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# Lecture 1: Zero-Input Response

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Schedule :: Intro :: Signal Models for Discrete-Time :: Signal Operations :: Difference Equations :: Zero-Input :: Zero-State :: Natural & Forced :: System Stability

When there is no input, we can write

or

(What happened for continuous time?) Similar. Let's try a simply case to get started:

Try a solution . (Comment on the difference from earlier case.) Substitute in and show that it works.

This works in the general case: subs. and show that it works. Substituting gives

or

For an interesting (nontrivial) solution, we will look for roots of

Write as

is thus the characteristic polynomial, and we look at its roots. The roots are , .

As before, we take all possible solutions in a linear combination:

Repeated roots: If

then

So how do things behave as a function of root location. (0.8, -0.8, -.5, 1.2, (0.8), 1, etc.) Same old, same old.

Complex roots: With roots , then

with and arbitrary constants (same steps as before).

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 16). Lecture 1: Zero-Input Response. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/1_4node4.html. This work is licensed under a Creative Commons License