ButterAnalog Routine

Summary

Design analog Butterworth type IIR prototype filter.

Declaration

Procedure ButterAnalog(Order: integer; z, p: TVec; out k: TSample);

Description

Design analog butterworth lowpass 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. 277):


                         k0
      H(s) = -------------------------
             (s - s[1])*...*(s - s[n])


      The poles of the filter are located at

      s[k] := Expj(Pi*(0.5+(2*k-1)/(2*n)));

      n = order of filter
      k = 1,...,n
      k0 = gain

The magnitude response is down 3dB at the cutoff frequency.

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

Example 1

Design an analog lowpass filter with cutoff frequency at 3 rad/sec.


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

    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.
        ButterAnalog(Order,z,p,k);  //design analog protype
        Wc := 3;
        LowpassToLowpass(z,p,k,WC);  //frequency transformation in s-domain
        ZeroPoleToTransferFun(num,den,z,p,k);
        FreqFr.Length := 1000;         //Define the frequency grid (logarithmic)
        LogRamp(FreqFr,-1,1); //between 0.1 (=10^(-1)) and 10 (=10^1) rad/sec
        FrequencyResponseS(num,den,FreqFr,Response); //Laplace
        DrawIt(Response); //Y axis linear, X axis logarithmic;
    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.
      ButterAnalog(Order,z,p,k);  //design analog protype
      Wc = 3;
      LowpassToLowpass(z,p,k,Wc);  //frequency transformation in s-domain
      ZeroPoleToTransferFun(num,den,z,p,k);
      FreqFr->Length = 1000;         //Define the frequency grid (logarithmic)
      LogRamp(FreqFr,-1,1); //between 0.1 (=10^(-1)) and 10 (=10^1) rad/sec
      FrequencyResponseS(num,den,FreqFr,Response); //Laplace
      DrawIt(Response); //Y axis linear, X axis logarithmic;
    }


  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.ButterAnalog(Order,z, p, out k);  //design analog protype
      Wc = 3; //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);
  }