(Time Invariance)
 
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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 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>.
 
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>.
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[[Image:Timeinvx_ECE301Fall2008mboutin.JPG]]
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One can show a system is time invarient by proving
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[[Image:Timeproof_ECE301Fall2008mboutin.JPG]]
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where <math>y_1(t)\!</math> and <math>y_2(t)\!</math> are equal.
  
 
== Example of a Time Invariant System ==
 
== Example of a Time Invariant System ==
Let <math>y(t)=2x(t)+2\!</math>.  The system is time invarient if for input <math>y(t)=2x(t-t_0)+2\!</math> the response is <math>y(t)=2x(t)+2\!</math>.
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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>.
 
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Proof:
 
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[[Image:Timinvar_ECE301Fall2008mboutin.JPG]]
  
 
== Example of a System that is not Time Invariant ==
 
== Example of a System that is not Time Invariant ==
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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.
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Proof:
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[[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

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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