614 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



interpolation problem may then be regarded as solved when a suitable 

 set of network singularities has been obtained; for quite different tech- 

 niques are used to design actual networks with these singularities. 



The interpolation problem may be attacked in a number of different 

 ways; and a variety of different techniciues are, in fact, needed to cover 

 the wide diversity of practical applications. The present topic is a fairly 

 general way of attacking the problem, based upon manipulations of two 

 series of Tchebycheff poljniomials. The two series represent expansions 

 of two functions of frequency — one, the ideal assigned gain or phase, 

 the other, the network approximation to the ideal. The interpolation 

 problem may be solved in this way because it is feasible, as ^^'ill be 

 shown, to determine network singularities from arbitrarily^ assigned 

 values of coefficients in the corresponding Tchebycheff polynomial series. 



The techniques to be described were derived originally from studies 

 of the so-called potential analogy; but they can now be developed most 

 easily without reference thereto. f In a sense they may be regarded as 

 extensions of familiar filter theory, using Tchebycheff polynomials, to 

 more general gain and phase functions. The extensions, however, depend 

 upon a number of new principles. Extensions of the filter theory applied 

 to more general problems have been noted in published papers; but 

 those noted have not used the specific general approach employed here.f 

 The wide applicability of this general approach will be illustrated by 

 specific examples. 



2. NETWORK AND TRANSMISSION FUNCTION 



It will be sufficient for our present purposes to limit the discussion to 

 the general 4-pole shown in Fig. 1. The 4-pole may be active or passive, 

 but it must be a stable finite network of linear lumped elements. E and 

 V are steady state ac voltages, E the driving voltage and T" the response. 

 The gain a and phase |S are here defined as the real and imaginary parts 

 of log V/E. 



For a finite network of lumped elements, a + ijS always has the fol- 

 lowing form: 



a + iff = log K ^^-^-'^■■- (1) 



t Tchebycheff polynomials are related to potential analogue charges on el- 

 lipses, as described in the author's paper "The Potential Analogue Method of 

 Network Synthesis''^ Section 15. 



X For the most part, they have used the potential analogy, in such a way that 

 Tchebycheff polynomials do not appear at all in general applications. For ex- 

 amples, see methods of Matthaei^, Bashkow', and Kuh^. 



