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

Revision as of 01:56, 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 $S(f)$ .

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

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

@@ Line 3: / Line 3: @@
 <math>\int_{- \infty}^{\infty} (|s(t)|)^2 dt</math> in time transforms to <math>\int_{- \infty}^{\infty} (|S(f)|)^2 df</math> in frequency
+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>
+Since <math> |e^{j 2 \pi f t}|= \sqrt{cos^2(2 \pi f t) + sin^2(2 \pi f t)}=1 </math>