(New page: == 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 sys...) |
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<math> x2(t) = x1(t-t0) </math> | <math> x2(t) = x1(t-t0) </math> | ||
<math> y2(t) = sin[x2(t)] = sin[x1(t-t0)] </math> | <math> y2(t) = sin[x2(t)] = sin[x1(t-t0)] </math> | ||
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Therefore it can be shown that: | Therefore it can be shown that: | ||
| + | |||
<math> y1(t-t0) = sin[x1(t-t0)] </math> | <math> y1(t-t0) = sin[x1(t-t0)] </math> | ||
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'''Example:''' Let <math> y1(t) = x1(2t) | '''Example:''' Let <math> y1(t) = x1(2t) | ||
<math> x2(t) = x1(t-t0) </math> | <math> x2(t) = x1(t-t0) </math> | ||
| − | <math> y2(t) = x2(2t) = x1(2(t-t0)) | + | <math> y2(t) = x2(2t) = x1(2(t-t0)) </math> |
| + | |||
Therefore it can be shown that: | Therefore it can be shown that: | ||
| − | <math> y1(t-t0) != x1(2t-2t0) | + | |
| + | <math> y1(t-t0) != x1(2t-2t0) </math> | ||
Revision as of 04:16, 19 September 2008
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) $
