654 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



equivalents. Then, in place of (81) or (94), we have 



Z C./ = -log g 



> k odd (98) 



Using n to represent n" + n', the total number of network singularities, 

 we may write N and D as follows, in place of (82) or (95) : 



N = 1 -\- Kiz + Koz^ + K^z • • • + K„z" 



(99) 

 D = I - Kyz + K22' - Kzz' •■■ i-TKnz'' 



Notice that A'' and D are related by 



D{z) = N{-z), (100) 



which is required by the form of the /3 equation in (21). 



To arrive at a design procedure most easily (but not the simplest 

 design procedure), one may express the assigned gain jS in the follo^\dng 

 w^ay (comparable to (58) and (96)) : 



i)8 = E CuTk , k odd 



J^Ckz" = -log^iCA./ 



Coefficients Kk may again be calculated by a modification of (61). This 

 time C-2k is replaced by Ck , wherever it appears in (61), and then all 

 even ordered Ck are made zero (since only odd terms appears in ^Z ^i^-^'' 

 of (101)). Note that even ordered Kk are not usually zero, even though 

 even ordered Ck are. 



The degrees of N and D, in (99), are such that we can make Ck = Ck 

 through terms of order k = 2n. This requires merely: 



AT ^" 



^ = E ^^' (102) 



As stated, the condition applies to Ck of both even and odd orders. 

 Since even ordered Ck are zero, it means that at least n even ordered 

 Ck will be zero, in addition to the match between n odd orders. (102) 

 is sufficient to determine an N and a D without reference to (100). If 

 the (equal) degrees w of (99) are assumed, however, the N and D deter- 

 mined by (102) will be found to obey (100) automatically (pro\-ided 

 E Kkz'' corresponds to an odd series ^ Ckz", as here assumed).! 

 A simpler method for computing the same N and D takes advantage 



t This was discovered bj' Mrs. M. D. Stoughton. 



