LinearSystems.MatchedZTransform Method

void MatchedZTransform(TVec z, TVec p, Double FS)

Transform the zeros and poles of a filter in s-domain to z-domain.

#	Name	Type	Description
1	z	TVec	source TVec
2	p	TVec	source TVec
3	FS	Double	scalar

Result: stored in self (calling object)

Remarks:

Transform the zeros Z and poles P of a filter from s-domain to z-domain, where FS is the sampling frequency. The transformation is defined as ([1], p. 224): s + a -> 1 - z^(-1)e^(-a/FS)

FS - sampling frequency
a - pole or zero

If the analog system has zeros with center frequencies greater then half the sampling frequency, their z-plane positions will be greatly aliased [1]. The transformation has the advantage of not affecting the phase response of the original transfer function.

References:

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

Examples

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,BW;
    double FS = 2;
    int Order = 5; //design a fifth order filter.

    IIRFilters.BesselAnalog(Order,z,p,out k);  //design analog protype
    Wc = 0.5;
    LinearSystems.LowpassToLowpass(z,p,ref k,Wc);  //frequency transformation in s-domain
    LinearSystems.MatchedZTransform(z, p, FS);
    k = k/LinearSystems.ComputeGain(z,p,1);
    z.Size(p.Length,false);
    z.SetVal(-1); //add missing zeros at -1
    LinearSystems.ZeroPoleToTransferFun(num,den,z,p,k);
    SignalUtils.FrequencyResponse(num,den,Response,64,false,TSignalWindowType.wtRectangular,0); //zero padding set to 64
    MtxVecTee.DrawIt(20*MtxExpr.Log10(MtxExpr.Abs(Response)),"Magnitude",false);
    MtxVecTee.DrawIt(MtxExpr.PhaseSpectrum(Response)*(180/Math.PI),"Phase",false);
}

See Also: LinearSystems.Bilinear