LinearSystems::Bilinear Function

LinearSystems Namespace

Overload List

#SignatureDescription
1void Bilinear(TVec *z, TVec *p, double &k, double FS = 2, bool AddZeros = true);Compute bilinear transformation.
2void Bilinear(TMtx *A, TVec *B, TVec *C, double &D, double FS = 2);Apply bilinear transform to a linear system represented in state-space form.

Overload 1: void Bilinear(TVec *z, TVec *p, double &k, double FS = 2, bool AddZeros = true);

Compute bilinear transformation.

#NameTypeDescription
1zTVec *
2pTVec *
3kdouble &
4FS = 2double
5AddZeros = truebool
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

Examples
#include "MtxExpr.h"
#include "Units.IIrFilters.h"
#include "Units.LinearSystems.h"
#include "Units.SignalUtils.h"

using namespace Dew;
using namespace Dew::Math;
using namespace Dew::Signal;
using namespace Dew::Signal::Units::IIRFilters;
using namespace Dew::Signal::Units::LinearSystems;
using namespace Dew::Signal::Units::SignalUtils;

void Example()
{
    Vector z, p, num, den, Response, b, c;
    int Order;
    double k, Wc, d;
    Matrix A;

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

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

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

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

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

    FrequencyResponse(num, den, Response, 64); //zero padding set to 64
}
Declared in Dew::Signal::Units::LinearSystems · Dew.Signal/Units.LinearSystems.h · Cross-compiler

Overload 2: void Bilinear(TMtx *A, TVec *B, TVec *C, double &D, double FS = 2);

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

#NameTypeDescription
1ATMtx *
2BTVec *
3CTVec *
4Ddouble &
5FS = 2double
Examples
#include "MtxExpr.h"
#include "Units.IIrFilters.h"
#include "Units.LinearSystems.h"
#include "Units.SignalUtils.h"

using namespace Dew;
using namespace Dew::Math;
using namespace Dew::Signal;
using namespace Dew::Signal::Units::IIRFilters;
using namespace Dew::Signal::Units::LinearSystems;
using namespace Dew::Signal::Units::SignalUtils;

void Example()
{
    Vector z(0);
    Vector p(0);
    Vector num(0);
    Vector den(0);
    Vector Response(0);
    Vector b(0);
    Vector c(0);
    double k, Wc, d;
    Matrix A(0, 0);
    double FS = 2;

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

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

    FrequencyResponse(num, den, Response, 64, false, TSignalWindowType::wtRectangular, 0); //zero padding set to 64

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

    FrequencyResponse(num, den, Response, 64, false, TSignalWindowType::wtRectangular, 0); //zero padding set to 64

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

    FrequencyResponse(num, den, Response, 64, false, TSignalWindowType::wtRectangular, 0); //zero padding set to 64

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

    FrequencyResponse(num, den, Response, 64, false, TSignalWindowType::wtRectangular, 0); //zero padding set to 64
}
See Also: LinearSystems::MatchedZTransform, LinearSystems::LowpassToLowpass, LinearSystems::LowpassToLowpassZ, LinearSystems::BilinearUnwarp, LinearSystems::BilinearPrewarp
Declared in Dew::Signal::Units::LinearSystems · Dew.Signal/Units.LinearSystems.h · Cross-compiler