(→Time Shifting Propery) |
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In CT, | In CT, | ||
| − | :<b><math> F(x(t-t_0)) = | + | :<b><math> F(x(t-t_0)) = e^{-jwt_0}X(w)</math></b> |
| + | |||
| + | Proof: | ||
| + | |||
| + | <math>F(x(t-t_0)) = \int_{-\infty}^{\infty}x(t-t_0)e^{-jwt}dt </math> | ||
| + | |||
| + | :Let <math>(t-t_0) = \tau </math> | ||
| + | :Therefore <math> d\tau = dt </math> | ||
| + | |||
| + | :<math> = \int_{-\infty}^{\infty}x(\tau)e^{-jw(\tau + t_0)}(d\tau) </math> | ||
| + | |||
| + | :<math> = \int_{-\infty}^{\infty}x(\tau)e^{-jw\tau}e^{-jwt_0}(d\tau) </math> | ||
| + | :Since <math> e^{-jwt_0} </math> does not depend on "<math> \tau </math>" it can be taken out of the integral. | ||
| + | |||
| + | :<math> =e^{-jwt_0} \int_{-\infty}^{\infty}x(\tau)e^{-jw\tau}(d\tau) </math> | ||
| + | Thus | ||
| + | :<math> = e^{-jwt_0}X(w) </math> | ||
Revision as of 09:25, 23 October 2008
Time Shifting Propery
In CT,
- $ F(x(t-t_0)) = e^{-jwt_0}X(w) $
Proof:
$ F(x(t-t_0)) = \int_{-\infty}^{\infty}x(t-t_0)e^{-jwt}dt $
- Let $ (t-t_0) = \tau $
- Therefore $ d\tau = dt $
- $ = \int_{-\infty}^{\infty}x(\tau)e^{-jw(\tau + t_0)}(d\tau) $
- $ = \int_{-\infty}^{\infty}x(\tau)e^{-jw\tau}e^{-jwt_0}(d\tau) $
- Since $ e^{-jwt_0} $ does not depend on "$ \tau $" it can be taken out of the integral.
- $ =e^{-jwt_0} \int_{-\infty}^{\infty}x(\tau)e^{-jw\tau}(d\tau) $
Thus
- $ = e^{-jwt_0}X(w) $
