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}
 ]