ASN4 - Fourier Transform property: Parseval's Theorem

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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

The magnitude of $s(t)$ 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 $|e^{j2\pi ft}|=\sqrt{co{s}^2(2\pi ft)+si{n}^2(2\pi ft)}=1$

The above equation of $|s(t)|$ simplifies to then $|s(t)|=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}S(f)df=|S(f)|$

Therefore,squaring the function and intergrating it in the time domain $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(|s(t)|{)}^{2}dt$ is to do the same in the frequency domain $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(|S(f)|{)}^{2}df$