Definition
A system is subjected to a time delay then fed into another system. The result of these two actions is recorded. The same (initial) system is then subjected first to another system (the same one as used to obtain the previous result) and then subjected to a time delay (again the same one used to obtain the first result). The results of these actions is again recorded and compared to the first recorded result. If the results are the same the system is time invariant. If the results are different the system in time variant.
Example
An example of a time invariant system is:
$ \ S_{1} = 2x(t + 3) + x(t - 8) $
$ \ S_{2} = x(t - t_{0}) $
$ \ S_{1} \rightarrow S_{2} = 2x(t - t_{0} + 3) + x(t - t_{0} - 8) $
$ \ S_{2} \rightarrow S_{1} = 2x(t - t_{0} + 3) + x(t - t_{0} - 8) $
Since the results are the same the system is time invariant.
An example of a time variant system is:
$ \ S_{1} = x(-t + 3) - x(-t - 8) $
$ \ S_{2} = x(t - t_{0}) $
$ \ S_{1} \rightarrow S_{2} = x(-t + t_{0} + 3) - x(-t + t_{0} - 8) $
$ \ S_{2} \rightarrow S_{1} = x(-t - t_{0} + 3) - x(-t - t_{0} - 8) $
Since the results are different they system is time variant.
