350 THF BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



We write this result alternatively in the form 



^t(^) = JlC2mT2m{0)) + iJ2C2m+l T2m+l{oo) (93) 



where Tom is the Tchebycheff polynomial of even order, 



T2m(co) = COS \2m sin~\co/coo)] (94) 



and T2m+i may be interpreted as a modified Tchebycheff polynomial of odd 

 order, particularly adapted to network synthesis problems, 



T2m+i{oi) = sin [(2m + 1) sin~\co/coo)]. (95) 



It is easy to verify that the T's are in fact polynomials in co/coo . For the 

 first few values of n we find 



Wo \coo/ 



Wo \coo/ \wc/ \coo/ 



In deahng with prescribed gain and phase functions for elliptic contours, 

 the simplest procedure is to expand the gain, not in an even power series, 

 but in a series of even Tchebycheff polynomials, while the phase is expanded 

 in a series of odd Tchebycheff polynomials. Such expansions are always 

 possible for analytic functions, and it should be pointed out that their 

 region of convergence is greater than that for a simple power series. An addi- 

 tional advantage of using the polynomials instead of the power series is that 

 the r's are orthogonal in the frequency range | co | < coo , while the various 

 terms of the power series are not. This increases the rapidity of convergence 

 and leads to a more efficient solution of the design problem. 



A simple illustration of the effect of contour shape on the accuracy of 

 the lumped charge approximation to the transmission function is shown in 

 Fig. 18. This refers to the constant gain filter we discussed, for a circular 

 contour, in Section 10. The granularity error for the circle (curve 1) is very 

 small at low frequencies, while for the two elHpses (curves 2 and 3) it is small, 

 but oscillatory, and the oscillations become larger as the ellipse becomes nar- 

 rower. On the other hand, at frequencies near the upper limit coo of the fre- 

 quency band, the granularity error is much smaller for the ellipses than 

 for the circle; in other words, the cut-off frequency is more sharply defined. 



16. The Expansion Theorem for General Contours 



The term by term correspondence between the Fourier expansion of the 

 charge on Ci and the expansion of the gain and phase functions as series 

 of polynomials holds also for general contour shapes. In the general case 



