Justin Kietzman - Time Shifting property of F.T and you ECE301Fall2008mboutin - Rhea

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)

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