LinearSystems.LowpassToLowpassZ Method

Overload List

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

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

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

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

Result: stored in self (calling object)

Remarks:

Freq is the cutoff frequency 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 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) -> (z^(-1) - a)/(1 - a z^(-1)) a = (sin((wc - wn)/2))/(sin((wc + wn)/2))

wc - old cutoff frequency
wn - new (desired) 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 LowpassToLowpassZ(TVec z, TVec p, ref Double k, Double Freq, 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	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 fifth order filter.

        IIRFilters.EllipticAnalog(Order,0.1,30, z, p, out k);  //design analog protype
        LinearSystems.Bilinear(z, p, ref k, FS,true);
        Wc = 0.5;
        LinearSystems.LowpassToLowpassZ(z, p, ref k, Wc, 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.LowpassToLowpassZ(num,den, Wc, 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);
    }