Time Invariance
A system is called time invariant if shifting it's input signal in time results in the same time shift propagated to its output.
Example of a Time Invariant System
Given the system $ y(t) = 10x(t) $:
First, apply the time delay to the input $ x(t) $: $ w_1(t) = x(t-t_0) $
Then feed $ w(t) $ into the system: $ z_1(t) = 10x(t-t_0) $
Now, try using the system first: $ w_2(t)=10x(t) $
Applying the time delay: $ z_2(t) = 10x(t-t_0) $
Since $ z_1(t) = z_2(t) $ the system is Time Invariant.
Example of a Time Variant System
Given the system $ y(t) = 6x(4t + 2) $:
First, apply the time delay to the input $ x(t) $: $ w_1(t) = x(t-t_0) $
Then feed $ w(t) $ into the system: $ z_1(t) = 6x(4t - t_0 + 2) $
Now, try using the system first: $ w_2(t)=6x(4t + 2) $
Applying the time delay: $ z_2(t) = 6x(4t - 4t_0 + 2) $
Since $ z_1(t) \ne z_2(t) $ the system is NOT Time Invariant.
