IDEAL FILTERS 



229 



the most economical network meeting given requirements consequently 

 cannot always be made without trial. The procedure may, however, 

 be considerably facilitated by a study of the curves and illustrative 

 material given in the sections which follow. The first two sections 

 show the quantitative relations to be expected when the theoretical 

 design parameters are adhered to strictly. The remaining sections 

 indicate modifications obtainable by making slight changes in the 

 theoretical parameters. 



Approximate Computation of Network Characteristics 

 When the frequency in which we are interested is not too close to 

 the transition interval, an approximate determination of the phase and 

 attenuation characteristic is most easily made from (23) and (24). 

 The ^'s appearing in these expressions are shown in the accompanying 

 table.^^ In addition to Am+i the table also supplies values of A^+z 



TABLE II 



Coefficients in Series Expansions for Approximation Errors in Phase and 

 Attenuation Characteristics 



and Am+s, for use if additional terms in the general expression (22) 

 are desired. 



A study of equations (23) and (24) shows that, aside from the 

 constant factor Am+i, each expression can be resolved into two factors 

 by means of which the contributions of the various design parameters 

 can be somewhat segregated. The first factor, a'^+^, is chiefly im- 

 portant in determining the effect of various choices of a and m on 



the approximation error, while the factor -; r^— -r + —r—, — ?^-^n 



L (1 — /) (1 +/)'""^ J 



expresses the variation of the network characteristics with frequency. 

 In order to facilitate design work the quantity 



20 logio 



A, 



i+i 



1 



1 



(1 -jy 



(1 +/)-+! J 



^' In preparing the table, coefficients of corresponding terms in the series expan- 

 sions for (17) and (20) have been combined, so that the coefficients as given represent 

 the accumulated errors of both approximations. 



