Short-time Fourier transform

Summary

The short-time Fourier transform of a function $f$ on $\R$ is a Fourier transform taken through a window. One multiplies $f$ by a window $g$ shifted to a point $x$, and transforms the product. It is a function of position $x$ and frequency $\omega$, and so it shows which frequencies are present near each time. It is the basic tool of time–frequency analysis; its squared modulus is the spectrogram. For a window in $L^2$ it is an isometry up to the factor $\norm{g}_2$, so no information is lost.