IDEAL FILTERS 111 



now quickly show how the desired impedance is to be obtained. It is 

 merely necessary to observe that for any filter there exists a comple- 

 mentary structure with the same arrangement of critical frequencies, 

 but having the transmitting and attenuating bands interchanged. 

 The complementary structure is found by replacing the Zy branch of 

 the original lattice by the inverse impedance Z/ = R-jZy. When 

 these are substituted in (1) and (2), the new transfer constant, 6', is 

 found to be 



\~7~' ~ 15 ^ZxZy = j^Z r, 

 and the new image impedance, Z i', 



Zi' = VzX^=i? Jf^= i?tanh|. 



\ Zj y L 



Thus, for any filter, the problem of adjusting the image impedance to 

 the constant R in its transmitting band is the same as the problem of 

 adjusting tanh 0/2 to 1 in the attenuating band of the complementary 

 filter. The latter problem, however, is merely a restatement of our 

 original requirement of high loss in attenuating bands and has already 

 been studied for various types of filters. 



It follows from this relation that the transfer constant expressions 

 which are appropriate for low-pass and band-pass filters furnish 

 suitable solutions for the impedance problem in high-pass and band- 

 elimination structures. We might also use our high-pass transfer 

 constant expression as a low-pass impedance characteristic except for 

 the difficulty previously mentioned that it requires an infinite number 

 of elements. This difficulty can be avoided, however, by observing 

 that by interchanging coils and condensers we can convert any low- 

 pass filter into a high-pass structure having the same characteristics 

 on a reciprocal frequency scale. We can thus use the finite low-pass 

 solutions to obtain the required finite high-pass filter having high 

 attenuation. For example, if we begin with a low-pass filter having 

 three evenly spaced critical frequencies and a half spaced cut-off the 

 resulting critical frequencies (including the cut-off) are in the ratio 

 1:7/6: 7/4 : 7/2. 



The device of inverting the frequency scale is, of course, not avail- 

 able to produce a finite high-pass filter having linear phase shift 

 throughout its transmission band since the linear phase property is 

 thereby destroyed. It can be used, however, to produce a finite 

 filter having linear phase shift for a limited region above its cut-off. 



