622 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



7. THE POWER SERIES IN Z 



Our applications to network synthesis depend upon a correspondence 

 which may be shown to exist between certain functions of z and certain 

 power series in z. The functions of z may be formulated in terms of 

 network singularities. The power series in z may be derived from the 

 Tchebycheff polynomial series in p representing the corresponding gain 

 and phase. 



The Tchebycheff polynomial expansion of a gain and phase function 

 may be written: 



« + //3 = Z C^-n (16) 



If a + ^/3 corresponds to a finite network, it may be represented by the 

 function of z in (13). At the same time, Tk may be represented by the 

 function of z in (6). With these changes, (16) becomes: 



log< 



f n(-.4)l 



n 1 



Same Rational 



Function in — 1/zj 



(17) 



^' + 'V 



The logarithm of the product of the two rational functions, in z and 

 — 1/z respectively, may be written as the sum of two logarithms. The 

 series in sums of z'' and { — l/zY may be written as the sum of two series. 

 Then 



log< 



f n(i-,^)] 



n(i-z' 



}+ log 



Same Rational 

 Function in — 1/z 



(18) 



+ Z^'^~''' 



The above expression equates the sum of two similar functions, in z 

 and — 1/z respectively, to the sum of two power series, also respectively 

 in z and —1/z. The theorem on which the synthesis methods are based 

 asserts that the functions and power series in z and — 1/z may be equated 

 separately, throughout the useful interval. That is: 



