660 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



directly to all these effects. They may still be useful, however, when 

 applied in the following way: 



Separate arrays of singularities are determined, which match the 

 separate effects to required accuracy, using any convenient methods. 

 Minimum network complexity is not required at this point. Combining 

 all the singularities in a single array gives an initial design which is 

 sufficiently accurate, but may use many more singularities than are 

 actually necessary. Tchebycheff polynomial metliods are now used to 

 obtain a simpler set of singularities, which approximates the initial set 

 to sufficient accuracy. This has been designated "boiling down" the 

 original set. 



In a problem of this sort the assigned characteristic has the network 

 form, as well as the network characteristic which is to approximate it. 

 (The example discussed in Sections 10 to 14 is also a problem of this 

 sort.) As a result, equations (20) and (21) apply to a and ^, as well as 

 to a and /5. This makes it possible to replace ^ KkZ *" and ^ Kkz'', of 

 (55), (56), (96) etc., by a finite rational fraction N/D. If both a and ^ 

 are to be approximated, the following is derived from (20). 



n (i - P) N 



E ife' = log K. ) -H- = -log ^ (114) 



The singularities Za , z^ correspond, of course, to the network singu- 

 larities which are to be boiled down. If only a, or only ^, is to be ap- 

 proximated, suitable modifications are readily derived from (21). 

 The boiling down is accomplished by requiring 



N ^ N ^ ^ 



where N/D is of lower total degree than N/D. If m = n" + n', and 

 n" — n' = or 1, the continued fraction method can again be used. 

 This requires expansion of N/D in continued fraction form, instead of 



An example of a boiled down set of singularities is illustrated in Fig. 13. 



28. GENERALIZATION OF THE USEFUL INTERVAL 



All the previous analysis applies to a useful frequency interval —o)c< 

 CO < 4-Wc . Its important characteristics are as follows: It is a single 

 continuous interval, with co = at its center. Useful intervals with other 



