IIRFilters.ChebyshevIAnalog Method

Design analog Chebyshev type I IIR prototype filter.

public ChebyshevIAnalog(int Order, double PassRipple, TVec z, TVec p, ref double k);

Remarks

Design analog Chebyshev type I lowpass prototype filter of order Order. Place the resulting transfer function in zero-pole form in Z (zeros), P (poles) and K (gain). PassRipple defines the ripple of the passband (dB). The cutoff frequency of the prototype filter is preset to 1 rad/sec, the unit circle. Chebyshevs type I filters are all-pole designs and are equiripple in the passband. The filter has all zeros in infinity. The design formulas are found in [1] p. 232:


Poles: p[k] = s[k] + j*W[k]

s[k] = -sinh(Phi)*sin((2*k-1)*Pi/(2*n))
W[k] =  cosh(Phi)*cos((2*k-1)*Pi/(2*n))

sinh(phi) =  0.5*(v - 1/v)
cosh(phi) =  0.5*(v + 1/v)

       1 + (1 + eps^2)^0.5
v = ( --------------------- )^(1/n)
              eps

n - order of the filter
k = 1,...,n

References:

[1] "Theory and application of digital signal processing, Lawrence R. Rabiner and Bernard Gold. Prentice-Hall, 1975".

Related Topics

ChebyshevIIFilter, LowpassToLowpass, Bilinear

C# Example

Design an analog highpass filter with cutoff frequency at 2 rad/sec with a 0.2dB ripple in the passband.

  using Dew.Math;
using Dew.Math.Editors;
using Dew.Math.Units;
using Dew.Signal;
using Dew.Signal.Units;
using Dew.Math.Tee;
using Dew.Signal.Tee;

 private void button1_Click(object sender, EventArgs e)
{
    Vector z = new Vector(0);
    Vector p = new Vector(0);
    Vector num = new Vector(0);
    Vector den = new Vector(0);
    Vector Response = new Vector(0);
    Vector FreqFr = new Vector(0);
    double k, Wc;
    int Order = 5; //design a fifth order filter.

    IIRFilters.ChebyshevIAnalog(Order,0.2,z, p, out k);  //design analog protype
    Wc = 2; //cutoff frequency
    LinearSystems.LowpassToLowpass(z, p, ref k, Wc);
    LinearSystems.ZeroPoleToTransferFun(num, den, z, p, k);
    FreqFr.Length = 1000;
    SignalUtils.LogRamp(FreqFr, -1, 1);

    SignalUtils.FrequencyResponseS(num, den, FreqFr, Response, 0);
    TeeChart.DrawIt(Response, "Frequency response", false);
}