Editing Fourier transform (section)

== Alternatives ==
In [[signal processing]] terms, a function (of time) is a representation of a signal with perfect ''time resolution'', but no frequency information, while the Fourier transform has perfect ''frequency resolution'', but no time information: the magnitude of the Fourier transform at a point is how much frequency content there is, but location is only given by phase (argument of the Fourier transform at a point), and [[standing wave]]s are not localized in time – a sine wave continues out to infinity, without decaying. This limits the usefulness of the Fourier transform for analyzing signals that are localized in time, notably [[transient (acoustics)|transients]], or any signal of finite extent.

As alternatives to the Fourier transform, in [[time–frequency analysis]], one uses time–frequency transforms or time–frequency distributions to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these. These can be generalizations of the Fourier transform, such as the [[short-time Fourier transform]], [[fractional Fourier transform]], Synchrosqueezing Fourier transform,<ref>{{cite journal |last1=Correia |first1=L. B. |last2=Justo |first2=J. F. |last3=Angélico |first3=B. A. |title=Polynomial Adaptive Synchrosqueezing Fourier Transform: A method to optimize multiresolution |journal=Digital Signal Processing |date=2024 |volume=150 |page=104526 |doi=10.1016/j.dsp.2024.104526|bibcode=2024DSPRJ.15004526C }}</ref> or other functions to represent signals, as in [[wavelet transform]]s and [[chirplet transform]]s, with the wavelet analog of the (continuous) Fourier transform being the [[continuous wavelet transform]].<ref name="Boashash-2003" />