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) $