Lay 4.8
Applications to Difference Equations

Data:
Color Names

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Visualizing a Sequence (Signal)

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Lay 4.8, Example 2,
Linear Independence

Show that three sequences are linearly independent.

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Lay 4.8, Example 3,
Filtering Signals

First Input Signal

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Conclusion : The output was shifted backward by one term.

Second Input Signal

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Conclusion : This filter killed the higher frequency signal.

Lay 4.8, Example 4,
Solutions of a Homogeneous Difference Equation

Define the difference equation.

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Define the auxiliary equation, and find its roots.

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Check these roots.

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Lay 4.8, Example 5,
Solution Space of a Homogeneous Equation

Find a basis for the set of all solutions of our homogeneous equation.

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Lay 4.8, Example 6,
Non-homogeneous Difference Equation

Consider a non-homogeneous difference equation.

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Verify a particular solution.

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Solve the associated homogeneous equation.
Define the auxiliary equation, and find its roots.

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Find a basis for the set of all solutions of the homogeneous equation.

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Assemble the solutions to the non-homogeneous equation by adding
a single particular solution of the non-homogeneous equation to
the general solution of the homogeneous equation.

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Lay 4.8, Example 7,
Reduction to Systems of First-Order Equations

Write the following equation as a first-order system.

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Define x[k], and observe.

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So, x[k + 1] == a.x[k].

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