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(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.
