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Matlab: Discrete Linear Systems
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Matlab: Discrete Linear Systems

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Matlab: Discrete Linear Systems

Matlab: Discrete Linear Systems

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Matlab: Discrete Linear Systems Matlab: Discrete Linear Systems Presentation Transcript

  • Discrete Linear Systems
  • Discrete Linear Systems
    A discrete linear system is a digital implementation of a linear time-invariant system. A linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case.
  • Discrete Linear Systems
    >> A=[8 -7 6 -5 6 -5 2 -3 2 -1];
    x=[0 1 0 0 0 0 0 0 0 0 ];
    B=conv(A,x);
    subplot(2,3,1); plot(A);
    subplot(2,3,2); plot(x);
    subplot(2,3,3); plot(B);
    x=[0 1 0 0 0 0 0 1 0 0 ];
    B=conv(A,x);
    subplot(2,3,4); plot(A);
    subplot(2,3,5); plot(x);
    subplot(2,3,6); plot(B);
  • Discrete Linear Systems
  • Discrete-Time System Models
    Transfer Function
    Zero-Pole-Gain
    State-Space
    Partial Fraction Expansion (Residue Form)
    Second-Order Sections (SOS)
    Lattice Structure
    Convolution Matrix
  • Discrete-Time System Models
    The transfer functionis a basic z-domain representation of a digital filter, expressing the filter as a ratio of two polynomials.
  • Discrete-Time System Models
    The factored or zero-pole-gain form of a transfer function is
  • Discrete-Time System Models
    It is always possible to represent a digital filter, or a system of difference equations, as a set of first-order difference equations. In matrix or state-space form, we can write the equations as
  • Discrete-Time System Models
    Each transfer function also has a corresponding partial fraction expansion or residue form representation, given by
  • Discrete-Time System Models
    Any transfer function H(z) has a second-order sections representation
  • Discrete-Time System Models
    The function latc2tf calculates the polynomial coefficients for a filter from its lattice (reflection) coefficients. Given the reflection coefficient vector k(above), the corresponding polynomial form is
    b = latc2tf(k)
    b = 1.0000 0.6149 0.9899 -0.0000 0.0031 -0.0082
    The lattice or lattice/ladder coefficients can be used to implement the filter using the function latcfilt.
  • Discrete-Time System Models
    Given any vector, the toolbox function convmtx generates a matrix whose inner product with another vector is equivalent to the convolution of the two vectors. The generated matrix represents a digital filter that you can apply to any vector of appropriate length; the inner dimension of the operands must agree to compute the inner product.
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