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# Matlab: Discrete Linear Systems

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Discrete Linear Systems using Matlab

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### Matlab: Discrete Linear Systems

1. 1. Discrete Linear Systems<br />
2. 2. 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 />
3. 3. Discrete Linear Systems<br />&gt;&gt; A=[8 -7 6 -5 6 -5 2 -3 2 -1];<br />x=[0 1 0 0 0 0 0 0 0 0 ];<br />B=conv(A,x);<br />subplot(2,3,1); plot(A);<br />subplot(2,3,2); plot(x);<br />subplot(2,3,3); plot(B);<br />x=[0 1 0 0 0 0 0 1 0 0 ];<br />B=conv(A,x);<br />subplot(2,3,4); plot(A);<br />subplot(2,3,5); plot(x);<br />subplot(2,3,6); plot(B);<br />
4. 4. Discrete Linear Systems<br />
5. 5. Discrete-Time System Models<br />Transfer Function<br />Zero-Pole-Gain<br />State-Space<br />Partial Fraction Expansion (Residue Form)<br />Second-Order Sections (SOS)<br />Lattice Structure<br />Convolution Matrix<br />
6. 6. 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 />
7. 7. Discrete-Time System Models<br />The factored or zero-pole-gain form of a transfer function is<br />
8. 8. 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 />
9. 9. Discrete-Time System Models<br />Each transfer function also has a corresponding partial fraction expansion or residue form representation, given by<br />
10. 10. Discrete-Time System Models<br />Any transfer function H(z) has a second-order sections representation<br />
11. 11. 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 />
12. 12. 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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