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== Definition of Time Invariance == | == Definition of Time Invariance == | ||
| − | + | A system <math>\,s(t)\,</math> is called time invariant if for any input signal <math>\,x(t)\,</math> yielding output signal <math>\,y(t)\,</math> and for any <math>\,t_o\in\mathbb{R}\,</math>, the response to <math>\,x(t-t_o)\,</math> is <math>\,y(t-t_o)\,</math>. | |
== Example of a Time Invariant System == | == Example of a Time Invariant System == | ||
| − | + | The following system is time invariant: | |
| + | |||
| + | <math>\,s(t)=2x(t-3)\,</math> | ||
| + | |||
| + | |||
| + | '''Proof:''' | ||
| + | |||
| + | We have a function <math>\,x(t)\,</math>. | ||
| + | |||
| + | After applying the function to the system <math>\,s(t)\,</math>, we get: | ||
| + | |||
| + | <math>\,y(t)=2x(t-3)\,</math> | ||
| + | |||
| + | Thus, | ||
| + | |||
| + | <math>\,y(t-t_o)=\,</math> | ||
| + | |||
| + | <math>\,2x((t-t_o)-3)=\,</math> | ||
| + | |||
| + | <math>\,2x(t-t_o-3)\,</math> | ||
| + | |||
| + | |||
| + | Now, apply <math>\,x(t-t_o)\,</math> to the system <math>\,s(t)\,</math>: | ||
| + | |||
| + | <math>\,2x((t-3)-t_o)\,</math> | ||
| + | |||
| + | <math>\,2x(t-3-t_o)\,</math> | ||
| + | |||
| + | |||
| + | Since these two are equal | ||
| + | |||
| + | <math>\,2x(t-t_o-3)=2x(t-3-t_o)\,</math> | ||
| + | |||
| + | the system is time invariant. | ||
== Example of a Time Variant System == | == Example of a Time Variant System == | ||
| − | + | The following system is time variant: | |
| + | |||
| + | <math>\,s(t)=2x(3t-3)\,</math> | ||
| + | |||
| + | |||
| + | '''Proof:''' | ||
| + | |||
| + | We have a function <math>\,x(t)\,</math>. | ||
| + | |||
| + | After applying the function to the system <math>\,s(t)\,</math>, we get: | ||
| + | |||
| + | <math>\,y(t)=2x(3t-3)\,</math> | ||
| + | |||
| + | Thus, | ||
| + | |||
| + | <math>\,y(t-t_o)=\,</math> | ||
| + | |||
| + | <math>\,2x(3(t-t_o)-3)=\,</math> | ||
| + | |||
| + | <math>\,2x(3t-3t_o-3)\,</math> | ||
| + | |||
| + | |||
| + | Now, apply <math>\,x(t-t_o)\,</math> to the system <math>\,s(t)\,</math>: | ||
| + | |||
| + | <math>\,2x((3t-3)-t_o)\,</math> | ||
| + | |||
| + | <math>\,2x(3t-3-t_o)\,</math> | ||
| + | |||
| + | |||
| + | Since these two are not equal | ||
| + | |||
| + | <math>\,2x(3t-3t_o-3)\not= 2x(3t-3-t_o)\,</math> | ||
| + | |||
| + | the system is time variant. | ||
Latest revision as of 20:05, 11 September 2008
Definition of Time Invariance
A system $ \,s(t)\, $ is called time invariant if for any input signal $ \,x(t)\, $ yielding output signal $ \,y(t)\, $ and for any $ \,t_o\in\mathbb{R}\, $, the response to $ \,x(t-t_o)\, $ is $ \,y(t-t_o)\, $.
Example of a Time Invariant System
The following system is time invariant:
$ \,s(t)=2x(t-3)\, $
Proof:
We have a function $ \,x(t)\, $.
After applying the function to the system $ \,s(t)\, $, we get:
$ \,y(t)=2x(t-3)\, $
Thus,
$ \,y(t-t_o)=\, $
$ \,2x((t-t_o)-3)=\, $
$ \,2x(t-t_o-3)\, $
Now, apply $ \,x(t-t_o)\, $ to the system $ \,s(t)\, $:
$ \,2x((t-3)-t_o)\, $
$ \,2x(t-3-t_o)\, $
Since these two are equal
$ \,2x(t-t_o-3)=2x(t-3-t_o)\, $
the system is time invariant.
Example of a Time Variant System
The following system is time variant:
$ \,s(t)=2x(3t-3)\, $
Proof:
We have a function $ \,x(t)\, $.
After applying the function to the system $ \,s(t)\, $, we get:
$ \,y(t)=2x(3t-3)\, $
Thus,
$ \,y(t-t_o)=\, $
$ \,2x(3(t-t_o)-3)=\, $
$ \,2x(3t-3t_o-3)\, $
Now, apply $ \,x(t-t_o)\, $ to the system $ \,s(t)\, $:
$ \,2x((3t-3)-t_o)\, $
$ \,2x(3t-3-t_o)\, $
Since these two are not equal
$ \,2x(3t-3t_o-3)\not= 2x(3t-3-t_o)\, $
the system is time variant.
