Editing Short-time Fourier transform (section)

=== Discrete-time STFT ===
{{See also|Modified discrete cosine transform}}
In the discrete time case, the data to be transformed could be broken up into chunks or frames (which usually overlap each other, to reduce artifacts at the boundary). Each chunk is [[Fourier transform]]ed, and the complex result is added to a matrix, which records magnitude and phase for each point in time and frequency.  This can be expressed as:

:<math>\mathbf{STFT}\{x[n]\}(m,\omega)\equiv X(m,\omega) = \sum_{n=0}^{N-1} x[n]w[n-m]e^{-i \omega n} </math>

likewise, with signal <math>x[n]</math> and window <math>w[n]</math>.  In this case, ''m'' is discrete and ω is continuous, but in most typical applications the STFT is performed on a computer using the [[fast Fourier transform]], so both variables are discrete and [[Quantization (signal processing)|quantized]].

The [[magnitude (mathematics)|magnitude]] squared of the STFT yields the [[spectrogram]] representation of the power spectral density of the function:

:<math>\operatorname{spectrogram}\{x(t)\}(\tau, \omega) \equiv |X(\tau, \omega)|^2 </math>

See also the [[modified discrete cosine transform]] (MDCT), which is also a Fourier-related transform that uses overlapping windows.

==== Sliding DFT ====
If only a small number of ω are desired, or if the STFT is desired to be evaluated for every shift ''m'' of the window, then the STFT may be more efficiently evaluated using a [[sliding DFT]] algorithm.<ref>E. Jacobsen and R. Lyons, [https://ieeexplore.ieee.org/document/1184347/;jsessionid=4C7542A520E95FD20371713367DD1C7F?arnumber=1184347 The sliding DFT], ''Signal Processing Magazine'' vol. 20, issue 2, pp. 74–80 (March 2003).</ref>