738 BELL SYSTEM TECHNICAL JOURNAL 



/i (cos ip) = />() + ^1 COS if + ^2 COS- ip -\- bz CO?, ip -\- ■■■+ bn cos" (p 



-'^^•' ^ (1 + x'Y 



+ 



= /(.v^)/.(.v^), 



where /2(.v ) = 



1 



(1 + .t2)»' 



Therefore, it is necessary to predistort the approximatefl function B{x'-) by 

 redefining the /(</?) corresponding to /(.v-) as 





my 



where 



and 



n 



Mf) = IZ '^'- cos A;(^ 



i=0 



yt-X 



i=0 



(1 + x^y 



= Mx), 



Hence, /i ''(/?), which corresponds to the approximating function /(.v-) multi- 



pUed by r- in the frequency domain, is the approximating function in 



\\ ~t~ X ) 

 the ip domain. In practice, the indicated predistortion of B{x-) may be carried 

 out either before or after the specification has been transformed to the <p 

 domain. Table I shows the relation of the Ak to the oa for n = 3 and n = 5. 



Table I 

 Relation of the Ak of/(.t^) to the au of/i(^) for « = 3 and m = 5 



« = 3 



Ao = ao + fli + ^2 + as 



Ai = 3ao + fli - 5a-2 - 15a; 



Ai = 3ao - fli - 5a-2 + 15a; 

 ^3 = oo - fli + an - flj 



» = 5 



^0 = flo + ai + 02 + fls + flj + ^5 

 /li = 5ao + 3ai — 3a2 — 13a3 — 21 Qi — ASa^ 

 A2 = lOao + 2ai - Hch - Ua^ + 42a4 + 210a3 

 As = lOao - 2ai - 140^ + Mas + 42a4 - 210a5 

 Ai = 5ao — 3ai — 3a2 + 13a3 — 27a4 + 45a5 

 Ai = ao — ai + a-2 — as + Oi — Oi 



