ASN6c - Fourier Transform property: Value at origin

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Value at Origin

This Fourier Transform property says $s(0)$ becomes $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}S(f)df$

$s(0)=s(t){|}_{t=0}={{ℱ}}^{-1}[S(f)]){|}_{t=0}$

Evaluating the inverse Fourier transform at time zero

$s(0)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}S(f)e^{j2\pi f(0)}df=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}S(f)df$

Therefore this process shows that $s(0)$ is $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}S(f)df$