352 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



where Prnip) is a polynomial of degree n whose coefficients depend only on 

 T(w), that is on the shape of the contour. 



By summing the above relations for all values of n we have the general 

 expansion theorem, 



Fiip) = 11 Cn Frnip), 



^ (98) 



q'W = ZjCn sin M. 



Thus if the assigned gain and phase functions can be expanded in terms 

 of the polynomials Pvn{p), appropriate to the given contour, then the 

 Fourier expansion of the charge on Ci can be written down immediately. 



17. High-pass and Band-pass Filters 



So far we have assumed that the contour in the />-plane is a simple closed 

 curve. This is adequate as long as the positive frequencies of interest extend 



REAL p 



Fig. 19 — Appropriate contour for a high-pass filter. 



from zero to a finite upper bound, ojo , as in low-pass filters. For high-pass 

 filters, in which the positive frequencies extend from a lower bound, coo , 

 to infinity, an appropriate shape of contour is shown in Fig. 19. However, 

 high-pass problems can always be reduced to the low-pass type by simply 

 using \/p as the variable instead of p. 



