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

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

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