96 BELL SYSTEM TECHNICAL JOURNAL 



formulae for element values of corresponding networks, using tandem sec- 

 tions of the simplest all-pass type (Fig. 3). 



Form of tiie tan { - ) Function I 



\2/ I. 



If fix and /So represent the phase shifts through the two constant resistance I 



networks of Fig. 1, then tan ( -^ 1 and tan ( ^ j must both be realizable I 



as the reactances of physical reactance networks. In other words, these 

 quantities must be odd rational functions of w with real coefficients and 

 must also meet various other special restrictions. If /3 is used to represent 



the phase difference 182 — /3i , the function tan ( - 1 must also be an odd 



rational function of oj with real coefficients. Because of the minus sign 



Fig. 3 — Simplest all-pass section. 



associated with ^i in the definition of /3, however, tan ( - J does not have to 

 meet the additional restrictions which must be imposed upon tan I J ) and 



tan ( ^ )• In a later part of the paper a method will be described by which 



a pair of physical phase shifting networks can be designed to produce any I 



tan ( - j function which is an odd rational function of co with real coefficients. j| 



In any range where the phase difference /3 approximates a constant, the | 



function tan [ - I will also approximate a constant. Hence, the present 1 



problem is really that of ai)proximating a constant over a given frequency 

 range with an odd rational function of w with real coefficients. In this prob- i' 

 lem, the degree of the function must be assumed to be prescribed as well 

 as the frequency range in which a good approximation is to be obtained, 

 for the degree of the function determines the complexity of the correspond- 

 ing network. 



W. Cauer shows how functions of certain types can be designed to approx- 



