(New page: Since <math>e^{2jt} \rightarrow system \rightarrow te^{-2jt}\!</math><br> <math>e^{-2jt} \rightarrow system \rightarrow te^{2jt}\!</math><br>) |
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<math>e^{2jt} \rightarrow system \rightarrow te^{-2jt}\!</math><br> | <math>e^{2jt} \rightarrow system \rightarrow te^{-2jt}\!</math><br> | ||
<math>e^{-2jt} \rightarrow system \rightarrow te^{2jt}\!</math><br> | <math>e^{-2jt} \rightarrow system \rightarrow te^{2jt}\!</math><br> | ||
| + | <br> | ||
| + | and using euler formula, we can replace exponent expressions with | ||
| + | |||
| + | Euler's formula: <math>e^{iy}=cos(y)+isin(y)\,</math><br> | ||
| + | <br> | ||
| + | <br> | ||
| + | They will change into:<br> | ||
| + | <math>e^{(2jt)} = cos{(2t)} + jsin{(2t)} --> system --> t*{(cos{(2t)} - jsin{(2t)})}\,</math><br> | ||
| + | <math>e^{(-2jt)} = cos{(2t)} - jsin{(2t)} --> system --> t*{(cos{(2t)} + jsin{(2t)})}\,</math><br><br> | ||
| + | |||
| + | It indicates that the system changes the expression on the middle of cos and sin. | ||
| + | While cos(2t) function can be found with this equation: | ||
| + | <br> | ||
| + | <math>\frac{1}{2}e^{(2jt)} + \frac{1}{2}e^{(-2jt)} =</math><br><math> \frac{1}{2}(cos{(2t)} + jsin{(2t)}) + \frac{1}{2}(cos{(2t)} - jsin{(2t)}) = cos{(2t)}</math><br><br> | ||
| + | |<br> | ||
| + | |<br> | ||
| + | |<br> | ||
| + | V<br><br> | ||
| + | <math>System \,</math><br><br> | ||
| + | |<br> | ||
| + | |<br> | ||
| + | |<br> | ||
| + | V<br> | ||
| + | <br> | ||
| + | <math>\frac{1}{2}(t*{(cos{(2t)} - jsin{(2t)})}) + \frac{1}{2}t*{(cos{(2t)} + jsin{(2t)})} = \frac{1}{2}tcos{(2t)} + \frac{1}{2}tcos{(2t)} = tcos(2t)</math><br> | ||
| + | <br> | ||
Latest revision as of 10:39, 16 September 2008
Since
$ e^{2jt} \rightarrow system \rightarrow te^{-2jt}\! $
$ e^{-2jt} \rightarrow system \rightarrow te^{2jt}\! $
and using euler formula, we can replace exponent expressions with
Euler's formula: $ e^{iy}=cos(y)+isin(y)\, $
They will change into:
$ e^{(2jt)} = cos{(2t)} + jsin{(2t)} --> system --> t*{(cos{(2t)} - jsin{(2t)})}\, $
$ e^{(-2jt)} = cos{(2t)} - jsin{(2t)} --> system --> t*{(cos{(2t)} + jsin{(2t)})}\, $
It indicates that the system changes the expression on the middle of cos and sin.
While cos(2t) function can be found with this equation:
$ \frac{1}{2}e^{(2jt)} + \frac{1}{2}e^{(-2jt)} = $
$ \frac{1}{2}(cos{(2t)} + jsin{(2t)}) + \frac{1}{2}(cos{(2t)} - jsin{(2t)}) = cos{(2t)} $
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$ System \, $
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V
$ \frac{1}{2}(t*{(cos{(2t)} - jsin{(2t)})}) + \frac{1}{2}t*{(cos{(2t)} + jsin{(2t)})} = \frac{1}{2}tcos{(2t)} + \frac{1}{2}tcos{(2t)} = tcos(2t) $
