NETWORK SYNTHESIS 639 



If we retain simulation with a network which has n natural modes, and 

 no frequencies of infinite loss, we must retain the formulation: 



a = 2-> C2kT2k 



z c../^ = -log Ki n (i - 1) , - = 1, ^^^^ 



n 



The corresponding formulation of the error is (in place of (33)): 



OC — a = 22 (^2fc — C2k)T2k 



Z (C2. - Cu)z"' = -log 



n(i-|)-^(^)^(-^)] 



(52) 



For Cofc = Cik , J<^ ^ m, the following special case of (28) is now required : 



1 - -2YR{z)R{-z) = 1 (53) 



Now the reciprocal of R(z)R( — z) has a power series expansion, in the 

 region of interest. (Recall Section 8.) 



It follows that (53) may be multiplied by this quantity, without 

 damaging the equality of power series coefficients. In other words (53) 

 is equivalent to: 



Let Kk be the coefficient of z'^ in the polynomial expansion of the left 

 hand side; and let Kk be the coefficient of z''' in the infinite series ex- 

 pansion of the right hand side. Then, 



Kl n (1 - I) = Ko + K,z' + • . . Kr.z''' 



(55) 



R(z)R{-z) 

 Substitution in (54) gives 



me 



K, + K,^ + • . . KJ- = E Kkz"' (56) 



In other w^ords, 



Kk = Kk , k ^ m (57) 



These relations are directly applicable to network synthesis, provided 



