634 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



of the detailed calculation of the p, . Otherwise, it may be necessary to 

 carry out several designs, in all detail, in order to obtain one satisfactory 

 design. 



The needed information about accuracy can in fact be obtained from 

 the error function H{z), which we formulated for general gain applica- 

 tions in (30), and for the present application in (38). The analysis 

 which yields (15) may be used to obtain a very similar expression for 

 a — a, in which R{z)R( — z) appears in combination with the rational 

 function of z from (15). It may be expressed in terms of the error func- 

 tion H(z) of (30), as follows: 



a - a = -log I 1 + H(z) I (40) 



When H{z) is zero, a — a is zero. When H(z) is small, a — a depends 

 on phase H{z) as much as on \ H{z) \ . When H{z) is a positive real, 

 a — a is negative. When H(z) is imaginary, a — a is very small. When 

 H{z) is a negative real, a — a is positive. When H(z) is complex, | a — a | 

 is always smaller than with a real H(z) of the same magnitude. The last 

 statement may be expressed as follows: 



- log {1 + 1 H(z) 1} ^ « - a ^ - log {1 - I H(z) \} (41) 



The left hand relation is an equality when phase H(z) is an even num- 

 ber of IT radians; the right hand side, when it is an odd number of x 

 radians. 



In the useful interval, where z = e'*, the H(z) corresponding to (36) 

 is as follows: 



H(z) = -(n + 2) (^y + (n 4- 1) (jj 



\H{z)\ = _2n+2 

 ^0 





in + 2)zt 



(42) 



phase H(z) = tt + (2w + 2)0 -f- phase ^ 1 — 7 — ZlToV^ ^^ * 



As CO varies from to Wc , varies by - radians. The corresponding 



phase of H(z) varies by (n -f l)7r radians, which means that H(z) is 

 successively positive real, imaginary, negative real, imaginary, through 

 n + 1 half cycles. This accounts for the oscillatory nature of the a — a 

 curve, illustrated in Fig. 7. 



The amplitudes of the oscillations are fixed by \ H(z) \ , which varies 

 relatively slowly. Specifically, the two logarithms in (41) determine 



