Difference between revisions of "HW2.D Jeff Kubascik ECE301Fall2008mboutin" - Rhea

Latest revision as of 19:05, 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.