Time Invariant System
A time invariant system means that the output of the system doesn't depend on the time. In other words, there is an input $ \,x(t) $ with an output $ \,y(t) $. If the input is shifted by time $ \,t0 $ so that $ \,x(t-t0) $ yields an output $ \,y(t-t0) $
Example of Time Invariance
Take $ \,x(t) = e^t $. This is run through a time delay, which delays it by t0, then through a system which transforms it to $ \,y(t) = 10x(t) $.
- $ \,x(t) = e^t ---> x(t-t0) = e^{(t-t0)} ---> y(t-t0) = 10e^{(t-t0)} $
Now the signal is run through the system first, then the time delay.
- $ \,x(t) = e^t ---> y1(t) = 10e^t ---> y1(t-t0) = 10e^{(t-t0)} $
$ \,y(t-t0) = y1(t-t0) $.
Thus, the system is time invariant.
