ASN4 - Fourier Transform property: Parseval's Theorem: Difference between revisions

Revision as of 01:59, 30 November 2009

$\int _{-\infty }^{\infty }(|s(t)|)^{2}dt$ in time transforms to $\int _{-\infty }^{\infty }(|S(f)|)^{2}df$ in frequency

First find the magnitude of $s(t)$ which is also the Inverse Fourier Transform of $S(f)$ .

$|s(t)|=F^{-1}[S(f)]=|\int _{-\infty }^{\infty }S(f)e^{j2\pi ft}df|$

Note that $|e^{j2\pi ft}|={\sqrt {cos^{2}(2\pi ft)+sin^{2}(2\pi ft)}}=1$

So then $|s(t)|=\int _{-\infty }^{\infty }S(f)df$

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 First find the magnitude of <math>s(t)</math> which is also the Inverse Fourier Transform of <math>S(f)</math>.
-<math> |s(t)|= F ^{-1}[S(f)]=|S(f)e^{j 2 \pi f t} df | </math>
+<math> |s(t)|= F ^{-1}[S(f)]=|\int_{- \infty}^{\infty}S(f)e^{j 2 \pi f t} df | </math>
-Since <math> |e^{j 2 \pi f t}|= \sqrt{cos^2(2 \pi f t) + sin^2(2 \pi f t)}=1 </math>
+Note that <math> |e^{j 2 \pi f t}|= \sqrt{cos^2(2 \pi f t) + sin^2(2 \pi f t)}=1 </math>
+So then <math>|s(t)|= \int_{- \infty}^{\infty}S(f) df</math>