View source for ASN6c - Fourier Transform property: Value at origin

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== Value at Origin ==

This Fourier Transform property says <math>s(0)\!</math> becomes <math>\int_{-\infty}^{\infty} S(f) df</math>

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


Evaluating the inverse Fourier transform at time zero


<math>s(0) = \int_{-\infty}^{\infty} S(f)e^{ j 2 \pi f(0) } df = \int_{-\infty}^{\infty} S(f) df</math>

Therefore this process shows that <math>s(0) \!</math> is <math>\int_{-\infty}^{\infty} S(f) df</math>