ASN4 - Fourier Transform property: Parseval's Theorem

Parseval's Theorem

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

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

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

Note that Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |e^{j 2 \pi f t}|= \sqrt{cos^2(2 \pi f t) + sin^2(2 \pi f t)}=1 }

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

Squareing the function and intergrating in the time main $\int _{-\infty }^{\infty }(|s(t)|)^{2}dt$ is to do the same in the frequency domain $\int _{-\infty }^{\infty }(|S(f)|)^{2}df$