IIRFilters.ButterAnalog Method

Design analog Butterworth type IIR prototype filter.

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

Remarks

Design analog butterworth lowpass prototype filter of order Order. Place the resulting transfer function in zero-pole form in Z (zeros), P (poles) and K (gain). The cutoff frequency of the prototype filter is preset to 1 rad/sec.

The filter has all zeros in infinity. The transfer function is defined as ([1], p. 277):


                   k0
H(s) = -------------------------
       (s - s[1])*...*(s - s[n])


The poles of the filter are located at

s[k] := Expj(Pi*(0.5+(2*k-1)/(2*n)));

n = order of filter
k = 1,...,n
k0 = gain

The magnitude response is down 3dB at the cutoff frequency.

References:

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

Related Topics

ButterFilter, LowpassToHighpass, Bilinear

C# Example

Design an analog lowpass filter with cutoff frequency at 3 rad/sec.

  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.ButterAnalog(Order,z, p, out k);  //design analog protype
    Wc = 3; //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);
}