228 BELL SYSTEM TECHNICAL JOURNAL 



To see this, it is merely necessary to observe that the set of rational 

 factors appearing in the low-pass image impedance expression de- 

 scribed in the preceding paragraph must approximate the reciprocal 

 of the cut-off factor at lower frequencies. We can therefore use such 

 a set of factors to replace the upper cut-off factor of a band-pass filter, 

 obtaining thereby a high-pass structure which approximates the ideal 

 characteristics over a portion of the transmitting band. 



If the cut-off factor of the low-pass filter transfer constant be 

 similarly replaced by rational factors, there results an all-pass "delay 

 network" having a constant impedance and a phase characteristic 

 linear below the original cut-off frequency. This network is of par- 

 ticular interest for its relation to the classic problem of the simulation 

 of a smooth line. As it stands, the network evidently simulates an 

 ideal dissipationless line. To include the effects of dissipation we 

 need merely add resistance and leakance to the coils and condensers in 

 the proportions in which they occur in the actual line. 



Part II — Design of Practical Filters 



Thus far we have been interested primarily in demonstrating that 

 an indefinitely close approximation to the ideal characteristics could be 

 obtained when all restrictions with respect to economy of elements 

 were removed. In practical designs, on the other hand, we wish to 

 approximate the ideal characteristics only within moderate limits, and 

 our interest centers upon the choice of the most economical network 

 which will prove satisfactory. We must now reappraise the theory 

 from this point of view. 



One question which must be examined is that of determining values 

 for m and a which will result in the most economical network meeting 

 a prescribed standard of performance. A second is concerned with 

 the possibility of changing the nature of the approximation with 

 respect either to the frequency, or to the relative emphasis laid upon 

 the phase and attenuation characteristics. In many practical designs 

 such changes can be obtained by slight modifications of the theoretical 

 design parameters and lead to corresponding economies in the use of 

 elements. In investigating both questions we must remember that 

 since a is no longer necessarily small, as it was in the theoretical 

 analysis, the frequency interval actually occupied by the transition 

 factors may be ap^preciable. Consequently it becomes important to 

 investigate the behavior of the network in this part of the frequency 

 range with more care than was hitherto necessary. 



The variety of possible design requirements precludes the possibility 

 of a thorough analytic treatment of these questions. The choice of 



