Editing Short-time Fourier transform (section)

== Forward STFT ==

=== Continuous-time STFT ===
Simply, in the continuous-time case, the function to be transformed is multiplied by a [[window function]] which is nonzero for only a short period of time.  The [[Fourier transform]] (a one-dimensional function) of the resulting signal is taken, then the window is slid along the time axis until the end resulting in a two-dimensional representation of the signal.  Mathematically, this is written as:

:<math>\mathbf{STFT}\{x(t)\}(\tau,\omega) \equiv X(\tau, \omega) = \int_{-\infty}^{\infty} x(t) w(t-\tau) e^{-i \omega t} \, d t </math>

where <math>w(\tau)</math> is the [[window function]], commonly a [[Window function#Hann and Hamming windows|Hann window]] or [[Window function#Gaussian window|Gaussian window]] centered around zero, and <math>x(t)</math> is the signal to be transformed (note the difference between the window function <math>w</math> and the frequency <math>\omega</math>). <math>X(\tau, \omega)</math> is essentially the Fourier transform of <math>x(t)w(t-\tau)</math>, a [[complex function]] representing the phase and magnitude of the signal over time and frequency. Often [[phase unwrapping]] is employed along either or both the time axis, <math>\tau</math>, and frequency axis, <math>\omega</math>, to suppress any [[jump discontinuity]] of the phase result of the STFT. The time index <math>\tau</math> is normally considered to be "''slow''" time and usually not expressed in as high resolution as time <math>t</math>. Given that the STFT is essentially a Fourier transform times a window function, the STFT is also called windowed Fourier transform or time-dependent Fourier transform.

=== 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>