Editing Uncertainty principle (section)

===Signal processing {{anchor|Gabor limit}}===
In the context of [[time–frequency analysis]] uncertainty principles are referred to as the '''Gabor limit''', after [[Dennis Gabor]], or sometimes the ''Heisenberg–Gabor limit''. The basic result, which follows from "Benedicks's theorem", below, is that a function cannot be both [[time limited]] and [[band limited]] (a function and its Fourier transform cannot both have bounded domain)—see [[Bandlimiting#Bandlimited versus timelimited|bandlimited versus timelimited]]. More accurately, the ''time-bandwidth'' or ''duration-bandwidth'' product satisfies
<math display="block">\sigma_{t} \sigma_{f} \ge \frac{1}{4\pi} \approx 0.08 \text{ cycles},</math>
where <math>\sigma_{t}</math> and <math>\sigma_{f}</math> are the standard deviations of the time and frequency energy concentrations respectively.<ref>{{cite book | last=Mallat | first=S. G. | title=A wavelet tour of signal processing: the sparse way | publisher=Elsevier/Academic Press | publication-place=Amsterdam ; Boston | date=2009 | isbn=978-0-12-374370-1|doi=10.1016/B978-0-12-374370-1.X0001-8|page=44}}</ref> The minimum is attained for a [[Gaussian function|Gaussian]]-shaped pulse ([[Gabor wavelet]]) [For the un-squared Gaussian (i.e. signal amplitude) and its un-squared Fourier transform magnitude <math>\sigma_t\sigma_f=1/2\pi</math>; squaring reduces each <math>\sigma</math> by a factor <math>\sqrt 2</math>.] Another common measure is the product of the time and frequency [[full width at half maximum]] (of the power/energy), which for the Gaussian equals <math>2 \ln 2 / \pi \approx 0.44</math> (see [[bandwidth-limited pulse]]).

Stated differently, one cannot simultaneously sharply localize a signal {{mvar|f}} in both the [[time domain]] and [[frequency domain]].

When applied to [[Filter (signal processing)|filters]], the result implies that one cannot simultaneously achieve a high temporal resolution and high frequency resolution at the same time; a concrete example are the [[Short-time Fourier transform#Resolution issues|resolution issues of the short-time Fourier transform]]—if one uses a wide window, one achieves good frequency resolution at the cost of temporal resolution, while a narrow window has the opposite trade-off.

Alternate theorems give more precise quantitative results, and, in time–frequency analysis, rather than interpreting the (1-dimensional) time and frequency domains separately, one instead interprets the limit as a lower limit on the support of a function in the (2-dimensional) time–frequency plane. In practice, the Gabor limit limits the ''simultaneous'' time–frequency resolution one can achieve without interference; it is possible to achieve higher resolution, but at the cost of different components of the signal interfering with each other.

As a result, in order to analyze signals where the [[Transient (acoustics)|transients]] are important, the [[Wavelet Transform|wavelet transform]] is often used instead of the Fourier.