ASN6c - Fourier Transform property: Value at origin: Difference between revisions

Revision as of 02:25, 30 November 2009

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

$s(0)=S(f)|_{f=0}=\int _{-\infty }^{\infty }s(t)e^{-j2\pi ft}dt=\int _{-\infty }^{\infty }s(t)dt$

And $\int _{-\infty }^{\infty }s(t)dt$ in time transforms in frequency to $\int _{-\infty }^{\infty }S(f)df$

The result is $s(0)$ transforms to $\int _{-\infty }^{\infty }S(f)df$

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 <math>s(0)= S(f)|_{f=0} = \int_{-\infty}^{\infty} s(t)e^{- j 2 \pi f t} dt = \int_{-\infty}^{\infty} s(t) dt</math>
-Thefore <math> s(0) = \int_{-\infty}^{\infty} s(t) dt</math> in time transformed to frequency is <math>\int_{-\infty}^{\infty} S(f) df</math>
+And <math> \int_{-\infty}^{\infty} s(t) dt</math> in time transforms in frequency to <math>\int_{-\infty}^{\infty} S(f) df</math>
+The result is <math>s(0)</math> transforms to <math>\int_{-\infty}^{\infty} S(f) df</math>