LinearSystems.LowpassToBandpassZ Method

Overload List

#	Signature	Description
1	procedure LowpassToBandpassZ(const Num: TVec; const Den: TVec; Freq: Double; BW: Double; PrototypeFreq: Double);	Apply frequency band transformation from lowpass to bandpass in the z-domain.
2	procedure LowpassToBandpassZ(const z: TVec; const p: TVec; var k: Double; Freq: Double; BW: Double; PrototypeFreq: Double);	The function returns modified z (zeros), p (poles) and k (gain).

Overload 1: procedure LowpassToBandpassZ(const Num: TVec; const Den: TVec; Freq: Double; BW: Double; PrototypeFreq: Double);

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

#	Name	Type	Description
1	Num	TVec
2	Den	TVec
3	Freq	Double	scalar
4	BW	Double	scalar
5	PrototypeFreq	Double	scalar

Result: stored in self (calling object)

Remarks:

Freq is the center frequency of the passband with width BW 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, BW 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) -> (-a2 + a1 z^(-1) - z^(-2))/(1 - a1 z^(-1) + a2 z^(-2)) beta = (cos((w2 + w1)/2))/(cos((w2 - w1)/2)) k1 = cot((w2 - w1)/2)tan(wc/2) a1 = (2 beta k1)/(k1 + 1), a2 = (k1 - 1)/(k1 + 1)

wc - old cutoff frequency
w2 - desired upper cutoff frequency
w1 - desired lower cutoff frequency

References:

[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

Examples

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

procedure TForm1.Button1Click(Sender: TObject);
var z,p,num,den,Response: Vector;
Order: integer;
k,Wc,w1,w2,BW: Double;
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.
    w1 := 0.2; //start of the passband at 0.2Hz.
    w2 := 0.5; //stop of the passband at 0.5Hz.
    Wc := Sqrt(w1*w2); //center frequency of the passband
    BW := w2-w1;  //passband width
    //    LowpassToBandpassZ(z,p,k,Wc,BW,BilinearUnwarp(1));
    //    ZeroPoleToTransferFun(num,den,z,p,k);

    //Alternative:
    //       ...
    ZeroPoleToTransferFun(num,den,z,p,k);
    LowpassToBandpassZ(num,den,Wc,BW,BilinearUnwarp(1));

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

Overload 2: procedure LowpassToBandpassZ(const z: TVec; const p: TVec; var k: Double; Freq: Double; BW: Double; PrototypeFreq: Double);

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

#	Name	Type	Description
1	z	TVec
2	p	TVec
3	k	Double
4	Freq	Double	scalar
5	BW	Double	scalar
6	PrototypeFreq	Double	scalar

Result: stored in self (calling object)