IDEAL FILTERS 217 



The discussion which follows has a two-fold objective. The first is 

 purely theoretical : to demonstrate that no matter how close the limits 

 of deviation from the ideal may be set, there is a finite physical net- 

 work all of whose characteristics meet these limits, except within a 

 certain "transition interval" about each cut-off, which transition 

 interval may also be taken as narrow as we please. This is by no 

 means trivial; for it is known that no network, finite or infinite, can 

 meet the ideal characteristics exactly} 



The second object is to guide the selection, from among the many 

 networks which would meet the requirements of a given practical 

 problem, of that one which meets them most economically. This 

 part of the paper contains a number of examples, among them some 

 which illustrate the use of slight empirical variations as a means of 

 obtaining the highest measure of economy when wide deviations from 

 the ideal are more tolerable in one respect than in others. The final 

 example, which is segregated as Part III, deals with a situation met 

 in picture transmission circuits, where the selectivity required is 

 frequently small, but the effects of phase distortion may be very 

 serious. Here a modification of the design technique leads to a filter 

 which has comparatively modest selectivity but which exhibits a 

 linear phase characteristic not only in the transmitting band but also 

 in the range of rising attenuation. 



Part I — Theoretical Analysis 



Since linear phase shift is not available from ladder networks, the 

 analysis will be based upon the more flexible lattice configuration. 

 Although the lattice lends itself particularly well to the theoretical 

 design problem, it is not so satisfactory for purposes of physical con- 

 struction. After the paper design has been made, therefore, it will 

 usually be desirable to convert it to a more suitable practical con- 

 figuration. This can be done by methods described elsewhere.^ 



We may greatly simplify the theoretical discussion by ignoring the 

 effects of parasitic dissipation — a simplification warranted by Mayer's 

 Theorem,'* which states that the attenuation resulting from dissipation 



2 This proposition is due to Dr. T. C. Fry, who showed that in a transducer possess- 

 ing the steady-state characteristics of an ideal filter, a signal would arrive at the 

 receiving terminals before it began to be impressed on the sending terminals. As 

 this is absurd, we must conclude that no such system exists. 



»H. W. Bode, "A General Theory of Electric Wave Filters," M.I.T. Journal of 

 Mathematics and Physics, November, 1934. A summary of this article appears in this 

 issue of the Bell System Technical Journal. 



* H. F. Mayer, "Uber die Dampfung von Siebketten im Durchlassigkeitsbereich," 

 E. N. T., October, 1925, p. 335. His results were later somewhat extended by Feige 

 and Holtzapfel, " Dampfung und Winkelmass von Vierpolen mit geringen Verlusten," 

 T. F. T., July, 1932, p. 179. Even these latter results are capable of considerable 

 generalization, so as to include other characteristics of the network besides the trans- 

 fer constant. 



