Defination
Time Invariant system is a system that doesn't depend on the time when a signal is inputed into the system.
In other words, the system doesn't have the variable $ t \, $ while the input is undergoing transformation.
Time invariant system : $ y(t) = 2x(t) \, $
Time variant system : $ y(t) = t^2x(t) \, $
Example of Time invariant system
$ y(t) = 2x(t) \, $
$ x_1(t) \, $ undergoes a time delay before it is inputted into the system : $ x_1(t) = x(t - \delta) \, $
It is then inputted into the system, $ y_1(t) = 2x_1(t) = 2x(t - \delta) \, $
If the system is shifted after the signal undergoes the transformation, then $ y_2(t) = y(t - \delta) = 2x(t - \delta) = y_1(t)\, $
The signal doesn't depend on the time when the signal is being inputted, thus it's time invariant
Example of time variant system
$ y(t) = t^2x(t) \, $
$ x_1(t) \, $ undergoes a time delay before it is inputted into the system : $ x_1(t) = x(t - \delta) \, $
It is then inputted into the system, $ y_1(t) = t^2x_1(t) = t^2x(t - \delta) \, $
If the system is shifted after the signal undergoes the transformation, then $ y_2(t) = y(t - \delta) = (t - \delta)^2x(t - \delta) \neq y_1(t)\, $
As the signal depends on the time when the signal is being inputted, it is a time variant system.
