LinearSystems.LowpassToBandpassZ Method

Overload List

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

Overload 1: void LowpassToBandpassZ(TVec Num, TVec Den, Double Freq, Double BW, Double PrototypeFreq)

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

#	Name	Type	Description
1	Num	TVec	source TVec
2	Den	TVec	source 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

Overload 2: void LowpassToBandpassZ(TVec z, TVec p, ref Double k, Double Freq, Double BW, Double PrototypeFreq)

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

#	Name	Type	Description
1	z	TVec	source TVec
2	p	TVec	source TVec
3	k	Double (ref)	output
4	Freq	Double	scalar
5	BW	Double	scalar
6	PrototypeFreq	Double	scalar

Result: stored in self (calling object)

Examples

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);
    double k, Wc;
    double FS = 2;
    int Order = 4; //design a fourth order filter.

    IIRFilters.EllipticAnalog(Order,0.1,30, z, p, out k);  //design analog protype
    LinearSystems.Bilinear(z, p, ref k, FS,true);
    double w1 = 0.2; //start of the passband at 0.2Hz.
    double w2 = 0.5; //stop of the passband at 0.5Hz.
    Wc = Math.Sqrt(w1*w2); //center frequency of the passband
    double BW = w2-w1;  //passband width
    LinearSystems.LowpassToBandpassZ(z, p, ref k, Wc, BW, LinearSystems.BilinearUnwarp(1,FS));
    LinearSystems.ZeroPoleToTransferFun(num,den, z, p, k);
    SignalUtils.FrequencyResponse(num, den, Response, 64, false, TSignalWindowType.wtRectangular, 0); //zero padding set to 64

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

    MtxVecTee.DrawIt(Response, "Frequency response", false);
    //MtxVecTee.DrawIt(20 * MtxExpr.Log10(MtxExpr.Abs(Response)), "Magnitude", false);
    //MtxVecTee.DrawIt(MtxExpr.PhaseSpectrum(Response) * (180 / Math.PI), "Phase", false);
}