LinearSystems.Bilinear Method

Overload List

#	Signature	Description
1	void Bilinear(TMtx A, TVec B, TVec C, ref Double D, Double FS)	Apply bilinear transform to a linear system represented in state-space form.
2	void Bilinear(TVec z, TVec p, ref Double k, Double FS, Boolean AddZeros)	Result = bilinear transformation

Overload 1: void Bilinear(TMtx A, TVec B, TVec C, ref Double D, Double FS)

Apply bilinear transform to a linear system represented in state-space form.

#	Name	Type	Description
1	A	TMtx	source TMtx
2	B	TVec	source TVec
3	C	TVec	source TVec
4	D	Double (ref)	output
5	FS	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);
        Vector b = new Vector(0);
        Vector c = new Vector(0);
        double k,Wc,d;
        Matrix A = new Matrix(0,0);
        double FS = 2;

        int Order = 5; //design a fifth order filter.

        IIRFilters.EllipticAnalog(Order,0.2,40,z,p,out k);  //design analog protype
        //passband ripple 0.2dB, stopband attenuation 40dB
        LinearSystems.Bilinear(z,p,ref k,FS,true);  //Sampling frequency  = 2
        Wc = 0.6; //request a cutoff at 0.6 Hz
        LinearSystems.LowpassToLowpassZ(z,p,ref k,Wc, LinearSystems.BilinearUnwarp(1,FS));  //frequency transformation in z-domain
        LinearSystems.ZeroPoleToTransferFun(num,den,z,p,k);

        SignalUtils.FrequencyResponse(num, den, Response, 64, false, TSignalWindowType.wtRectangular, 0); //zero padding set to 64
        MtxVecTee.DrawIt(Response, "Design method 0", false);

//Alternative 1:
    IIRFilters.EllipticAnalog(Order,0.2,40,z,p,out k);  //design analog protype
    LinearSystems.Bilinear(z,p,ref k,FS,true);  //Sampling frequency  = 2
    LinearSystems.ZeroPoleToTransferFun(num,den,z,p,k);
    Wc = 0.6; //request a cutoff at 0.6 Hz
    LinearSystems.LowpassToLowpassZ(num,den,Wc,LinearSystems.BilinearUnwarp(1,FS));  //frequency transformation in z-domain

    SignalUtils.FrequencyResponse(num, den, Response, 64, false, TSignalWindowType.wtRectangular, 0); //zero padding set to 64
    MtxVecTee.DrawIt(Response, "Design method 1", false);

//Alternative 2:
    IIRFilters.EllipticAnalog(Order,0.2,40,z,p,out k);  //design analog protype
    Wc = LinearSystems.BilinearPrewarp(0.6,FS); //request a cutoff at 0.6 Hz
    LinearSystems.LowpassToLowpass(z,p,ref k,Wc);  //frequency transformation in s-domain
    LinearSystems.Bilinear(z,p,ref k, FS,true);  //Sampling frequency  = 2
    LinearSystems.ZeroPoleToTransferFun(num,den,z,p,k);

    SignalUtils.FrequencyResponse(num, den, Response, 64, false, TSignalWindowType.wtRectangular, 0); //zero padding set to 64
    MtxVecTee.DrawIt(Response, "Design method 2", false);

//Alternative 3:
    IIRFilters.EllipticAnalog(Order,0.2,40,z,p,out k);  //design analog protype
    LinearSystems.ZeroPoleToStateSpace(A,b,c,out d,z,p,k);
    Wc = LinearSystems.BilinearPrewarp(0.6,FS); //request a cutoff at 0.6 Hz
    LinearSystems.LowpassToLowpass(A,b,c,ref d,Wc); //frequency transformation in s-domain
    LinearSystems.Bilinear(A,b,c,ref d,2);
    LinearSystems.StateSpaceToZeroPole(z,p,out k,A,b,c,d);
    LinearSystems.ZeroPoleToTransferFun(num,den,z,p,k);

    SignalUtils.FrequencyResponse(num,den,Response,64,false,TSignalWindowType.wtRectangular,0); //zero padding set to 64
    MtxVecTee.DrawIt(Response,"Design method 3",false);
    }

Overload 2: void Bilinear(TVec z, TVec p, ref Double k, Double FS, Boolean AddZeros)

Compute bilinear transformation.

#	Name	Type	Description
1	z	TVec	source TVec
2	p	TVec	source TVec
3	k	Double (ref)	output
4	FS	Double	scalar
5	AddZeros	Boolean

Result: stored in self (calling object)

Remarks:

Apply bilinear transform to transfer function represented in zero-pole form. FS defines the sampling frequency. If z.Length < p.Length the routine will not add zeroes at -1 to match the number of poles unless AddZeros is True.

Bilinear transformation is defined with the mapping: s -> 2/T (1 - z^(-1))/(1 + z^(-1)), T = 1/FS Bilinear transform requires that the amplitude response of a continuous system is piecewise constant and can not be used to transform an analog differentiator to a digital filter (for example). Bilinear transform also does not preserve the impulse or the phase response [1].

References:

[1] Theory and application of digital signal processing, Lawrence R. Rabiner and Bernard Gold. Prentice-Hall, 1975