XKTNNOUK SV.XrilKSlS (),j3 



E(iuatioii.s (84), (85), (8()) may now be modified, for the new A^, D 

 and Kk , by merely using z in place of z' wherever it occurs (including 

 z'' in place of 2"^).t The modifications of equations (84), (85), and the 

 truncation of (86) after a„, now It^ad to (\- = Ck , A" ^ m, instead of the 

 previous C-jt = C-ik . This means that //; must be twice as great to match 

 coefficients out to the same actual ordiMs. This is to be expected since 

 now one half of our design parameters are used to appioximate phase 

 ^, leaving only half for approximating gain a. Eciuation (89) must be 

 changed not only in n^gai-d to the orders of Ck , Ck , but also in regard 

 to the factors | in (94), (9()). This gives 



2 (aitto • ■ ■ am)" a,n+iKo 



The most important change is in regard to the zeros and poles z^ . 

 The polynomials N and D now determine z^ and z„ directly, instead of 

 their squares. There is no opportunity to adjust the sign of Re z„ by 

 choosing the correct sign of y/zT^. When non-physical singularities ap- 

 pear, adjustments of high order coefficients may be tried. Section 23 

 applies provided z is replaced by z. If the specification of the problem 

 permits added delay, linear phase may be added to a + ?^ to increase 

 the probability of phj^sical singularities!. (Addition of linear phase 

 changes only Ci , in ^ CkTk . A negative change in Ci increases the 

 delay.) 



25. APPROXIMATION' TO AX ASSIGXED PHASE ^§ 



Sometimes it is recjuired to approximate an assigned phase, without 

 regard to gain. More commonly, it is required to approximate an as- 

 signed phase, using an "all-pass" network, which has a theoretically 

 zero gain. These two problems, however, are very nearly identical, due 

 to circumstances explained at the end of this section. 



For approximation to phase only, we go back to the /3 equation in 

 (21). As before, products of factors {z — z^) are replaced by polynomial 



t and = in place of = . 



X The well known relation between the gain and j)huse of any j)hysical network 

 (See for instance Bode'") may give some information regarding the reasonable- 

 ness of 3. It must be remembered, however, that departures of network gain a, 

 from the assigned gain a, outside the useful interval, may affect the permissible 

 phase /3, within the interval. 



§ Up to the present, apjjlications to phase problems have not been developed 

 to the same extent as for gain. Techni<iues liave l)oen explorefl, however, to deter- 

 mine how such applications may in fact be carried out. 



