652 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



systematized, to a considerable extent, by using the error formula (88), 

 and other relations between the coefficients of the continued fraction, 

 and the rational fraction truncations of various orders, f 



24. APPROXIMATION TO BOTH GAIN AND PHASE 



The applications described in previous sections relate to approxima- 

 tions to prescribed gain, a, without regard to the associated phase. 

 Quite similar methods apply, however, to the simultaneous approxima- 

 tion of gain and phase. 



The starting point is equation (20). Replacing products of factors by 

 polynomials gives, in place of (81): 



a -{- i^ = ^ CkTk, k even and odd 



ZiCkz' = -log'^ 



The polynomials are now as follows, in place of (82) : 



N = lu + K['z ■ ■ • K"»2"" 



D = 1 4- Kiz • • ■ Kn'Z 



(If only natural modes are to be used, the suitable replacement for the 

 first equation or (55) is here obtauied merely by using D = \, and K^ , 

 z, z„ , in place of their squares.) 



A comparable expression is needed for the assigned gain and phase 

 a. + Zj8. In place of (50), we now repeat (22), and redefine the coeffi- 

 cients Kk in accordance with 



a + ZjS = ^ CfcTfc , fc even and odd 



E iC,/- - log R{z) = -log E Kkz' 



The definition of Kk has been changed in such a way that it is now 

 related to Ck/2 exactly as it was previously (in 58) related to C^k . 

 Equations (59), (61) may be applied to calculating the Kk by merely 

 substituting therein a Ck/2 for every C2k ■ 



t For example, a simple recursion formula may be used to assemble the polj^- 

 nomials A'^ and D which correspond to truncation of the continued fraction (86) 



at a number of different points. Specifically, P„ = P„_i -\ Pn-2 , where P 



a„a„_i 



is either N or D and P„ corresponds to truncation of the continued fraction after 



the term in a„ . The formula holds for n ^ 2. 



