LowpassToLowpassZ Routines

Summary

Apply frequency band transformation from lowpass to lowpass in the z-domain.

Declaration

Procedure LowpassToLowpassZ(num, den: TVec; Freq: TSample; PrototypeFreq: TSample);

Description

Freq is the cutoff frequency of the new filter. The function returns modified num and den. PrototypeFreq is the cutoff frequency of the prototype lowpass filter after it has been mapped to z-domain. Freq and PrototypeFreq must be between 0 and 1 (Sampling frequency = 2). The transformation is defined with the following mapping ([1] p. 260 and [2] p. 434, [3] p. 352):

                     z^(-1) - a
      z^(-1) --->  --------------
                    1 - a*z^(-1)

            sin((wc - wn)/2)
      a = -------------------
            sin((wc + wn)/2)

      wc - old cutoff frequency
      wn - new (desired) cutoff frequency

Declaration

Procedure LowpassToLowpassZ(z, p: TVec; var k: TSample; Freq, PrototypeFreq: TSample);

Description

The function returns modified z (zeros), p (poles) and k (gain).

See Also
[1] "Theory and application of digital signal processing, Lawrence R. Rabiner and Bernard Gold. Prentice-Hall, 1975".
[2] "Discrete-time signal processing, Oppenheim and Schafer, Prentice-Hall, 1989".
[3] "Digital signal processing, Vinay K. Ingle and John G. Proakis, Brooks-Cole, 2000".
Bilinear
RationalSubstitution

Example 1

Elliptic lowpass filter design. The cutoff frequency of an analog filter prototype transformed in to z domain is obtained with BilinearUnwarp method. The analog prototype filter has a normalized cutoff frequency at 1 rad/sec.


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

    procedure TForm42.Button1Click(Sender: TObject);
    var z,p,num,den,Response: Vector;
        Order: integer;
        k,Wc: TSample;
    begin
        Order := 4;//design a fourth order filter.
        EllipticAnalog(Order,0.1,30,z,p,k);  //design analog protype
        Bilinear(z,p,k,2); //bilinear with sampling frequency 2Hz.
        Wc := 0.5;   //desired cutoff at 0.5 Hz.

        LowpassToLowpassZ(z,p,k,Wc,BilinearUnwarp(1)); //move cutoff to 0.5 Hz.
        ZeroPoleToTransferFun(num,den,z,p,k);

    //Alternative:
    //       ...
    //      ZeroPoleToTransferFun(num,den,z,p,k);
    //      LowpassToLowpassZ(num,den,Wc,BilinearUnwarp(1)); //move cutoff to 0.5 Hz.

        FrequencyResponse(num,den,Response,64);
        DrawIt(Response);
    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,Response;
      int	Order;
      double k,Wc;

      Order = 4;//design a fourth order filter.
      EllipticAnalog(Order,0.1,30,z,p,k);  //design analog protype
      Bilinear(z,p,k,2); //bilinear with sampling frequency 2Hz.
      Wc = 0.5;   //desired cutoff at 0.5 Hz.

      LowpassToLowpassZ(z,p,k,Wc,BilinearUnwarp(1)); //move cutoff to 0.5 Hz.
      ZeroPoleToTransferFun(num,den,z,p,k);

    //Alternative:
    //       ...
    //      ZeroPoleToTransferFun(num,den,z,p,k);
    //      LowpassToLowpassZ(num,den,Wc,BilinearUnwarp(1)); //move cutoff to 0.5 Hz.

      FrequencyResponse(num,den,Response,64);
      DrawIt(Response);
    }


  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);
      double k, Wc;
      double FS = 2;
      int Order = 4; //design a fifth order filter.

      IIRFilters.EllipticAnalog(Order,0.1,30, z, p, out k);  //design analog protype
      LinearSystems.Bilinear(z, p, ref k, FS,true);
      Wc = 0.5;
      LinearSystems.LowpassToLowpassZ(z, p, ref k, Wc, LinearSystems.BilinearUnwarp(1,FS));
      LinearSystems.ZeroPoleToTransferFun(num,den, z, p, k);
      SignalUtils.FrequencyResponse(num, den, Response, 64, false, TSignalWindowType.wtRectangular, 0); //zero padding set to 64

//Alternative:
//            ...
//            LinearSystems.ZeroPoleToTransferFun(num,den, z, p, k);
//            LinearSystems.LowpassToLowpassZ(num,den, Wc, LinearSystems.BilinearUnwarp(1,FS));

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