# Short-time Fourier transform

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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

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

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

${\displaystyle s(\tau ,\omega )=\int _{-\infty }^{\infty }s(t)w(t-\tau )e^{-i\omega t}\,dt}$.[1]

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

## References

1. Cohen, L., 1995, Time-Frequency Analysis: Prentice-Hall