646 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



It must be remembered, however, that this formulation is permissible 

 only if the approximations, inherent in (75), are justified when z = z^ 

 (as Avell as in the useful interval). Taking the logarithm of each side 

 gives: 



log {Xo + • • • Kj-:"] 



, , f- K:\ (78) 



= log {-G) + 2n log 2. + log ^Ko + • • • - 



Equation (58) may now be applied, to replace the logarithms by 

 power series, provided the truncation of the series is again justified, and 

 provided the convergence of ^ CikZ^ is also proper, at both z = z, and 

 z = \]za . This gives 



-E Cuzl' = -E <^2.Af + 2n log z. + log i-G) (79) 



(Summations ^ are all with respect to A;; and there is one equation for 

 each <T.) This is a suitable rule, for obtaining an ] « — a | with equal 

 maxima, whenever the approximations are in fact unimportant. It is 

 not at all clear, however, just when the approximations become sig- 

 nificant. 



Kuh uses the potential analogy approach for the same sort of syn- 

 thesis problem. He spaces the natural modes along a p-plane contour 

 defined in fairly complicated potential analogy terms. It can be shown, 

 however, that mapping his potentials from p plane to z plane leads to 

 (79). 



When the network is to approximate a in the useful interval, but is 

 not required to supply selectivity at other frequencies, the approxima- 

 tion (74) is usually untenable. It is generally necessary to retain the 

 exact formulation (73). 



When selectivity is not required, the phase excursion of H{z), in the 

 useful interval, can usually be increased to that of 2"""'"" (as in (67), 

 corresponding to Cik = Cik , k ^ n). As a step toward meeting the first 

 condition of (45), one may then write (73) as follows: 



J^n+l+k 2k . "T^ ^ ^ 



•sr-y -tvn+l+fc 2fc ■ "S^ n 



2^ ^ ^ -r Z^ ^k ^2k 

 H{z) = -Kn+xz""-' 



An+1 1 Z 



Z Kk^"" (80) 



r\ Kn+l-k — Kn+l-k , . , , 



Qk = ^ , /c = 1, • • • , n + 1 



The coefficients Kk and -^ are fixed by a. The only arbitrary 



■tV„+i 



