626 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



terms of order m. If both series have converged to small remainders 

 when k = m, this criterion will surely make a -\- i^ a good approxima- 

 tion to « + ^^^-t 111 terms of the coefficients, the criterion requires: 



C, = C, , k S m (25) 



If (25) is applied to the second equation of (24), the power series is 

 zero through terms of order m. In other words, the logarithm, equated 

 to the series, will approximate zero in the power series, or "maximally 

 flat" manner, to order m. The logarithm is zero when the expression in 

 brackets is unity. Further, the logarithm \vi\\ approximate zero in the 

 maximally flat manner when, and only when the bracket approximates 

 unity in the maximally flat manner. Thus a condition which is equivalent 

 to (25) is the following: 



1 n(i-7) 



i \ W .^(^) ^ 1 ^ ^m,.l _^ ^.+2 . . . (26) 



n(.-|,) 



This may be represented symbolically by 



1 nfi-T^) _ . 



^ V ^- / .ii(^) = 1 (27) 



^-nd- 



where = is used to indicate equality through power series terms of 

 order w. 



When (27) is applied to network sjaithesis, the singularities 2<, , and 

 scale factor Kz are the unknowns, while R{z) is known. If m is equal to 

 the total number of Zo , (27) will determine the network function com- 

 pletely. When jn is smaller, (27) will furnish m + 1 relations between 

 the network parameters (including the zero order condition), which 

 may be combined with specifications of other sorts. Since (27) is equiv- 

 alent to (25), this procedure amounts to the determination of network 

 parameters which will yield assigned values of the coefficients, Ck = Ck , 

 k ^ m, in the Tchebycheff polynomial expansion of a + i(3. 



Equation (27) applies when both gain and phase are to be approx- 

 imated. For approximation to gain only, or to phase only, similar rela- 

 tions may be derived from (21) and (23). Only even ordered Tchebycheff 



t When both residues are relatively large, the approximation may still be 

 good, for the remainders maj' be quite similar, and the error will be their differ- 

 ence. In practical applications, this is a not uncommon situation. 



