DescriptionElliptic Filter s-Plane Response (8th Order).svg	English: Log-absolute complex frequency (s=σ+jω) response of an 8th order elliptic filter. Values less than 0.0001 are black; greater than 10 are in white. ξ = 1.05, ε = 0.5 and ω0 = 1.
Date	30 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[xi_] := 
  Module[{g, num, den}, g = Sqrt[4*xi^2 + (4*xi^2*(xi^2 - 1))^(2/3)];
   num = 2*xi^2*Sqrt[g];
   den = Sqrt[8*xi^2*(xi^2 + 1) + 12*g*xi^2 - g^3] - Sqrt[g^3];
   num/den];
xz2[xi_] := xi^2/xp2[xi];

t[xi_] := Sqrt[1 - 1/xi^2];

(*Use particular values for low-order functions*)
r1[xi_, x_] := x;
r2[xi_, x_] := ((t[xi] + 1)*x^2 - 1)/((t[xi] - 1)*x^2 + 1);
r3[xi_, x_] := 
  x*((1 - xp2[xi])*(x^2 - xz2[xi]))/((1 - xz2[xi])*(x^2 - xp2[xi]));

(*Use nesting property for higher-degree functions*)
r4[xi_, x_] := r2[r2[xi, xi], r2[xi, x]];
r8[xi_, x_] := r4[r2[xi, xi], r2[xi, x]];

ellgain[xi_, w_, w0_, ep_] := 1/Sqrt[1 + ep^2*r8[xi, w/w0]^2];

DensityPlot[
 w0 = 1;
 ep = 0.5;
 xi = 1.05;
 min = 0.0001;
 max = 10;
 
 Log[Abs[
   ellgain[xi, sig + I*w, w0*I, ep]
   ]],
 {sig, -4, 4},
 {w, -4, 4},
 PlotRange -> {Log[min], Log[max]},
 PlotPoints -> 100,
 ColorFunction -> GrayLevel,
 ClippingStyle -> {Black, White}
 ]