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Just as for the case, the multiplicity-M scaling function and scaling coefficients are unique and are simply the solution of the basic recursiveor refinement equation [link] . However, the wavelets and wavelet coefficients are no longer unique or easy to design in general.
We now have the possibility of a more general and more flexible multiresolution expansion system with the M-band scaling function and wavelets. There are now signal spaces spanned by the wavelets at each scale . They are denoted
for . For example with ,
and
or
In the limit as , we have
Our notation for in Chapter: A multiresolution formulation of Wavelet Systems is
This is illustrated pictorially in [link] where we see the nested scaling function spaces but each annular ring is now divided into subspaces, each spanned by the wavelets at that scale. Compare [link] with Figure: Scaling Function and Wavelet Vector Spaces for the classical case.
The expansion of a signal or function in terms of the M-band wavelets now involves a triple sum over , and .
where the expansion coefficients (DWT) are found by
and
We now have an M-band discrete wavelet transform.
Theorem 35 If the scaling function satisfies the conditions for existence and orthogonality and the wavelets are defined by [link] and if the integer translates of these wavelets span the orthogonal compliments of , all being in , i.e., the wavelets are orthogonal to the scaling function at the samescale; that is, if
for , then
for all integers and for .
Combining [link] and [link] and calling gives
as necessary conditions on for an orthogonal system.
Unlike the case, for there is no formula for and there are many possible wavelets for a given scaling function.
Mallat's algorithm takes on a more complex form as shown in [link] . The advantage is a more flexible system that allows a mixture of linear and logarithmic tiling of thetime–scale plane. A powerful tool that removes the ambiguity is choosing the wavelets by “modulated cosine" design.
[link] shows the frequency response of the filter band, much as Figure: Frequency Bands for the Analysis Tree did for . Examples of scaling functions and wavelets are illustrated in [link] , and the tiling of the time-scale plane is shown in [link] . [link] shows the time-frequency resolution characteristics of a four-band DWT basis. Notice how it is different from theStandard, Fourier, DSTFT and two-band DWT bases shown in earlier chapters. It gives a mixture of a logarithmic and linear frequency resolution.
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