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BELL SYSTEM TECHNICAL JOURNAL 



shown in Table III. An illustration of the results obtainable with the 

 present method using a large number of critical frequencies, is furnished 

 by Fig. 14, since the curve can evidently be interpreted as a representa- 

 tion of Z ijR for a certain high-pass filter. 



TABLE III 



Impedance Controlling Frequencies Corresponding to Characteristics 



OF Fig. 13 



1.000 



0.999 



0.998 



0.996 



0995 



1.00 125 1.50 1.75 2.00 2 25 2 50 2.75 3 00 3.25 3 50 



FREQUENCY 



Fig. 14 — Tanh djl for low pass filter with m = 2, a = 1/12. 



Weighting the Approximation 



In the limiting case in which a is very small while m is fixed, the 

 methods we have followed give the best obtainable results with respect 

 to both attenuation and phase, for the errors in both characteristics 

 depend upon higher order terms and become negligible as a approaches 

 zero. In a practical design, for which a is finite, on the other hand, 

 it will frequently be desirable to make slight adjustments in such 

 parameters as the transition factors or the constant multiplier of the 

 tanh djl expression in order to take some account of the higher order 

 terms. The effect may be either to improve both the phase and 

 attenuation characteristics or, more usually, to improve one at the 

 expense of the other. 



The general nature of the problem is illustrated by Fig. 14, which 



