ChebyshevIIAnalog Routine

Summary

Design analog Chebyshev type I IIR prototype filter.

Declaration

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

Description

Design analog Chebyshev type II lowpass prototype filter of order Order. Place the resulting transfer function in zero-pole form in Z (zeros), P (poles) and K (gain). Ripple defines the StopRipple (dB) of the stopband.The cutoff frequency of the prototype filter is preset to 1 rad/sec, the unit circle. Chebyshevs type II filters have poles and zeros and are equiripple in the stopband. The design formulas are found in [1] p. 232:


                                Wr
      Zeros: z[k] = j* ---------------------
                       cos((2*k-1)/(2*n)*Pi)

      Poles: p[k] = s[k] + j*W[k]

                    Wr*a[k]
      s[k]  = -----------------
               a[k]^2  + b[k]^2

                   -Wr*b[k]
      W[k]  = -----------------
               a[k]^2  + b[k]^2

      a[k] = -sinh(Phi)*sin((2*k-1)*Pi/(2*n))
      b[k] =  cosh(Phi)*cos((2*k-1)*Pi/(2*n))

      sinh(phi) =  0.5*(v - 1/v)
      cosh(phi) =  0.5*(v + 1/v)

      v = (A  + (A^2 - 1)^0.5)^(1/n),    A = 1/sr^2

      n - order of the filter
      k = 1,...,n
      Wr - stopband edge
      sr - stopband ripple

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

Example 1

Design an analog bandpass filter with passband between 2 and 3 rad/sec and with 20dB ripple in the stopband.


    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.
        ChebyshevIIAnalog(Order,20,z,p,k);  //design analog protype
        Wc := Sqrt(2*3);
        BW := 3-2;
        LowpassToBandpass(z,p,k,WC,BW);  //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.
      ChebyshevIIAnalog(Order,20,z,p,k);  //design analog protype
      Wc = Sqrt(2*3);
      BW = 3-2;
      LowpassToBandpass(z,p,k,Wc,BW);  //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.ChebyshevIIAnalog(Order,20,z, p, out k);  //design analog protype
      Wc = Math.Sqrt(3*2); //cutoff frequency
      double BW = 3 - 2;
      LinearSystems.LowpassToBandpass(z, p, ref k, Wc,BW);
      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);
  }