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<math> 1/2 * t * e^{-2jt} + 1/2 * t * e^{2jt} </math> | <math> 1/2 * t * e^{-2jt} + 1/2 * t * e^{2jt} </math> | ||
| + | |||
| + | Simplifying this yields | ||
| + | |||
| + | <math> 1/2 * t * (\cos(-2t) + i\sin(-2t)) + 1/2 * t * (\cos(2t) + i\sin(2t)) </math> | ||
Revision as of 18:59, 17 September 2008
A linear system’s response to $ e^{2jt} $ is $ t*e^{-2jt} $, and its response to $ e^{-2jt} $ is $ t*e^{2jt} $.
What is the system’s response to $ \cos(2t) $?
Well, if we convert $ \cos(2t) $ using euler's formula, we get $ 1/2 * e^{2jt} + 1/2 * e^{-2jt} $.
Since the system is linear, we can assume that with constants of 1/2,
$ 1/2 * x_1(t) + 1/2*x_2(t) => 1/2*y_1(t)+1/2*y_2(t) $
So our result is
$ 1/2 * t * e^{-2jt} + 1/2 * t * e^{2jt} $
Simplifying this yields
$ 1/2 * t * (\cos(-2t) + i\sin(-2t)) + 1/2 * t * (\cos(2t) + i\sin(2t)) $
