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BELL SYSTEM TECHNICAL JOURNAL 



attenuating ranges and the allowable width of the transition interval. 

 It can best be determined by inspection of results given later. 



The result of solving the equations (21) for the ratios Cjja for values 

 of m between 1 and 5 is given in the following table. 



TABLE I 

 Spacing of Transition Factors 



The first of these solutions corresponds to a single frequency, the 

 cut-off, in the transition interval. It follows the uniformly spaced 

 critical frequencies of the practical transmission band at one-half the 

 uniform spacing, a. The other solutions represent networks having, 

 in addition to the cut-off, rational factors which vanish in the transi- 

 tion interval. 



When these values of the c's are used in equation (8), with due regard 

 for (6) and (18), the form of Q is completely determined. For example, 

 the frequency pattern corresponding to the case m = 3, is illustrated 

 by Fig. 2. 



d da 





-^- 



(A-2)a (A-Ooc 



TRANSMITTING BAND 

 AB<0.0025 RAD. (A >6) 



1 + OC 



. TRANSITION 

 INTERVAL 



ATTENUATING BAND 

 ATTENUATION >50DB(A5 5) 



Fig. 2 — Location of transition factors with in = 3. 



Nature of the Approximation 

 How closely we approach ideal characteristics by this method 

 depends on how nearly log Q is represented by the first m terms of (17) 

 and (20). In both cases the series of omitted terms can be written 

 in the form 





20C 



m+2 



1 



+ 



1 



(I -/)'"+^ ' (I +/)'"+^ 



+ 



(22) 



