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THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



phase separately. These may be derived from (14), and take the form: 



« = X CkTk j 



11 I ^ ^'2 J > k, even 

 CkZ = logA, 7 -^ 



n(i-4.)J 



E c,z' = log n 



(21) 



> k, odd 



(The absence of factors | in ^ Ct^*", as compared with (20), reflects the 

 factors 2 associated with a and /3 in (14).) 



8. REPRESENTATION OF ASSIGNED GAIN AND PHASE 



In synthesis problems, the network gain or phase, a or 13, is to ap- 

 proximate an assigned (ideal) gain or phase, say a or ^. To make effec- 

 tive use of the 2-plane analysis, in network synthesis, we need to describe 

 a and jS by relations analogous to (20) and (21), which express a and /3 

 in 2-plane terms. These relations, while similar to (20) and (21), must 

 take a more general form (since a or ^ need only be approximately the 

 gain or phase of a finite network). For our present purposes, the ap- 

 propriate relations are those noted below. 



Let a + i^ be any function of p which has the following properties: 

 It must be analytic throughout the useful interval. Further, there are 

 to be no singularities within a (p-plane) distance e of the useful interval, 

 where e is finite (but may be small). Finally, at real frequencies, a and 

 tj8 are to equal respectively the even and odd parts of a + i^. 



Under the conditions stated, a + i^ may always be expanded in 

 terms of our Tchebycheff polynomials Tk . Let ^ CkTk be the expansion. 

 To obtain a parallel to (20), we may form (arbitrarily) a power series 

 XI hCkz''. Then we may defiyie a function R{z) by identifying log R(z) 

 with the power series. All this adds up to the following, comparable to 

 (20): 



« + ?^ = Z CkTk 

 E hCkz' = log R{z) 



(22) 



