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

Pascal
procedure MatchedZTransform(const z: TVec; const p: TVec; FS: Double = 1); overload;
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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. 

References:  

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

See Also
Example

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: Double;
    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 "MtxExpr.hpp"
    #include "MtxVecEdit.hpp"
    #include "MtxVecTee.hpp"
    #include "SignalUtils.hpp"
    #include "IirFilters.hpp"
    #include "LinearSystems.hpp"

    void __fastcall TForm1::BitBtn1Click(TObject *Sender)
    {
      sVector 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);
    }
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