IIRFilters.ButterFilter Method

IIrFilters

Overload List

#SignatureDescription
1function ButterFilter(Order: Integer; CutoffFreq: TDoubleArray; FilterType: TFilterType; Analog: Boolean; A: TMtx; B: TVec; C: TVec; out d: Double): Double;The resulting transfer function is returned in the state-space form with A,B,C,D variables.
2function ButterFilter(Order: Integer; CutoffFreq: TDoubleArray; FilterType: TFilterType; Analog: Boolean; Num: TVec; Den: TVec; IirFrequencyTransform: TIirFrequencyTransform): Double;The resulting transfer function is returned in the numerator/denumerator form with num and den.
3function ButterFilter(Order: Integer; CutoffFreq: TDoubleArray; FilterType: TFilterType; Analog: Boolean; z: TVec; p: TVec; out k: Double; IirFrequencyTransform: TIirFrequencyTransform): Double;Design a complete Butterworth IIR filter of the given Order, cutoff(s) CutoffFreq and band type FilterType, returning the transfer function in zero-pole form (z, p, k): H(z)=k (prod_i (z-z_i))/(prod_i (z-p_i)). Set Analog=True for an s-plane design or Analog=False for a z-plane (digital) design; for a digital design CutoffFreq lies in $(0,1)$ with sampling frequency 2 (so 1 = Nyquist). FilterType is lp/hp (1 cutoff) or bp/bs (2 cutoffs); a bandpass/bandstop design has twice the prototype order. IirFrequencyTransform selects when the frequency-band transform is applied (state-space-analog, zero-pole-analog or zero-pole-discrete). Digital poles satisfy |p_i|<1 (stable). The function result is the natural 3 dB cutoff W_c. The other overloads return the same filter as num/den (ba), second-order sections (sos) or state space (A,B,C,D). NOTE: a digital cutoff outside $(0,1)$ is NOT range-checked and yields a degenerate filter.
4function ButterFilter(Order: Integer; CutoffFreq: TDoubleArray; FilterType: TFilterType; Analog: Boolean; sos: TVec; IirFrequencyTransform: TIirFrequencyTransform): Double;The resulting transfer function is returned in the second order section form stored in the sos variable.

Overload 1: function ButterFilter(Order: Integer; CutoffFreq: TDoubleArray; FilterType: TFilterType; Analog: Boolean; A: TMtx; B: TVec; C: TVec; out d: Double): Double;

The resulting transfer function is returned in the state-space form with A,B,C,D variables.

#NameTypeDescription
1OrderInteger
2CutoffFreqTDoubleArray
3FilterTypeTFilterType
4AnalogBoolean
5ATMtxsource TMtx
6BTVecsource TVec
7CTVecsource TVec
8dDouble

Returns: Double

See Also: IIRFilters.ButterOrder, IIRFilters.ChebyshevIFilter, IIRFilters.ChebyshevIIFilter, IIRFilters.EllipticFilter, IIRFilters.BesselFilter

Overload 2: function ButterFilter(Order: Integer; CutoffFreq: TDoubleArray; FilterType: TFilterType; Analog: Boolean; Num: TVec; Den: TVec; IirFrequencyTransform: TIirFrequencyTransform): Double;

The resulting transfer function is returned in the numerator/denumerator form with num and den.

#NameTypeDescription
1OrderInteger
2CutoffFreqTDoubleArray
3FilterTypeTFilterType
4AnalogBoolean
5NumTVecsource TVec
6DenTVecsource TVec
7IirFrequencyTransformTIirFrequencyTransform

Returns: Double

Overload 3: function ButterFilter(Order: Integer; CutoffFreq: TDoubleArray; FilterType: TFilterType; Analog: Boolean; z: TVec; p: TVec; out k: Double; IirFrequencyTransform: TIirFrequencyTransform): Double;

Design a complete Butterworth IIR filter of the given Order, cutoff(s) CutoffFreq and band type FilterType, returning the transfer function in zero-pole form (z, p, k): H(z)=k (prod_i (z-z_i))/(prod_i (z-p_i)). Set Analog=True for an s-plane design or Analog=False for a z-plane (digital) design; for a digital design CutoffFreq lies in (0,1)(0,1) with sampling frequency 2 (so 1 = Nyquist). FilterType is lp/hp (1 cutoff) or bp/bs (2 cutoffs); a bandpass/bandstop design has twice the prototype order. IirFrequencyTransform selects when the frequency-band transform is applied (state-space-analog, zero-pole-analog or zero-pole-discrete). Digital poles satisfy |p_i|<1 (stable). The function result is the natural 3 dB cutoff W_c. The other overloads return the same filter as num/den (ba), second-order sections (sos) or state space (A,B,C,D). NOTE: a digital cutoff outside (0,1)(0,1) is NOT range-checked and yields a degenerate filter.

#NameTypeDescription
1OrderInteger
2CutoffFreqTDoubleArray
3FilterTypeTFilterType
4AnalogBoolean
5zTVecsource TVec
6pTVecsource TVec
7kDouble
8IirFrequencyTransformTIirFrequencyTransform

Returns: Double

Remarks:

Design Butterworth filter of Order with CutoffFreq frequencies and of FilterType type. Set Analog to True, to request and analog filter design in s-plane or set it to false to obtain a digital filter design in z-plane. The CutoffFreq must be in range between 0 and 1 (Sampling frequency = 2) in case of a digital filter design.

IIrFrequencyTransform specifies when and how will the frequency band transformation be applied. The resulting transfer function is returned in the zero-pole form, with z,p,k variables.

Examples
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,Bw,Wc: double;
WcArray: TDoubleArray; //modified 3dB frequency
begin
    SetLength(WcArray,2);
    Order := ButterOrder([0.2,0.3,0.6,0.7],0.2,50,ftBandstop,WcArray);
    ButterFilter(Order,WcArray,ftBandstop,false,num,den);

    // Alternative 1. Specify the order and the 3 dB frequencies explicitely:
    //
    //     ButterFilter(5,[0.2,0.7],ftBandstop,false,num,den);


    // Alternative 2. Specifying the 3 dB frequencies explicitely
    // will result in 3 dB ripple (and not 0.2 as requested) in the passband,
    // but one coulde always move the 3 dB frequencies a little:
    //
    //     ButterFilter(5,[0.22,0.68],ftBandstop,false,num,den);

    // Alternative 3. Specifying the order explicitely
    // will not ensure 50 dB attenuation in the edges of the stopband,
    // but one can increase filter order:
    //
    //     ButterFilter(10,[0.22,0.68],ftBandstop,false,num,den);

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

Overload 4: function ButterFilter(Order: Integer; CutoffFreq: TDoubleArray; FilterType: TFilterType; Analog: Boolean; sos: TVec; IirFrequencyTransform: TIirFrequencyTransform): Double;

The resulting transfer function is returned in the second order section form stored in the sos variable.

#NameTypeDescription
1OrderInteger
2CutoffFreqTDoubleArray
3FilterTypeTFilterType
4AnalogBoolean
5sosTVecsource TVec
6IirFrequencyTransformTIirFrequencyTransform

Returns: Double

Remarks:

The sos variable can be passed directly to the IirInitBQ filter initialization routine. Second order section form delivers substantially higher numerical stability and range than filtering with num/den form which is used by the IirInit routine.