(New page: Given:<br> <math>e^{2jt} \rightarrow SYSTEM \rightarrow te^{-2jt}</math><br> <math>e^{-2jt} \rightarrow SYSTEM \rightarrow te^{2jt}</math><br> Solve: <br> <math>cos(2t) \rightarrow SYSTEM...) |
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Given:<br> | Given:<br> | ||
| − | <math>e^{2jt} \rightarrow SYSTEM \rightarrow te^{-2jt}</math><br> | + | <math>e^{2jt} \rightarrow SYSTEM \rightarrow te^{-2jt}</math><br><br> |
<math>e^{-2jt} \rightarrow SYSTEM \rightarrow te^{2jt}</math><br> | <math>e^{-2jt} \rightarrow SYSTEM \rightarrow te^{2jt}</math><br> | ||
Solve: <br> | Solve: <br> | ||
| − | <math>cos(2t) \rightarrow SYSTEM \rightarrow ?</math><br> | + | <math>cos(2t) \rightarrow SYSTEM \rightarrow ?</math><br><br> |
| + | At this point, we must use Euler's relation to expand cos(2t) into exponentials. Then, we will be able to use the given inputs and corresponding outputs to come to a conclusion.<br><br> | ||
| + | <math>cos(2t) = \frac{e^{2jt}+e^{-2jt}}{2}</math><br><br> | ||
| + | <math>cos(2t) \rightarrow SYSTEM \rightarrow \frac{1}{2}te^{-2jt}+\frac{1}{2}te^{2jt}</math><br><br> | ||
| + | The above answer can then be simplified. | ||
Latest revision as of 10:09, 17 September 2008
Given:
$ e^{2jt} \rightarrow SYSTEM \rightarrow te^{-2jt} $
$ e^{-2jt} \rightarrow SYSTEM \rightarrow te^{2jt} $
Solve:
$ cos(2t) \rightarrow SYSTEM \rightarrow ? $
At this point, we must use Euler's relation to expand cos(2t) into exponentials. Then, we will be able to use the given inputs and corresponding outputs to come to a conclusion.
$ cos(2t) = \frac{e^{2jt}+e^{-2jt}}{2} $
$ cos(2t) \rightarrow SYSTEM \rightarrow \frac{1}{2}te^{-2jt}+\frac{1}{2}te^{2jt} $
The above answer can then be simplified.
