232 BELL SYSTEM TECHNICAL JOURNAL 



If 



-TT < ^0 + ^2 < -^, 



Ds- Da= --. 



001 



If 



^ < ^0 + ^2 < TT, 



Ds- Da = - 



Wl 



Thus the delay as derived from the system characteristic alone may be 

 identical with the aperiodic delay based on maximum absolute value or it 



may differ from it by ±— , that is by half a period. Which condition 



COl 



holds depends on the interrelation of the phase functions which characterize 

 the signal spectrum at the input and the transmission of the system, and 

 not on either of these functions alone. 



If the attenuation is not uniform, ai cannot be neglected and the expres- 

 sion for the output signal becomes more complicated. Both the amplitude 

 and phase in (5) vary with time in a manner which depends on the value 

 chosen for 8. The expression becomes fairly simple, however, for the case 

 where ai is ver>^ large, as in anomalous dispersion and in highly resonant 

 systems. Then, even when 8 is small, we may assume that 



cosh (5ai) = exp (± 8a\), 



sinh (Sai) = ± exp (± 8ai), 



according as ai ^ 0. 



The amplitude factor in (5) then becomes 



M exp ( — ai ± 8ai) 



Here the exponent is equal to the value of a at that edge of the segment of 

 the spectrum where the amplitude is greatest. The amplitude is sym- 

 metrical about T = 0, that is, about / = 0i , at which point it has its maxi- 

 mum value. Hence the instant of maximum envelope is still given by the 

 slope of the phase, frequency curve, as when cti is small. However, the 

 maximum is now extremely flat and its sharpness no longer depends directly 

 on 8. Over the range of values of r for which r^ < < ar, the amplitude is 



