640 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



the coefficients Kk can be calculated. Formulae for their calculation 

 may be derived in the following way: 



If (55) is substituted in (50), the result is-.f 



a = T. C-2kT2k (58) 



Z Cuz'' = - log Z Kj' 

 When 2 = 0, the second equation reduces to: 



Ko = e-^0 (59) 



If the functions of z are differentiated, a simple rearrangement gives: 



E kK,i'-' = [ - Z kC.J'-'] [Z Kj'] (60) 



The right hand side may be expanded as a single power series, and then 

 like powers on the two sides may be equated separately. The result is: 



ivi = — C 2ivo 



2Zo = -C-2K1 - 2CiKo (61) 



SKz = — C2K0 — 2CiKi — SC^Ko 



Synthesis calculations may now be carried out in the following stages. 

 The assigned gain a. is expanded as a Tchebycheff polynomial series, to 

 determine coefficients ^2^ , say through order k = n. The equations (61) 

 are then used to calculate coefficients Kk , also through order n. Each 

 successive coefficient is computed in terms of those previously deter- 

 mined. Note that the Kk , k ^ n, are fixed by the same number of 

 Cik — that is, orders k ^ n. 



Equation (57) is now applied to identify Kk with Kk , k ^ w. If all 

 the network degrees of freedom are to be used to get Cok = Cok , index 

 m = n, and (57) determines the polynomial in (55) completely. Other- 

 wise, m < w, and coefficients K„,+i to Kn are to be adjusted in accord- 

 ance with specifications of other kinds. When all the Kk have been 

 determined, the singularities z„^ are found by root extraction methods, 

 applied to the right hand side of the first equation of (55). 



The previous example might have been carried out in these terms, 

 but happened to be simpler in the terms used. If (32) is regarded as a 

 special case of (50), and if (32) is simplified (for purposes illustration) 

 by using K, = 1, the corresponding R(z)R( — z) becomes simply 



f 1 - ^y Then Z Kkz'' is 



t We may think of these equations as defining an infinite network, with na- 

 tural modes only, which would match the assigned gain a exactly. 



