ASN4 - Fourier Transform property: Parseval's Theorem

Parseval's Theorem

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

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

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

Note that $|e^{j2\pi ft}|=\sqrt{co{s}^2(2\pi ft)+si{n}^2(2\pi ft)}=1$

So then $|s(t)|=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}S(f)df$