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== Time Invariant Problem == | == Time Invariant Problem == | ||
| + | <math>Y(t) = x(t - 1)</math> | ||
| + | <math>S_1 = Y(t) = x(t - 1)</math> | ||
| + | |||
| + | <math>S_2 = Y(t) = x(t - t_o)</math> | ||
| + | |||
| + | <math>x(t) -> S1 -> S2 -> x(t - t_o - 1)</math> | ||
| + | |||
| + | <math>x(t) -> S2 -> S1 -> x(t - t_o - 1)</math> | ||
| + | |||
| + | <math> x(t - t_o - 1) = x(t - t_o - 1)</math> | ||
| + | |||
| + | Since they are equal it is time invariant. | ||
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<math> x(t - t_o - 1) - x(1 - t + t_o) =/= x(t - t_o - 1) - x(1 - t - t_o)</math> | <math> x(t - t_o - 1) - x(1 - t + t_o) =/= x(t - t_o - 1) - x(1 - t - t_o)</math> | ||
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| + | Since they are not equal it is time variant. | ||
Latest revision as of 14:17, 12 September 2008
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.
