Editing Wavelet (section)

== Comparisons with Fourier transform (continuous-time) ==
The wavelet transform is often compared with the [[Fourier transform]], in which signals are represented as a sum of sinusoids. In fact, the [[Fourier transform]] can be viewed as a special case of the continuous wavelet transform with the choice of the mother wavelet
<math>\psi (t) = e^{-2 \pi i t}</math>.
The main difference in general is that wavelets are localized in both time and frequency whereas the standard [[Fourier transform]] is only localized in [[frequency]]. The [[short-time Fourier transform]] (STFT) is similar to the wavelet transform, in that it is also time and frequency localized, but there are issues with the frequency/time resolution trade-off.

In particular, assuming a rectangular window region, one may think of the STFT as a transform with a slightly different kernel
<math display="block">\psi (t) = g(t-u) e^{-2 \pi i t}</math>
where <math>g(t-u)</math> can often be written as <math display="inline">\operatorname{rect}\left(\frac{t-u}{\Delta_t}\right)</math>, where <math>\Delta_t</math> and ''u'' respectively denote the length and temporal offset of the windowing function. Using [[Parseval's theorem]], one may define the wavelet's energy as
<math display="block">E = \int_{-\infty}^{\infty} |\psi (t)|^2\, dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} |\hat{\psi}(\omega)|^2 \, d\omega</math>
From this, the square of the temporal support of the window offset by time ''u'' is given by
<math display="block">\sigma_u^2 = \frac{1}{E}\int |t-u|^2 |\psi(t)|^2 \, dt</math>

and the square of the spectral support of the window acting on a frequency <math>\xi</math>
<math display="block">\hat{\sigma}_\xi^2 =\frac{1}{2\pi E} \int |\omega-\xi|^2|\hat{\psi}(\omega)|^2 \, d\omega</math>

Multiplication with a rectangular window in the time domain corresponds to convolution with a <math>\operatorname{sinc}(\Delta_t\omega)</math> function in the frequency domain, resulting in spurious [[ringing artifacts]] for short/localized temporal windows. With the continuous-time Fourier transform, <math>\Delta_t \to \infty</math> and this convolution is with a delta function in Fourier space, resulting in the true Fourier transform of the signal <math>x(t)</math>. The window function may be some other [[Apodizing|apodizing filter]], such as a [[Gaussian filter|Gaussian]]. The choice of windowing function will affect the approximation error relative to the true Fourier transform.

A given resolution cell's time-bandwidth product may not be exceeded with the STFT. All STFT basis elements maintain a uniform spectral and temporal support for all temporal shifts or offsets, thereby attaining an equal resolution in time for lower and higher frequencies. The resolution is purely determined by the sampling width.

In contrast, the wavelet transform's [[Multiresolution analysis|multiresolutional]] properties enables large temporal supports for lower frequencies while maintaining short temporal widths for higher frequencies by the scaling properties of the wavelet transform. This property extends conventional time-frequency analysis into time-scale analysis.{{sfn|Mallat|2009|loc=Chpt. 7}} [[File:Time frequency atom resolution.png|thumb|STFT time-frequency atoms (left) and DWT time-scale atoms (right). The time-frequency atoms are four different basis functions used for the STFT (i.e. '''four separate Fourier transforms required'''). The time-scale atoms of the DWT achieve small temporal widths for high frequencies and good temporal widths for low frequencies with a '''single''' transform basis set.]]

The discrete wavelet transform is less computationally [[complexity|complex]], taking [[Big O notation|O(''N'')]] time as compared to O(''N''&nbsp;log&nbsp;''N'') for the [[fast Fourier transform]] (FFT). This computational advantage is not inherent to the transform, but reflects the choice of a logarithmic division of frequency, in contrast to the equally spaced frequency divisions of the FFT which uses the same basis functions as the discrete Fourier transform (DFT).<ref>The Scientist and Engineer's Guide to Digital Signal Processing By Steven W. Smith, Ph.D. chapter 8 equation 8-1: http://www.dspguide.com/ch8/4.htm</ref> This complexity only applies when the filter size has no relation to the signal size. A wavelet without [[compact support]] such as the [[Shannon wavelet]] would require O(''N''<sup>2</sup>). (For instance, a logarithmic Fourier Transform also exists with O(''N'') complexity, but the original signal must be sampled logarithmically in time, which is only useful for certain types of signals.<ref>{{cite journal|url=http://homepages.dias.ie/~ajones/publications/28.pdf |title=Logarithmic Fourier Transform|last1=Haines|first1=VG. V. |last2=Jones|first2=Alan G.|year=1988|issue=92|pages=171–178|journal=[[Geophysical Journal]]|doi=10.1111/j.1365-246X.1988.tb01131.x|doi-access=free |s2cid=9720759 }}</ref>)