662 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



formation, replacing (5), is: 



2coc 



p = 



z 



With this transformation, the whole of the previous 0-plane analysis 

 may be applied at once to high-pass useful intervals, except where 

 linear phase is involved. 



The situation is more complicated in regard to "band-pass" intervals. 

 If the useful interval includes the frequencies between coci and coc2 , the 

 complete useful interval (p-plane mapping of | 2 | = 1) must include 

 also the "image" frequencies, between — Wd and — coc2 . Otherwise, 

 conjugate complex 2;-plane singularities Za will not lead to conjugate 

 network singularities p^ . When there are two disjoint parts of the use- 

 ful interval, the appropriate relation between -p and z is relatively com- 

 plicated. Up to the present, no corresponding technique has been dis- 

 covered for approximating assigned phases over band-pass intervals, 

 in Tchebycheff polynomial terms. Gain approximations can be handled, 

 however, and for a quite simple reason. Gain functions are even func- 

 tions, and behave in the p plane much as gain-and-phase functions 

 behave in the p plane. In the p plane, -co and -fco are identical, and a 

 band-pass useful interval is a single segment of the co" axis. 



For gain approximations over a band -pass interval, (5) may be re- 

 placed by: 



P' = ) ^ (117) 



The three coefficients, a, h, c are subject to two conditions, stemming 

 from the requirement that the interval | 2 | = 1 must map onto the 

 interval cod < co < coc2 • This leaves one arbitrary degree of freedom. 

 Its choice may be related to ordinary least squares approximations in 

 the foUomng way: 



If a = ^ CokTik , the first n terms approximate a in the least squares 

 sense. In other words, the integrated square of the error is a minimum, 

 relative to all possible choices of the first n coefficients Cik , provided 

 the integration extends over the useful frequency interval, and in- 

 cludes an appropriate "weight function". When (117) relates z to p, 

 the arbitrary degree of freedom in the choice of the constants a, h, c 

 permits selection of any one of a, family of weight functions. Conventional 



