Editing Short-time Fourier transform (section)

=== 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.