CONSTANT PHASE DIFFERENCE 



97 



imate unity in prescribed frequency ranges.^ These functions, however, are 

 not odd rational functions of frequency but are irrational functions appro- 

 priate to represent filter image impedances or the hyperbolic tangents or 

 cotangents of filter transfer constants. It turns out, however, that they 

 can be transformed into odd rational functions of the desired type by a 

 simple transformation of the variable. 



Each of Cauer's functions is said to approximate a constant in the Tcheby- 

 cheff sense, which means that in the prescribed range of good approximation 

 the maximum departure from the approximated constant is as small as is 

 permitted by the specifications on the frequency range and the degree of 

 the function. Each function also has the property of exhibiting series of 

 equal maxima and equal minima in the range of good approximation, such 

 as those indicated in the illustrative /3 curve'* of Fig. 4. 



LU 

 O 



lllUJ 



u. O 



IL LU 



60 80 100 



200 400 600 1000 2000 



FREQUENCY IN CYCLES PER SECOND 



10,000 



Fig. 4 — Example of a phase difference characteristic. 



Of the various forms in which Cauer's Tchebycheff functions F can be 

 expressed, the following form is the one appropriate for showing how odd 

 rational functions of frequency can be obtained: 



When 11 is odd 



'2s - 



(1) 



F = U\/\ - X2 it ^ 





When n is even 



F = 



U 



n['-™'(^'^''*)^ 



n 



■[l-.„'gA-.*).V'] 



' "Ein Interpolationsprohlem niit Funktionen mit Positivem Realteil," Mathematische 

 Zeitschrift, 38, 1-44 (1933). 



* The data for the illustrative curve were obtained from a trial design carried out by 

 P. W. Rounds. 



