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

Revision as of 13:28, 19 December 2009

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

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

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

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 This Fourier Transform property says <math>s(0)</math> transforms to <math>\int_{-\infty}^{\infty} S(f) df</math>
-<math>s(0)= s(t)|_{t=0} = \int_{-\infty}^{\infty} s(t)e^{ j 2 \pi f t} dt = \int_{-\infty}^{\infty} s(t) dt</math>
+<math>s(0)= s(t)|_{t=0} = \int_{-\infty}^{\infty} S(f)e^{ j 2 \pi (0) t} dt = \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>
+<math>s(0)= s(t)|_{t=0} = \int_{-\infty}^{\infty} S(f)e^{ j 2 \pi (0) t} dt = \int_{-\infty}^{\infty} S(f) df</math>
 The result is <math>s(0)</math> transforms to <math>\int_{-\infty}^{\infty} S(f) df</math>