MatchedZTransform Routine

Summary

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

Declaration

Procedure MatchedZTransform(z, p: TVec; FS: TSample = 1);

Description

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.

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

Example 1

A bessel analog lowpass filter is converted to z-domain by using the matched Z transform. The analog filter has a normalized cutoff frequency at 1 rad/sec.


    uses MtxExpr, Math387, MtxVec, MtxVecTee, MtxVecEdit,
         LinearSystems, IirFilters, SignalUtils;

    procedure TForm1.Button1Click(Sender: TObject);
    var z,p, num,den, FreqFr,Response: Vector;
        Order: integer;
        k,Wc,BW: TSample;
    begin
        Order := 5; //design a fifth order filter.
        BesselAnalog(Order,z,p,k);  //design analog protype
        Wc := 0.5; //request a cutoff at 0.5 rad/sec
        LowpassToLowpass(z,p,k,Wc);
        MatchedZTransform(z,p,2);  //Sampling frequency  = 2
        k := k/ComputeGain(z,p,1);
        z.Size(p.Length,false);
        z.SetVal(-1); //add missing zeros at -1
        ZeroPoleToTransferFun(num,den,z,p,k);
        FrequencyResponse(num,den,Response,64); //zero padding set to 64
        DrawIt(20*Log10(Abs(Response)),'Magnitude');
        DrawIt(PhaseSpectrum(Response)*(180/Pi),'Phase');
    end;


    #include "MtxVecCPP.h" //MtxVecCPP.cpp must be included in the project
    #include "MtxVecEdit.hpp"
    #include "MtxVecTee.hpp"
    #include "SignalUtils.hpp"
    #include "IirFilters.hpp"
    #include "LinearSystems.hpp"

    void __fastcall TForm1::BitBtn1Click(TObject *Sender)
    {
      Vector z,p, num,den, FreqFr,Response;
      int Order;
      double k,Wc,BW;

      Order = 5; //design a fifth order filter.
      BesselAnalog(Order,z,p,k);  //design analog protype
      Wc = 0.5; //request a cutoff at 0.5 rad/sec
      LowpassToLowpass(z,p,k,Wc);
      MatchedZTransform(z,p,2);  //Sampling frequency  = 2
      k = k/ComputeGain(z,p,1);
      z->Size(p->Length,false);
      z->SetVal(-1); //add missing zeros at -1
      ZeroPoleToTransferFun(num,den,z,p,k);
      FrequencyResponse(num,den,Response,64); //zero padding set to 64
      DrawIt(20*Log10(Abs(Response)),"Magnitude",false);
      DrawIt(PhaseSpectrum(Response)*(180/PI),"Phase",false);
    }


  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
     TeeChart.DrawIt(20*MtxExpr.Log10(MtxExpr.Abs(Response)),"Magnitude",false);
     TeeChart.DrawIt(MtxExpr.PhaseSpectrum(Response)*(180/Math.PI),"Phase",false);
  }