



where 



— - < '^0 = ^1 — COlb'l + koTT < -. 



^ _ ^>i + ^1 + (^0 + h)7r 



-t ai 



COl 



"2 < ^3 - ^'1 + 5i - mib[ + ^i') + (^'0 + h)ir < 2- 



