IIRFilters.BesselAnalog Method

Design analog Bessel type IIR prototype filter.

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

Remarks

Design analog Bessel 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. 230):


          d0
H(s) = --------
         Bn(s)

      (2*n)!               n
d0 = ------- ,    Bn(s) = Sum(d[k]*s^k),  k = 0,...,n
      2^n*n!              k=0

          (2*n-k)!
d[k] = --------------    , n = order of the filter
        2^(n-k)*(n-k)!


Filter poles must be scaled with d0^(1/n)

Roots of the Bessel polynomial Bn(s) are found with the PolyRoots routine. Bessel lowpass filters are charachterized by the property that the group delay is maximally flat at the origing of the s-plane. ([1], p. 228).

References:

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

Related Topics

BesselFilter, LowpassToHighpass, Bilinear

C# Example

Design an analog lowpass filter with cutoff at 1.1 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.BesselAnalog(Order,z, p, out k);  //design analog protype
    Wc = 1.1; //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);
}