NETWORK SYXTIIESIS 643 



The coefficient """^^ ^^ in (72) is a sort of average of the enve- 



lopes of llie ripples in a — a. The variability of the envelopes, across 

 the useful interval, depends upon higher order coefficients, in compari- 

 son with the leading term. Calculation of higher order coefficients is 

 relatively complicated. 



19. APPROXIMATION IN THE TCHEBYCHEFF SENSE 



The criteria (45) carry over to general assigned gains, as conditions 

 on H(z) which, if realized, are usually sufficient to establish approxima- 

 tion in the Tchebycheff sense. For this purpose we must use the H{z) of 

 (04), rather than (G7) or (68) (which correspond explicitly to Cu = C2k , 

 k ^ m). In terms of the polynomial and series representations of (55), 

 the H(z) of (64) becomes: 



/C. + g.z' ■ ■ ■ K.z''- 



"^'^ = E^? " ^ ^'^^ 



The followang somewhat special problem is easily solved, in these 

 terms, and has a direct bearing on various quite different synthesis 

 techniques: A network is to be designed which combines the functions 

 of an equalizer or simulator, with those of a filter, or selective network. 

 In the useful interval, an assigned gain variation a is to be approximated 

 in the Tchebycheff sense. At higher frequencies, there is to be a rapidly 

 increasing loss, or "sharp filter cut-off." The number of natural modes, 

 n, is to be more than sufficient to match a to the required accuracy, in 

 the absence of a selectivity re(iuirement, the latitude being used to pro- 

 duce the required sharp cut-off. In particular, n is to be large enough so 

 that an n term match of Tchebycheff coefficients produces errors that 

 are negligible compared with those accepted as a price of the sharp 

 cut-off. 



On the above assumption of an ample n, the infinite series ^ K^z^ 

 in (73) may be truncated after the term of order n, and the errors due to 

 the truncation may he neglected in calculating the design error a — a.] 

 Then (73) becomes 



„, , g. + g.z' ■ ■ ■ K^z" 



t The truncated series is merely the polynomial on the left side of (56) which 

 tvould he obtained if the filter selectivity were ignored, and m were given the 

 maximum value, n. 



