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.