Editing Fast wavelet transform (section)

== Forward DWT ==
For the [[discrete wavelet transform]] (DWT), 
one computes [[recursion|recursively]], starting with the coefficient sequence <math>s^{(J)}</math> and counting down from ''k = J &minus; 1'' to some ''M < J'',

[[Image:Wavelets_-_DWT.png|thumb|390px|single application of a wavelet filter bank, with filters g=a<sup>*</sup>, h=b<sup>*</sup>]]
:<math>
s^{(k)}_n:=\frac12 \sum_{m=-N}^N a_m s^{(k+1)}_{2n+m}
</math> or <math>
s^{(k)}(z):=(\downarrow 2)(a^*(z)\cdot s^{(k+1)}(z))
</math>
and 
:<math>
d^{(k)}_n:=\frac12 \sum_{m=-N}^N b_m s^{(k+1)}_{2n+m}
</math> or <math>
d^{(k)}(z):=(\downarrow 2)(b^*(z)\cdot s^{(k+1)}(z))
</math>,

for ''k = J &minus; 1, J &minus; 2, ..., M'' and all <math>n\in\Z</math>. In the Z-transform notation:
[[Image:Wavelets_-_Filter_Bank.png|thumb|400px|recursive application of the filter bank]]
:* The [[downsampling|downsampling operator]] <math>(\downarrow 2)</math> reduces an infinite sequence, given by its [[Z-transform]], which is simply a [[Laurent series]], to the sequence of the coefficients with even indices, <math>(\downarrow 2)(c(z))=\sum_{k\in\Z}c_{2k}z^{-k}</math>. 
:* The starred Laurent-polynomial <math>a^*(z)</math> denotes the [[adjoint filter]], it has ''time-reversed'' adjoint coefficients, <math>a^*(z)=\sum_{n=-N}^N a_{-n}^*z^{-n}</math>. (The adjoint of a real number being the number itself, of a complex number its conjugate, of a real matrix the transposed matrix, of a complex matrix its hermitian adjoint).
:* Multiplication is polynomial multiplication, which is equivalent to the convolution of the coefficient sequences.

It follows that

:<math>P_k[f](x):=\sum_{n\in\Z} s^{(k)}_n\,\varphi(2^kx-n)</math>

is the orthogonal projection of the original signal ''f'' or at least of the first approximation <math>P_J[f](x)</math> onto the [[linear subspace|subspace]] <math>V_k</math>, that is, with sampling rate of 2<sup>''k''</sup> per unit interval. The difference to the first approximation is given by

:<math>P_J[f](x)=P_k[f](x)+D_k[f](x)+\dots+D_{J-1}[f](x), </math>

where the difference or detail signals are computed from the detail coefficients as

:<math>D_k[f](x):=\sum_{n\in\Z} d^{(k)}_n\,\psi(2^kx-n), </math>

with <math>\psi</math> denoting the ''mother wavelet'' of the wavelet transform.