(New page: As mentioned in the problem, the response of :<math>e^{2jt}\,</math> is :<math>te^{-2jt}\,</math> Suppose we let <math>y(t)</math> be the response of <math>x(t)</math>, in order to...) |
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| − | Therefore, the | + | Therefore, using the Euler formula: |
| − | :<math> | + | :<math>cos(2t) = \frac{1}{2}(e^{-2jt}+e^{2jt})\,</math> |
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| + | and the two responses mentioned above, the response of <math>cos(2t)</math> is: | ||
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| + | :<math>\frac{1}{2}(te^{2jt}+te^{-2jt}) = tcos(2t)\,</math> | ||
Latest revision as of 20:39, 18 September 2008
As mentioned in the problem, the response of
- $ e^{2jt}\, $
is
- $ te^{-2jt}\, $
Suppose we let $ y(t) $ be the response of $ x(t) $, in order to make $ x(t) $ produce the output corresponding to $ y(t) $, we need to multiply the input by $ t $ and make the $ t $ of $ x $ negative. ie.
- $ y(t) = tx(-t)\, $
This can be confirmed by the second condition, which is
- $ te^{2jt}\, $
is the response of
- $ e^{-2jt}\, $
Therefore, using the Euler formula:
- $ cos(2t) = \frac{1}{2}(e^{-2jt}+e^{2jt})\, $
and the two responses mentioned above, the response of $ cos(2t) $ is:
- $ \frac{1}{2}(te^{2jt}+te^{-2jt}) = tcos(2t)\, $
