∫ − ∞ ∞ ( | s ( t ) | ) 2 d t {\displaystyle \int _{-\infty }^{\infty }(|s(t)|)^{2}dt} in time transforms to ∫ − ∞ ∞ ( | S ( f ) | ) 2 d f {\displaystyle \int _{-\infty }^{\infty }(|S(f)|)^{2}df} in frequency
First find the magnitude of s ( t ) {\displaystyle s(t)} which is also the Inverse Fourier Transform of S ( f ) {\displaystyle S(f)} .
| s ( t ) | = F − 1 [ S ( f ) ] = | ∫ − ∞ ∞ S ( f ) e j 2 π f t d f | {\displaystyle |s(t)|=F^{-1}[S(f)]=|\int _{-\infty }^{\infty }S(f)e^{j2\pi ft}df|}
Note that | e j 2 π f t | = c o s 2 ( 2 π f t ) + s i n 2 ( 2 π f t ) = 1 {\displaystyle |e^{j2\pi ft}|={\sqrt {cos^{2}(2\pi ft)+sin^{2}(2\pi ft)}}=1}
So then | s ( t ) | = ∫ − ∞ ∞ S ( f ) d f {\displaystyle |s(t)|=\int _{-\infty }^{\infty }S(f)df}
in time transforms to 
in frequency
which is also the Inverse Fourier Transform of 
.
![{\displaystyle |s(t)|=F^{-1}[S(f)]=|\int _{-\infty }^{\infty }S(f)e^{j2\pi ft}df|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53cd503e791b406b36de0e4949c7f7f97d1f6926)
