DescriptionElliptic Filter Response (4th Order, Detail).svg	English: The response of a 4-th order elliptic filter (detail of the transistion band). Also shown is the maximum extent of the passband ripples, where ε is the ripple factor and Ln is the discrimination factor.
Date	29 January 2009
Source	Own work
Author	Inductiveload
Permission (Reusing this file)	Public domainPublic domainfalsefalse I, the copyright holder of this work, release this work into the public domain. This applies worldwide. In some countries this may not be legally possible; if so: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

xp2[zeta_] :=
  Module[{g, num, den},
   g = Sqrt[4*zeta^2 + (4*zeta^2*(zeta^2 - 1))^(2/3)];
   num = 2*zeta^2*Sqrt[g];
   den = Sqrt[8*zeta^2*(zeta^2 + 1) + 12*g*zeta^2 - g^3] - Sqrt[g^3];
   num/den
   ];
xz2[zeta_] := zeta^2/xp2[zeta];

t[zeta_] := Sqrt[1 - 1/zeta^2];

r1[zeta_, x_] := x;
r2[zeta_, x_] := ((t[zeta] + 1)*x^2 - 1)/((t[zeta] - 1)*x^2 + 1);
r3[zeta_, x_] := 
  x*((1 - xp2[zeta])*(x^2 - xz2[zeta]))/((1 - xz2[zeta])*(x^2 - 
        xp2[zeta]));
r4[zeta_, x_] :=
  Module[{num, den},
   num = (1 + t[zeta]) (1 + Sqrt[t[zeta]])^2*x^4 - 
     2 (1 + t[zeta]) (1 + Sqrt[t[zeta]])*x^2 + 1;
   den = (1 + t[zeta]) (1 - Sqrt[t[zeta]])^2*x^4 - 
     2 (1 + t[zeta]) (1 - Sqrt[t[zeta]])*x^2 + 1;
   num/den
   ];

ellgain[xi_, w_, w0_, ep_] := 1/Sqrt[1 + ep^2*r4[xi, w/w0]^2];

Plot[
 ellgain[1.05, w, 1, 0.5],
 {w, 0, 3}]