IDEAL FILTERS 245 



Part III — Filters with Linear Phase Shift 

 Through the Cut-off 



It was the conclusion of the theoretical discussion that any desired 

 approximation to ideal filter characteristics may be obtained from a 

 finite network, so long as a finite transition interval separates trans- 

 mitting from attenuating bands. The transition interval can be 

 taken small at pleasure, but very small transition intervals are associ- 

 ated with networks of many natural frequencies and numerous 

 elements. We have already seen how considerable economies in 

 meeting a given attenuation requirement could be obtained if the phase 

 requirement were subordinated or removed entirely. We now con- 

 sider the contrary case, in which major emphasis is placed upon the 

 phase characteristic of the filter. Filters of this type are of practical 

 interest in picture transmission systems since instruments used in the 

 reproduction of images seem to be much more sensitive to the effects 

 of phase distortion than the ear. The selectivity required from filters 

 used in such systems is comparatively modest, but phase linearity is 

 required not only in the practical transmission band but also through 

 the transition interval into the region of rising attenuation. 



In one important particular the present problem differs from those 

 previously considered. In the present analysis we can no longer 

 regard the adjustment of Z/ and the adjustment of d as independent 

 problems. On the contrary, in the attenuating region the contribution 

 of d to the phase shift is constant and we must therefore rely upon 

 reflection effects to maintain the desired linear characteristic. More- 

 over, near the cut-off d must be very carefully adjusted with respect 

 to Z/ in order that the contributions to the phase shift from reflection 

 and interaction effects may preserve the linearity through the transition 

 band also. The added restrictions imposed by the extension of the 

 phase requirement require a revision of the frequency spacings already 

 found, and set limits upon the approximation to ideal characteristics 

 obtainable from reactive networks of reasonable complexity. 



Use of Reflection Effects to Produce Linear Phase 

 In the practical transmission band, Z/ can be adjusted to approxi- 

 mate R sufficiently closely to make reflection and interaction effects 

 negligible. Therefore, in this range the total insertion phase is the 

 same as the transfer constant phase, and, as before, is to be obtained 

 from a chain of uniformly spaced critical frequencies in tanh djl. In 

 the practical attenuating band, on the other hand, we find that the 

 imaginary part of d is either or it, while interaction effects can be 



