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Time Invariant. | Time Invariant. | ||
| − | A system is time-invariant as long as the system shows certain fixed behaviors over time. For example, when x(t) shifts by a constant, y(t) should shift by the same constant.<br> | + | A system is time-invariant as long as the system shows certain fixed behaviors over time. For example, when x(t) shifts by a constant, y(t) should shift by the same constant.<br><br> |
<math> y = x(t) </math><br> | <math> y = x(t) </math><br> | ||
| − | <math> x2 = x(t-t0) </math><br> | + | <math> x2 = x(t-t0) </math><br><br> |
Then<br> | Then<br> | ||
<math> y(t-t0) = x(t-t0)</math> | <math> y(t-t0) = x(t-t0)</math> | ||
<br> | <br> | ||
| − | Also, the following should satisfy. <br> | + | Also, the following should satisfy. <br><br> |
<math> y = x(t) </math><br> | <math> y = x(t) </math><br> | ||
<math> x2 = x(2t) </math><br> | <math> x2 = x(2t) </math><br> | ||
Then<br> | Then<br> | ||
<math> y(2t) = x(2t) </math><br> | <math> y(2t) = x(2t) </math><br> | ||
| + | |||
| + | == Example of Time-Invariant System == | ||
| + | <br> | ||
| + | <math> y = sin(x) = sin(x(t)) </math><br><br> | ||
| + | When x(t) shifts by a constant t0, <br> | ||
| + | <math> x2(t) = x(t-t0) </math><br><br> | ||
| + | Then y(t) responds accordingly to the shift. <br> | ||
| + | <math> y2(t) = sin(x2) = sin(x(t-t0)) </math><br> | ||
| + | While maintaining, <math> y2(t) = y(t-t0) </math><br> | ||
| + | |||
| + | == Example of Time-Variant System == | ||
| + | <br> | ||
| + | <math> y = t * sin(x) = t * sin(x(t)) </math><br><br> | ||
| + | When x(t) shifts by a constant t0, <br> | ||
| + | <math> x2(t) = x(t-t0) </math><br><br> | ||
| + | Then y(t) does not respond accordingly to the shift. <br> | ||
| + | <math> y2(t) = t * sin(x2) = t * sin(x(t-t0)) </math><br> | ||
| + | Which is not equal to <math> y2(t) = y(t-t0) = (t - t0) * sin(x(t-to)) </math><br> | ||
Latest revision as of 14:57, 11 September 2008
Time Invariant.
A system is time-invariant as long as the system shows certain fixed behaviors over time. For example, when x(t) shifts by a constant, y(t) should shift by the same constant.
$ y = x(t) $
$ x2 = x(t-t0) $
Then
$ y(t-t0) = x(t-t0) $
Also, the following should satisfy.
$ y = x(t) $
$ x2 = x(2t) $
Then
$ y(2t) = x(2t) $
Example of Time-Invariant System
$ y = sin(x) = sin(x(t)) $
When x(t) shifts by a constant t0,
$ x2(t) = x(t-t0) $
Then y(t) responds accordingly to the shift.
$ y2(t) = sin(x2) = sin(x(t-t0)) $
While maintaining, $ y2(t) = y(t-t0) $
Example of Time-Variant System
$ y = t * sin(x) = t * sin(x(t)) $
When x(t) shifts by a constant t0,
$ x2(t) = x(t-t0) $
Then y(t) does not respond accordingly to the shift.
$ y2(t) = t * sin(x2) = t * sin(x(t-t0)) $
Which is not equal to $ y2(t) = y(t-t0) = (t - t0) * sin(x(t-to)) $
