Time invariance is where the output is not effected by time. As the book puts it the behavior and characteristics of the system are fixed over time.
Time Invariant Problem
$ Y(t) = x(t - 1) $
$ S_1 = Y(t) = x(t - 1) $
$ S_2 = Y(t) = x(t - t_o) $
$ x(t) -> S1 -> S2 -> x(t - t_o - 1) $
$ x(t) -> S2 -> S1 -> x(t - t_o - 1) $
$ x(t - t_o - 1) = x(t - t_o - 1) $
Since they are equal it is time invariant.
Time Variant Problem
(Time variant problem was taken from the in class "exercise" section I posted)
$ Y(t) = x(t - 1) - x(1 - t) $
$ S_1 = Y(t) = x(t - 1) - x(1 - t) $
$ S_2 = Y(t) = x(t - t_o) $
$ x(t) -> S1 -> S2 -> x(t - t_o - 1) - x(1 - t + t_o) $
$ x(t) -> S2 -> S1 -> x(t - t_o - 1) - x(1 - t - t_o) $
$ x(t - t_o - 1) - x(1 - t + t_o) =/= x(t - t_o - 1) - x(1 - t - t_o) $
Since they are not equal it is time variant.
