ASN6c - Fourier Transform property: Value at origin

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Value at Origin

This Fourier Transform property says $s(0)\!$ becomes $\int _{-\infty }^{\infty }S(f)df$

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

Evaluating the inverse Fourier transform at time zero

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

This shows how $s(0)\!$ becomes $\int _{-\infty }^{\infty }S(f)df$