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Just as for the M = 2 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 M - 1 signal spaces spanned by the M - 1 wavelets at each scale j . They are denoted

W , j = Span k { ψ ( M j t + k )

for = 1 , 2 , , M - 1 . For example with M = 4 ,

V 1 = V 0 W 1 , 0 W 2 , 0 W 3 , 0

and

V 2 = V 1 W 1 , 1 W 2 , 1 W 3 , 1

or

V 2 = V 0 W 1 , 0 W 2 , 0 W 3 , 0 W 1 , 1 W 2 , 1 W 3 , 1 .

In the limit as j , we have

L 2 = V 0 W 1 , 0 W 2 , 0 W 3 , 0 W 1 , 1 W 2 , 1 W 3 , 1 W 3 , .

Our notation for M = 2 in  Chapter: A multiresolution formulation of Wavelet Systems is W 1 , j = W j

This is illustrated pictorially in [link] where we see the nested scaling function spaces V j but each annular ring is now divided into M - 1 subspaces, each spanned by the M - 1 wavelets at that scale. Compare [link] with Figure: Scaling Function and Wavelet Vector Spaces for the classical M = 2 case.

Vector Space Decomposition for a Four-Band Wavelet System
Vector Space Decomposition for a Four-Band Wavelet System, W j

The expansion of a signal or function in terms of the M-band wavelets now involves a triple sum over , j , and k .

f ( t ) = k c ( k ) φ k ( t ) + k = - j = 0 = 1 M - 1 M j / 2 d , j ( k ) ψ ( M j t - k )

where the expansion coefficients (DWT) are found by

c ( k ) = f ( t ) φ ( t - k ) d t

and

d , j ( k ) = f ( t ) M j / 2 ψ ( M j t - k ) d t .

We now have an M-band discrete wavelet transform.

Theorem 35 If the scaling function φ ( t ) satisfies the conditions for existence and orthogonality and the wavelets are defined by [link] and if the integer translates of these wavelets span W , 0 the orthogonal compliments of V 0 , all being in V 1 , i.e., the wavelets are orthogonal to the scaling function at the samescale; that is, if

φ ( t - n ) ψ ( t - m ) d t = 0

for = 1 , 2 , , M - 1 , then

n h ( n ) h ( n - M k ) = 0

for all integers k and for = 1 , 2 , , M - 1 .

Combining [link] and [link] and calling h 0 ( n ) = h ( n ) gives

n h m ( n ) h ( n - M k ) = δ ( k ) δ ( m - )

as necessary conditions on h ( n ) for an orthogonal system.

Filter Bank Structure for a Four-Band Wavelet System
Filter Bank Structure for a Four-Band Wavelet System, W j

Unlike the M = 2 case, for M > 2 there is no formula for h ( n ) 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 M = 2 . 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.

Frequency Responses for the Four-Band Filter Bank
Frequency Responses for the Four-Band Filter Bank, W j
A Four-Band Six-Regular Wavelet System
A Four-Band Six-Regular Wavelet System: Φ

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Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
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