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Discrete Linear Systems<br />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.<br />
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Discrete-Time System Models<br />The transfer functionis a basic z-domain representation of a digital filter, expressing the filter as a ratio of two polynomials.<br />
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Discrete-Time System Models<br />The factored or zero-pole-gain form of a transfer function is<br />
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Discrete-Time System Models<br />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<br />
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Discrete-Time System Models<br />Each transfer function also has a corresponding partial fraction expansion or residue form representation, given by<br />
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Discrete-Time System Models<br />Any transfer function H(z) has a second-order sections representation<br />
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Discrete-Time System Models<br />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<br />b = latc2tf(k)<br /> b = 1.0000 0.6149 0.9899 -0.0000 0.0031 -0.0082 <br />The lattice or lattice/ladder coefficients can be used to implement the filter using the function latcfilt.<br />
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Discrete-Time System Models<br />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.<br />
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