# 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

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

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
2. Whaley, J., 2017, Oil in the Heart of South America, https://www.geoexpro.com/articles/2017/10/oil-in-the-heart-of-south-america], accessed November 15, 2021.
3. Wiens, F., 1995, Phanerozoic Tectonics and Sedimentation of The Chaco Basin, Paraguay. Its Hydrocarbon Potential: Geoconsultores, 2-27, accessed November 15, 2021; https://www.researchgate.net/publication/281348744_Phanerozoic_tectonics_and_sedimentation_in_the_Chaco_Basin_of_Paraguay_with_comments_on_hydrocarbon_potential
4. Alfredo, Carlos, and Clebsch Kuhn. “The Geological Evolution of the Paraguayan Chaco.” TTU DSpace Home. Texas Tech University, August 1, 1991. https://ttu-ir.tdl.org/handle/2346/9214?show=full.