Editing Daubechies wavelet (section)

==Properties==

In general the Daubechies wavelets are chosen to have the highest number ''A'' of vanishing moments, (this does not imply the best smoothness) for given support width (number of coefficients) 2''A''.<ref>I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992, p. 194.</ref> There are two naming schemes in use, D''N'' using the length or number of taps, and db''A'' referring to the number of vanishing moments. So D4 and db2 are the same wavelet transform.

Among the 2<sup>''A''−1</sup> possible solutions of the algebraic equations for the moment and orthogonality conditions, the one is chosen whose scaling filter has extremal phase. The wavelet transform is also easy to put into practice using the [[fast wavelet transform]]. Daubechies wavelets are widely used in solving a broad range of problems, e.g. self-similarity properties of a signal or [[fractal]] problems, signal discontinuities, etc.

The Daubechies wavelets are not defined in terms of the resulting scaling and wavelet functions; in fact, they are not possible to write down in [[closed form expression|closed form]]. The graphs below are generated using the [[cascade algorithm]], a numeric technique consisting of inverse-transforming [1 0 0 0 0 ... ] an appropriate number of times.

{| class="wikitable"
|Scaling and wavelet functions
|[[Image:Daubechies4-functions.svg|360px]]
|[[Image:Daubechies12-functions.png|360px]]
|[[Image:Daubechies20-functions.png|360px]]
|-
|Amplitudes of the frequency spectra of the above functions
|[[Image:Daubechies4-spectrum.svg|360px]]
|[[Image:Daubechies12-spectrum.png|360px]]
|[[Image:Daubechies20-spectrum.png|360px]]
|}
Note that the spectra shown here are not the frequency response of the high and low pass filters, but rather the amplitudes of the continuous Fourier transforms of the scaling (blue) and wavelet (red) functions.

Daubechies orthogonal wavelets D2–D20 resp. db1–db10 are commonly used. Each wavelet has a number of ''zero moments'' or ''vanishing moments'' equal to half the number of coefficients. For example, D2 has one vanishing moment, D4 has two, etc. A vanishing moment limits the wavelets ability to represent [[polynomial]] behaviour or information in a signal. For example, D2, with one vanishing moment, easily encodes polynomials of one coefficient, or constant signal components. D4 encodes polynomials with two coefficients, i.e. constant and linear signal components; and D6 encodes 3-polynomials, i.e. constant, linear and [[quadratic polynomial|quadratic]] signal components. This ability to encode signals is nonetheless subject to the phenomenon of ''scale leakage'', and the lack of shift-invariance, which arise from the discrete shifting operation (below) during application of the transform. Sub-sequences which represent linear, [[quadratic polynomial|quadratic]] (for example) signal components are treated differently by the transform depending on whether the points align with even- or odd-numbered locations in the sequence. The lack of the important property of [[translational invariance|shift-invariance]], has led to the development of several different versions of a [[shift invariant wavelet transform|shift-invariant (discrete) wavelet transform]].