Homework 2_ECE301Fall2008mboutin - A - B - C - D - E

Time Invariance

A system is time-invariant if for any input $ x(t)\! $ and any $ t_0\! $ (where $ t_0\! $ is a real number) the response to the shifted input $ x(t-t_0)\! $ is $ y(t-t_0)\! $.

One can show a system is time invarient by proving

Timeproof ECE301Fall2008mboutin.JPG

where $ y_1(t)\! $ and $ y_2(t)\! $ are equal.

Example of a Time Invariant System

Let $ y(t)=2x(t)\! $. The system is time invarient if for input $ x(t-t_0)\! $ the response is $ 2x(t-t_0)\! $.

Proof:
Timinvar ECE301Fall2008mboutin.JPG

Example of a System that is not Time Invariant

Let $ y(t)=2tx(t)\! $. Because the two outputs are not equal, the system is not time invariant. Rather, it is called time variant.

Proof:
Timvar ECE301Fall2008mboutin.jpg

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