Editing Wavelet (section)

{{Short description|Function for integral Fourier-like transform}}
{{For|the concept in physics|Wave packet}}

A '''wavelet''' is a [[wave]]-like [[oscillation]] with an [[amplitude]] that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for [[signal processing]].
[[File:Seismic Wavelet.svg|thumb|Seismic wavelet]]

For example, a wavelet could be created to have a frequency of [[middle C]] and a short duration of roughly one tenth of a second. If this wavelet were to be [[convolution|convolved]] with a signal created from the recording of a melody, then the resulting signal would be useful for determining when the middle&nbsp;C note appeared in the song. Mathematically, a wavelet correlates with a signal if a portion of the signal is similar. [[Correlation]] is at the core of many practical wavelet applications.

As a mathematical tool, wavelets can be used to extract information from many kinds of data, including [[audio signal]]s and images. Sets of wavelets are needed to analyze data fully. "Complementary" wavelets decompose a signal without gaps or overlaps so that the decomposition process is mathematically reversible. Thus, sets of complementary wavelets are useful in [[Wavelet compression|wavelet-based compression]]/decompression algorithms, where it is desirable to recover the original information with minimal loss.

In formal terms, this representation is a [[wavelet series]] representation of a [[square-integrable function]] with respect to either a [[Complete orthogonal system#Incomplete orthogonal sets|complete]], [[orthonormal]] set of [[basis function]]s, or an [[Complete orthogonal system#Incomplete orthogonal sets|overcomplete]] set or [[frame of a vector space]], for the [[Hilbert space]] of square-integrable functions. This is accomplished through [[Coherent states in mathematical physics#The group-theoretical approach|coherent states]].

In [[classical physics]], the diffraction phenomenon is described by the [[Huygens–Fresnel principle]] that treats each point in a propagating [[wavefront]] as a collection of individual spherical wavelets.<ref>Wireless Communications: Principles and Practice, Prentice Hall communications engineering and emerging technologies series, T.&nbsp;S. Rappaport, Prentice Hall, 2002, p.&nbsp;126.</ref> The characteristic bending pattern is most pronounced when a wave from a [[Coherence (physics)|coherent]] source (such as a laser) encounters a slit/aperture that is comparable in size to its [[wavelength]]. This is due to the addition, or [[Interference (wave propagation)|interference]], of different points on the wavefront (or, equivalently, each wavelet) that travel by paths of different lengths to the registering surface. Multiple, [[diffraction grating|closely spaced openings]] (e.g., a [[diffraction grating]]), can result in a complex pattern of varying intensity.