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Suppose we have a continuous time-interval $I=[a,b]$, and a signal $x \colon I \to \mathbb{R}$.

A procedure that is sometimes carried out (e.g. when doing bispectral analysis) is to partition $I$ into $k$ small sub-intervals $[a\!=\!a_0,a_1],[a_1,a_2],\ldots,[a_{k-2},a_{k-1}],[a_{k-1},a_k\!=\!b]$ and take the Fourier transform of $x$ over each of these intervals, $$ \hat{x}(n,\xi) \ := \ \int_{a_{n-1}}^{a_n} x(t)e^{-2\pi i \xi t} \, dt \, , \hspace{8mm} n \in \{1,\ldots,k\}. $$

Is there a name for $\hat{x}(n,\xi)$ (e.g. something like "time-segmented Fourier transform")?

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    $\begingroup$ Short term Fourier Transform (STFT) $\endgroup$ – Stanley Pawlukiewicz Feb 15 '18 at 23:48
  • $\begingroup$ I have only ever seen "STFT" used to refer to a continuously sliding window. Is it also used to refer to a finite number of back-to-back time-windows? (I have edited the question a little to clarify this aspect of my question.) $\endgroup$ – Julian Newman Feb 16 '18 at 0:20
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    $\begingroup$ The amount of overlap is up to the user and includes the case of no overlap $\endgroup$ – Stanley Pawlukiewicz Feb 16 '18 at 0:57
  • $\begingroup$ Stanley's correct, but personally i have never seen an implementation of the STFT that did not have 50% overlap except in one case that i saw 75% overlap. the "continuous sliding window" is a sorta nice theoretical tool (because nothing actually implemented in computers is continuous nor infinite) and i have seen that theoretical STFT used to sorta derive a Gabor wavelet transform. $$ $$ also i would be very careful with those non-overlapping adjacent rectangular windows (that appear to not necessarily be equal width). appears like a copulating female canine to mess with the math. $\endgroup$ – robert bristow-johnson Feb 16 '18 at 2:05
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    $\begingroup$ In the 2D world, FFTs and DCTs of a grid of non-overlapping rectangular windows are often used for block-based analysis and compression of graphics and image data. $\endgroup$ – hotpaw2 Feb 16 '18 at 17:53
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The Short Term Fourier Transform (STFT)

https://en.m.wikipedia.org/wiki/Short-time_Fourier_transform

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For your question, the windows are finite and non overlapping.

The continuous time case is probably more of a mathematical device that could be approximated by diffraction over some periodic, channelized structure.

The discrete time case is very commonly used, although as RJB pointed out, usually with overlap.

If your signal consisted of flat noise and a set of sinusoids, for large DFT segments, the bins would be approximately statistically independent in time and frequency for a non overlapping window, so there are applications where overlapping is not used.

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