OptimalFir.remez Function

Design an optimal equiripple FIR filter with Parks-McClellan algorithm.

Pascal

function remez(const h: TVec; const bands: array of Double; const gains: array of Double; const weights: array of Double; FilterType: TRemezType; out err: Double; FS: Double = 2; ConstantRipple: boolean = false): integer;

File

OptimalFir

Parameters

Parameters	Description
h	An array of length h.Length on entry and contains the filter impulse response on exit.
bands	Defines the frequency bands.
gains	Defines, if the band is a stopband or a passband.
weights	Array contains the ratios between required ripples for different bands.
FilterType	Choose between bandpass, hilbert, differentator and integrator.
err	Contains the maximum ripple error upon return.
FS	Specifies the sampling frequency.
ConstantRipple	If True, the ripple will be constant and not weighted to give constant percentage error in case of the following filter types: rmtDifferentiator, rmtIntegrator, rmtDoubleDifferentiator, rmtDoubleIntegrator

Description

Designs an equiripple (optimal) FIR filter. The routine will not always converge. Parameters have to be specified, for which the filter exists. Most common causes for trouble are:

too wide transition bands
transition bands not of equal width.
too strong attenuation (of the stopband) or to small ripple (of the passband) specified.
wrong filter length (odd/even).

The length of the filter can be estimated with the RemezLength routine. RemezLength routine will also properly adjust the error weights. An extensive explanation of the algorithm can be found in [1] Ch. 7.6, p. 462.

A few need to know things about FIR filters:

The length of the filter is defined as: n = Order-1. (Order is the order of the polynomial and n is the number of coefficients.)
highpass or a bandstop filter or any filter with the passband at FS/2 has to have odd length (even order). RemezLength routine automatically adjusts filter length.
a filter with negative symmetry also shifts the phase by 90 degrees. The routine automatically assumes negative symmetry, if the FilterType is different from rmtBandpass.
Required stopband attenuation is usually specified in dB. To obtain the required ripple, the following formula can be used: Ripple = Exp10(Att/-20);
Passband ripple can also be specified in dB. To obtain the required linear ripple, the following formula can be used: Ripple = (1-Exp10(PassRippldB/-20))/2
The passband ripple "rp" specified for the passband is a +/-rp specification. Total ripple is 2*rp.
the amount of ripple is a function of filter length. (longer filters give less ripple).
if different bands have different required ripple, this can be addressed by adjusting the error weights array.

The Fortran source code can be found in [2] p. 198.

References:

Example

A sample lowpass filter design with sampling frequency 8000Hz, transition band between at 1500Hz and 2000Hz with 60 dB attenuation (20 * Log10(0.001)) and 0.001 ripple in the passband. To try out other setups comment out the lowpass and comment in the desired filter.

  uses MtxExpr, Math387, MtxVec, SignalUtils, MtxVecTee, MtxVecEdit, OptimalFIR;

  procedure TForm1.Button1Click(Sender: TObject);
  var z,Response,Weights: Vector;
      n: integer;
      err: Double;
  begin
        n := RemezLength([0, 1500, 2000, 4000], [1, 0], [0.001, 0.001],Weights, 8000);
        z.Size(n);
        Remez(z,[0, 1500, 2000, 4000], [1, 0], TDoubleArray(Weights), rmtBandpass,err,8000);

  //Design a highpass
  //      n := RemezLength([0, 1500, 2000, 4000], [0, 1], [0.001, 0.001],Weights, 8000);
  //      z.Size(n);
  //      Remez(z,[0, 1500, 2000, 4000], [0, 1], TDoubleArray(Weights), rmtBandpass,err,8000);

  //Design a bandpass (Sampling frequency = 2)
  //      n := RemezLength([0, 0.15, 0.25, 0.45, 0.55, 1], [0, 1, 0], [0.001, 0.001, 0.001],Weights);
  //      z.Size(n);
  //      Remez(z,[0, 0.15, 0.25, 0.45, 0.55, 1], [0, 1, 0], TDoubleArray(Weights), rmtBandpass, err);

  //Design a bandstop (Sampling frequency = 2)
  //      n := RemezLength([0, 0.15, 0.25, 0.45, 0.55, 1], [1, 0, 1], [0.001, 0.001, 0.001],Weights);
  //      z.Size(n);
  //      Remez(z,[0, 0.15, 0.25, 0.45, 0.55, 1], [1, 0, 1], TDoubleArray(Weights), rmtBandpass, err);

  //Design a multiband (Sampling frequency = 2)
  //      n := RemezLength([0.00, 0.10, 0.20, 0.30, 0.40,
  //                        0.45, 0.55, 0.60, 0.70, 1.00], [1, 0, 1, 0, 1], [0.001, 0.001, 0.001, 0.001, 0.001],Weights);
  //      z.Size(n);
  //      Remez(z,[0.00, 0.10, 0.20, 0.30, 0.40,
  //                      0.45, 0.55, 0.60, 0.70, 1.00], [1, 0, 1, 0, 1], TDoubleArray(Weights), rmtBandpass, err);

  //Design a hilbert (type III) transformer  (Sampling frequency = 2)
  //        n := RemezLength([0, 0, 0.1, 0.9, 1, 1], [0,1,0], [0.01,0.01,0.01],Weights);
  //        if not Odd(n) then Inc(n); //odd length type III filter
  //        z.Size(n);
  //        Remez(z,[0.1, 0.9], [1], [1], rmtHilbert,err);

  //Design a hilbert (type IV) transformer  (Sampling frequency = 2)
  //        n := RemezLength([0, 0, 0.1, 0.9, 1, 1], [0,1,0], [0.01,0.01,0.01],Weights);
  //        if Odd(n) then Inc(n); //even length type IV filter
  //        z.Size(n);
  //        Remez(z,[0.1, 1], [1], [1], rmtHilbert,err);

  //Design a differentiator (type III) (Sampling frequency = 2)
  //        n := RemezLength([0, 0, 0.1, 0.9, 1, 1], [0,1,0], [0.01,0.01,0.01],Weights);
  //        if not Odd(n) then Inc(n); //odd length type III filter
  //        z.Size(n);
  //        Remez(z,[0.1, 0.9], [1], [1], rmtDifferentiator,err);

  h//Design a differentiator (type IV) (Sampling frequency = 2)
  //        n := RemezLength([0, 0, 0.1, 0.9, 1, 1], [0,1,0], [0.01,0.01,0.01],Weights);
  //        if Odd(n) then Inc(n); //even length type IV filter
  //        z.Size(n);
  //        Remez(z,[0.1, 1], [1], [1], rmtDifferentiator,err);

  //Design a double differentiator (Sampling frequency = 2)
  //        n := RemezLength([0, 0, 0.05, 0.95, 1, 1], [0,1,0], [0.01,0.01,0.01],Weights);
  //        if Odd(n) then Inc(n); //even length type IV filter
  //        z.Size(n);
  //        Remez(z,[0.05, 1], [1], [1], rmtDoubleDifferentiator,err);

  //Design an integrator (Sampling frequency = 2)
  //        n := RemezLength([0, 0, 0.05, 0.95, 1, 1], [0,1,0], [0.01,0.01,0.01],Weights);
  //        if Odd(n) then Inc(n); //even length type IV filter
  //        z.Size(n);
  //        Remez(z,[0.05, 1], [1], [1], rmtIntegrator,err);

  //Design a double integrator  (Sampling frequency = 2)
  //        n := RemezLength([0, 0, 0.05, 0.95, 1, 1], [0,1,0], [0.01,0.01,0.01],Weights);
  //        if Odd(n) then Inc(n); //even length type IV filter
  //        z.Size(n);
  //        Remez(z,[0.05, 1], [1], [1], rmtDoubleIntegrator,err);

        FrequencyResponse(z,nil,Response,8);
        DrawIt(Response);
  end;

  #include "MtxExpr.hpp"
  #include "MtxVecEdit.hpp"
  #include "MtxVecTee.hpp"
  #include "SignalUtils.hpp"
  #include "OptimalFIR.hpp"

  void __fastcall TForm1::BitBtn1Click(TObject *Sender)
  {
      sVector z, Response, Weights;
      int n;
      double err;

      n = RemezLength(OPENARRAY(double,(0, 1500, 2000, 4000)), OPENARRAY(double,(1, 0)), OPENARRAY(double,(0.001, 0.001)),Weights, 8000);
      z.Size(n);
      remez(z,OPENARRAY(double,(0, 1500, 2000, 4000)), OPENARRAY(double,(1, 0)), Weights.PValues1D(0), Weights.Length-1, rmtBandPass,err,8000);

    //Design a highpass
  //        n = RemezLength(OPENARRAY(double,(0, 1500, 2000, 4000)), OPENARRAY(double,(0, 1)), OPENARRAY(double,(0.001, 0.001)),Weights, 8000);
  //        z.Size(n);
  //        remez(z,OPENARRAY(double,(0, 1500, 2000, 4000)), OPENARRAY(double,(0, 1)), Weights.PValues1D(0), Weights.Length-1, rmtBandPass,err,8000);

    //Design a bandpass (Sampling frequency = 2)
  //        n = RemezLength(OPENARRAY(double,(0, 0.15, 0.25, 0.45, 0.55, 1)), OPENARRAY(double,(0, 1, 0)), OPENARRAY(double,(0.001, 0.001, 0.001)),Weights);
  //        z.Size(n);
  //        remez(z,OPENARRAY(double,(0, 0.15, 0.25, 0.45, 0.55, 1)), OPENARRAY(double,(0, 1, 0)), Weights.PValues1D(0), Weights.Length-1, rmtBandPass, err);

    //Design a bandstop (Sampling frequency = 2)
  //        n = RemezLength(OPENARRAY(double,(0, 0.15, 0.25, 0.45, 0.55, 1)), OPENARRAY(double,(1, 0, 1)), OPENARRAY(double,(0.001, 0.001, 0.001)),Weights);
  //        z.Size(n);
  //        remez(z,OPENARRAY(double,(0, 0.15, 0.25, 0.45, 0.55, 1)), OPENARRAY(double,(1, 0, 1)), Weights.PValues1D(0), Weights.Length-1, rmtBandPass, err);

    //Design a multiband (Sampling frequency = 2)
  //        n = RemezLength(OPENARRAY(double,(0.00, 0.10, 0.20, 0.30, 0.40, 0.45, 0.55, 0.60, 0.70, 1.00)),
  //                              OPENARRAY(double,(1, 0, 1, 0, 1)), OPENARRAY(double,(0.001, 0.001, 0.001, 0.001, 0.001)),Weights);
  //        z.Size(n);
  //        remez(z,OPENARRAY(double,(0.00, 0.10, 0.20, 0.30, 0.40, 0.45, 0.55, 0.60, 0.70, 1.00)),
  //                    OPENARRAY(double,(1, 0, 1, 0, 1)), Weights.PValues1D(0), Weights.Length-1, rmtBandPass, err);

    //Design a hilbert (type III) transformer  (Sampling frequency = 2)
  //          n = RemezLength(OPENARRAY(double,(0, 0, 0.1, 0.9, 1, 1)), OPENARRAY(double,(0,1,0)), OPENARRAY(double,(0.01,0.01,0.01)),Weights);
  //          if ((n%2) == 0) n++; //odd length type III filter
  //          z.Size(n);
  //          remez(z,OPENARRAY(double,(0.1, 0.9)), OPENARRAY(double,(1)), OPENARRAY(double,(1)), rmtHilbert,err);

    //Design a hilbert (type IV) transformer  (Sampling frequency = 2)
  //          n = RemezLength(OPENARRAY(double,(0, 0, 0.1, 0.9, 1, 1)), OPENARRAY(double,(0,1,0)), OPENARRAY(double,(0.01,0.01,0.01)),Weights);
  //          if (!((n%2) == 0)) n++; //odd length type IV filter
  //          z.Size(n);
  //          remez(z,OPENARRAY(double,(0.1, 1)), OPENARRAY(double,(1)), OPENARRAY(double,(1)), rmtHilbert,err);

    //Design a differentiator (type III) (Sampling frequency = 2)
  //          n = RemezLength(OPENARRAY(double,(0, 0, 0.1, 0.9, 1, 1)), OPENARRAY(double,(0,1,0)), OPENARRAY(double,(0.01,0.01,0.01)),Weights);
  //          if ((n%2) == 0) n++; //odd length type III filter
  //          z.Size(n);
  //          remez(z,OPENARRAY(double,(0.1, 0.9)), OPENARRAY(double,(1)), OPENARRAY(double,(1)), rmtDifferentiator,err);

    //Design a differentiator (type IV) (Sampling frequency = 2)
  //          n = RemezLength(OPENARRAY(double,(0, 0, 0.1, 0.9, 1, 1)), OPENARRAY(double,(0,1,0)), OPENARRAY(double,(0.01,0.01,0.01)),Weights);
  //          if (!((n%2) == 0)) n++; //even length type IV filter
  //          z.Size(n);
  //          remez(z,OPENARRAY(double,(0.1, 1)), OPENARRAY(double,(1)), OPENARRAY(double,(1)), rmtDifferentiator,err);

    //Design a double differentiator (Sampling frequency = 2)
  //          n = RemezLength(OPENARRAY(double,(0, 0, 0.05, 0.95, 1, 1)), OPENARRAY(double,(0,1,0)), OPENARRAY(double,(0.01,0.01,0.01)),Weights);
  //          if (!((n%2) == 0)) n++; //even length type IV filter
  //          z.Size(n);
  //          remez(z,OPENARRAY(double,(0.05, 1)), OPENARRAY(double,(1)), OPENARRAY(double,(1)), rmtDoubleDifferentiator,err);

  //  Design an integrator (Sampling frequency = 2)
  //          n = RemezLength(OPENARRAY(double,(0, 0, 0.05, 0.95, 1, 1)), OPENARRAY(double,(0,1,0)), OPENARRAY(double,(0.01,0.01,0.01)),Weights);
  //          if (!((n%2) == 0)) n++; //even length type IV filter
  //          z.Size(n);
  //          remez(z,OPENARRAY(double,(0.05, 1)), OPENARRAY(double,(1)), OPENARRAY(double,(1)), rmtIntegrator,err);

    //Design a double integrator  (Sampling frequency = 2)
  //          n = RemezLength(OPENARRAY(double,(0, 0, 0.05, 0.95, 1, 1)), OPENARRAY(double,(0,1,0)), OPENARRAY(double,(0.01,0.01,0.01)),Weights);
  //          if (!((n%2) == 0)) n++; //even length type IV filter
  //          z.Size(n);
  //          remez(z,OPENARRAY(double,(0.05, 1)), OPENARRAY(double,(1)), OPENARRAY(double,(1)), rmtDoubleIntegrator,err);

      FrequencyResponse(z,NULL,Response,8);
      DrawIt(Response);

  }

Group

Functions

Links

OptimalFir, Functions, Example, See Also

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