Short-time Fourier transform

The short-time Fourier transform (STFT) is a spectral decomposition method that involves calculating the Fourier transform of a windowed portion of a time-varying signal as the window slides down the time axis.

Definition

The STFT requires a windowed portion of the signal, such that

$ s(t)w(t-\tau )={\begin{cases}s(t)&\tau \approx t\\0&otherwise\\\end{cases}} $

where $ s(t) $ is the original signal and $ w(t-\tau ) $ is the window function centered at time $ \tau $. The Fourier transform is computed for the windowed portion, for each $ \tau $, as follows:

$ s(\tau ,\omega )=\int _{-\infty }^{\infty }s(t)w(t-\tau )e^{-i\omega t}\,dt $.

The result is a representation of the one-dimensional signal in a two-dimensional time-frequency domain.