A time-invariant system
For any input signal x(t), a system yelids y(t). Now, suppose input signal shifted t0, x(t-t0). Then output signal also shifted t0, y(t-t0). Then we can say a system is time-invariant.
Example of a tume-invariant system
x(t) = $ e^t $
Output signal y(t) can be $ 10e^t $ by system
Prove.
$ e^t $ is changed to $ e^{(t-t0)} $ by time delay.
$ e^{(t-t0)} -> 10e^{(t-t0)} $ by system.
$ e^t -> 10e^t $ by system.
$ 10e^t -> 10e^{(t-t0)} $ by time delay.
The output signals are same. Then we can say that the system is time-invariant.
Example of a time-variant system
Input signal is $ x(t). $
Output signal y(t) can be $ x(2t) $ by system.
Prove.
$ x(t) $ is changed to $ x{(t-t0)} $ by time delay.
$ x{(t-t0)} -> x{(2t-t0)} $ by system.
$ x(t) $ is changed to $ x(2t) $ by system.
$ x(2t) is changed to x{(2(t-t0))} $ by time delay.
The output signals are same. Then we can say that the system is time-variant.
