240 



BELL SYSTEM TECHNICAL JOURNAL 



The relationship between the phase and attenuation characteristics 

 can be seen in another light if we observe that the improvement in 

 attenuation which comes from the use of several transition factors is 

 due essentially to a progressive decrease in the interval between 

 critical frequencies as the cut-off is approached. In the final solution, 

 for example, the intervals between critical frequencies are initially 

 almost equal to the constant interval a. Thus in this solution, the 

 interval between Ja and /.4+1 is 0.992a: and that between /,i+i and 

 Ja+i is 0.945q;. As the cut-off is approached, however, the interval 

 gradually decreases to about 0.2a. In the transition interval, con- 

 sequently, the phase characteristic is originally almost linear and 

 curves upward sharply near the cut-off. Thus if the phase requirement 

 is not severe we can consider that the first part of the transition region 

 falls within the practical transmitting band, thereby securing a better 

 attenuation characteristic than would be possible if the spacing of 

 critical frequencies in the transmitting band were strictly uniform. 

 A sketch of the phase characteristics through the transition interval 



(n 2340 



O 



UJ 



^ 2160 



1!0 !1.5 120 12.5 13.0 



FREQUENCY IN UNITS OF a 



Fig. 15 — Transfer constant phase shift in the transition interval; a = 1/12. 



for the networks corresponding to Figs. 5 and 6 is shown by Fig. 15. 

 The last evenly spaced critical frequency falls at lla. 



In the extreme case when no phase requirement is imposed, it is 

 reasonable to expect that the best attenuation characteristic will be 



