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].

Created by Wolfram Mathematica 6.0 (29 February 2008) |