232 BELL SYSTEM TECHNICAL JOURNAL 



m + (l/a) it would therefore appear that the most economical 

 structure meeting given requirements will be obtained by using a large 

 m in combination with a large a. This procedure is, however, re- 

 stricted by two considerations. The first is chiefly theoretical. Since 

 the series we have been using is merely asymptotic, the successive 

 terms obtained by choosing progressively higher m's eventually grow 

 larger. For ordinary values of a, however, the value of m at which 

 the series begins to diverge lies beyond the range of practical interest. 

 A more important limitation is the fact that as we increase the number 

 of transition factors, the width of the transition interval, as measured 

 in terms of a, also increases. Thus, the spread between the last uni- 

 formly spaced critical frequency and the cut-off, which is a/2 for m = 1 

 and about 2a for ni = 3, has risen to more than 3.5a for m = 5. In 

 each case a certain additional allowance is of course required for the 

 region of rising attenuation beyond the cut-off. When the transition 

 interval is fixed on an absolute frequency scale, therefore, the per- 

 missible values of m will depend upon the choice of a. Unless the 

 transition interval is unusually broad only low values of w will be of 

 practical interest. 



Illustrative Characteristics 



The curves shown in Figs. 3 and 4 are not of use in the neighborhood 

 of the transition interval. To supplement them, therefore, exact 

 computations on a number of typical structures have been made. 

 One set was obtained by choosing a — 1/12 and computing the 

 characteristics corresponding to various w's. The resulting phase 

 characteristics are shown by Fig. 5. Since the departures from 

 linearity are too small to be noticeable when the characteristics as a 

 whole are drawn, the figure shows only the departures themselves in 

 terms of an envelope similar to that used for Fig. 4. The curves are 

 drawn approximately as far as the last evenly spaced critical frequency 

 which marks the practical limit of the range within which a high degree 

 of phase linearity is to be expected. Since the curves vary rapidly 

 in this vicinity, however, the fact that they are merely envelopes is 

 important in determining the exact performance of the structure. 

 Curves of the phase characteristics in the transition interval will be 

 given later. 



The attenuation characteristics are shown by Fig. 6. As ni is 

 increased, the cut-off moves to successively higher frequencies because 

 of the progressively broader intervals consumed by the transition 

 factors. Once past the cut-off, however, the curves for large values of 

 m rise more rapidly and quickly cross the others. 



