(New page: <math>\,cos(2t)</math> can be written as <math>\ {e^{-2jt} + e^{2jt} \over 2}</math> based on Euler's forumla: :<math>\cos x = \mathrm{Re}\{e^{ix}\} ={e^{ix} + e^{-ix} \over 2}</math> Sin...) |
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| − | :<math>\frac{e^{2t}}{2} \to t | + | :<math>\frac{e^{2t}}{2} \to t\cdot\frac{e^{-2t}}{2}</math> and <math>\frac{e^{-2t}}{2} \to t\cdot\frac{e^{2t}}{2}</math> |
| − | :<math>t | + | :<math>t\cdot\frac{e^{-2jt}}{2} + t\cdot\frac{e^{2jt}}{2}</math> |
| − | :<math> t | + | :<math> t\cdot\frac{e^{-2jt}+e^{2jt}}{2} </math> |
| − | <math>\therefore \ t | + | <math>\therefore \ t \cdot cos(2t)</math> |
Latest revision as of 18:31, 19 September 2008
$ \,cos(2t) $ can be written as $ \ {e^{-2jt} + e^{2jt} \over 2} $ based on Euler's forumla:
- $ \cos x = \mathrm{Re}\{e^{ix}\} ={e^{ix} + e^{-ix} \over 2} $
Since the system is linear, and one of the properties of linear system is that:
- Input $ \,ax_1(t)+bx_2(t) $ equals to the output $ \, ay_1(t)+by_2(t) $
and
- $ \frac{e^{2t}}{2} \to t\cdot\frac{e^{-2t}}{2} $ and $ \frac{e^{-2t}}{2} \to t\cdot\frac{e^{2t}}{2} $
- $ t\cdot\frac{e^{-2jt}}{2} + t\cdot\frac{e^{2jt}}{2} $
- $ t\cdot\frac{e^{-2jt}+e^{2jt}}{2} $
$ \therefore \ t \cdot cos(2t) $
