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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.
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Discrete-Time System Models Transfer Function Zero-Pole-Gain State-Space Partial Fraction Expansion (Residue Form) Second-Order Sections (SOS) Lattice Structure Convolution Matrix
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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.
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Discrete-Time System Models The factored or zero-pole-gain form of a transfer function is
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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
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Discrete-Time System Models Each transfer function also has a corresponding partial fraction expansion or residue form representation, given by
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Discrete-Time System Models Any transfer function H(z) has a second-order sections representation
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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.
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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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