Lay 4.8
Applications to Difference Equations
Data:
Color Names
Visualizing a Sequence (Signal)
Lay 4.8, Example 2,
Linear Independence
Show that three sequences are linearly independent.
Lay 4.8, Example 3,
Filtering Signals
First Input Signal
Conclusion : The output was shifted backward by one term.
Second Input Signal
Conclusion : This filter killed the higher frequency signal.
Lay 4.8, Example 4,
Solutions of a Homogeneous Difference Equation
Define the difference equation.
Define the auxiliary equation, and find its roots.
Check these roots.
Lay 4.8, Example 5,
Solution Space of a Homogeneous Equation
Find a basis for the set of all solutions of our homogeneous equation.
Lay 4.8, Example 6,
Non-homogeneous Difference Equation
Consider a non-homogeneous difference equation.
Verify a particular solution.
Solve the associated homogeneous equation.
Define the auxiliary equation, and find its roots.
Find a basis for the set of all solutions of the homogeneous equation.
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.
Lay 4.8, Example 7,
Reduction to Systems of First-Order Equations
Write the following equation as a first-order system.
Define x[k], and observe.
So, x[k + 1] == a.x[k].
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