Time Invariance
Definition of a time invariant system: "A system is Time Invariant if the behavior and characteristics of the system are fixed over time... Specifically, a system is time invariant if a time shift in the input signal results in an identical time shift in the output signal." - (Oppenheim Willsky pgs. 50-51)
Example: Let $ y1(t) = sin[x1(t)] $
$ x2(t) = x1(t-t0) $ $ y2(t) = sin[x2(t)] = sin[x1(t-t0)] $
Therefore it can be shown that:
$ y1(t-t0) = sin[x1(t-t0)] $
Definition of a time variant system: Any system that does not follow the characteristics of a time invariant system is considered to be Time Variant. Specifically, a system is time variant if a time shift in the input signal results in some different time shift in the output signal.
Example: Let $ y1(t) = x1(2t) <math> x2(t) = x1(t-t0) $
$ y2(t) = x2(2t) = x1(2(t-t0)) $
Therefore it can be shown that:
$ y1(t-t0) != x1(2t-2t0) $
