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| + | [[Homework 2_ECE301Fall2008mboutin]] - [[HW2-A Phil Cannon_ECE301Fall2008mboutin|'''A''']] - [[HW2-B Phil Cannon_ECE301Fall2008mboutin|'''B''']] - [[HW2-C Phil Cannon_ECE301Fall2008mboutin|'''C''']] - [[HW2-D Phil Cannon_ECE301Fall2008mboutin|'''D''']] - [[HW2-E Phil Cannon_ECE301Fall2008mboutin|'''E''']] | ||
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== Time Invariance == | == Time Invariance == | ||
| − | A system is time-invariant if for any | + | A system is time-invariant if for any input <math>x(t)\!</math> and any <math>t_0\!</math> (where <math>t_0\!</math> is a real number) the response to the shifted input <math>x(t-t_0)\!</math> is <math>y(t-t_0)\!</math>. |
| + | <br> | ||
| + | <br> | ||
| + | |||
| + | One can show a system is time invarient by proving | ||
| + | <br> | ||
| + | <br> | ||
| + | [[Image:Timeproof_ECE301Fall2008mboutin.JPG]] | ||
| + | <br> | ||
| + | <br> | ||
| + | where <math>y_1(t)\!</math> and <math>y_2(t)\!</math> are equal. | ||
| + | |||
| + | == Example of a Time Invariant System == | ||
| + | Let <math>y(t)=2x(t)\!</math>. The system is time invarient if for input <math>x(t-t_0)\!</math> the response is <math>2x(t-t_0)\!</math>. | ||
| + | <br> | ||
| + | <br> | ||
| + | Proof: | ||
| + | <br> | ||
| + | [[Image:Timinvar_ECE301Fall2008mboutin.JPG]] | ||
| − | [[Image: | + | == Example of a System that is not Time Invariant == |
| + | Let <math>y(t)=2tx(t)\!</math>. Because the two outputs are not equal, the system is not time invariant. Rather, it is called time variant. | ||
| + | <br> | ||
| + | <br> | ||
| + | Proof: | ||
| + | <br> | ||
| + | [[Image:Timvar_ECE301Fall2008mboutin.jpg]] | ||
Latest revision as of 16:05, 11 September 2008
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
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:
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:
