A finite impulse congugate mirror filter bank is caracterized by a filter h which satisfies
where h(w) is a trigonometric polynomial (or, equivalently, a polynomial with respect to the shift and sign operators). The p frequency shift amounts to a change of sign every two coefficient. Moreover, this continuous time transfer must vanish up to the order p at p in order to have a wavelet with p vanishing moments.
The synthesis of such filters can be done using several methods. The best known filters are Daubechies's compactly supported filters. An outline of the construction method is available.
There remains to check that the filter h does generate a scaling function. To do so, it is enough to verify that the transfer of h does not vanish on [-p/2,p/2] (theorem by Mallat and Meyer). The construction on Daubechies'compactly supported orthogonal wavelets is presented here.
The coefficients of the conjugatemirror filter h can be obtained in Wavelab, are freeware Matlab toolbox, using the function MakeONFilter.
The same filters are used to implement
the
dyadid
wavelet transform