0.7 Generalizations of the basic multiresolution wavelet system  (Page 6/28)

We next define the $k^{th}$ moments of ${\psi }_{\ell }\left(t\right)$ as

and the $k^{th}$ discrete moments of $h_{\ell }\left(n\right)$ as

Theorem 36 (Equivalent Characterizations of K-Regular M-Band Filters) A unitary scaling filter is K-regular if and only if the following equivalentstatements are true:

1. All moments of the wavelet filters are zero, ${\mu }_{\ell }\left(k\right)=0$ , for $k=0,1,\cdots ,(K-1)$ and for $\ell =1,2,\cdots ,(M-1)$
2. All moments of the wavelets are zero, $m_{\ell }\left(k\right)=0$ , for $k=0,1,\cdots ,(K-1)$ and for $\ell =1,2,\cdots ,(M-1)$
3. The partial moments of the scaling filter are equal for $k=0,1,\cdots ,(K-1)$
4. The frequency response of the scaling filter has zeros of order $K$ at the $M^{th}$ roots of unity, $\text{ω}=2\pi \ell /M$ for $\ell =1,2,\cdots ,M-1$ .
5. The magnitude-squared frequency response of the scaling filter is flat to order 2K at $\text{ω}=0$ . This follows from [link] .
6. All polynomial sequences up to degree $(K-1)$ can be expressed as a linear combination of integer-shifted scaling filters.
7. All polynomials of degree up to $(K-1)$ can be expressed as a linear combination of integer-shifted scaling functions for all $j$ .

This powerful result [link] , [link] is similar to the $M=2$ case presented in Chapter: Regularity, Moments, and Wavelet System Design . It not only ties the number of zero moments to the regularity but also to the degree of polynomials that canbe exactly represented by a sum of weighted and shifted scaling functions. Note the location of the zeros of $H\left(z\right)$ are equally spaced around the unit circle, resulting in a narrower frequency response than for thehalf-band filters if $M=2$ . This is consistent with the requirements given in [link] and illustrated in [link] .

Sketches of some of the derivations in this section are given in the Appendix or are simple extensions of the $M=2$ case. More details are given in [link] , [link] , [link] .

M-band scaling function design

Calculating values of $\phi \left(n\right)$ can be done by the same methods given in Section: Calculating the Basic Scaling Function and Wavelet . However, the design of the scaling coefficients $h\left(n\right)$ parallels that for the two-band case but is somewhat more difficult [link] .

One special set of cases turns out to be a simple extension of the two-band system. If the multiplier $M=2^m$ , then the scaling function is simply a scaled version of the $M=2$ case and a particular set of corresponding wavelets are those obtained by iterating the waveletbranches of the Mallat algorithm tree as is done for wavelet packets described in [link] . For other values of $M$ , especially odd values, the situation is more complex.

M-band wavelet design and cosine modulated methods

For $M>2$ the wavelet coefficients $h_{\ell }\left(n\right)$ are not uniquely determined by the scaling coefficients, as was the case for $M=2$ . This is both a blessing and a curse. It gives us more flexibility in designing specific systems,but it complicates the design considerably. For small $N$ and $M$ , the designs can be done directly, but for longer lengths and/or for large $M$ , direct design becomes impossible and something like the cosine modulateddesign of the wavelets from the scaling function as described in Chapter: Filter Banks and Transmultiplexers , is probably the bestapproach [link] , [link] , [link] , [link] , [link] [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] .

Wavelet packets

The classical $M=2$ wavelet system results in a logarithmic frequency resolution. The low frequencies have narrow bandwidths and the highfrequencies have wide bandwidths, as illustrated in Figure: Frequency Bands for the Analysis Tree . This is called “constant-Q" filtering and is appropriate for someapplications but not all. The wavelet packet system was proposed by Ronald Coifman [link] , [link] to allow a finer and adjustable resolution of frequencies at high frequencies. It also gives a rich structure thatallows adaptation to particular signals or signal classes. The cost of this richer structureis a computational complexity of $O(Nlog(N\left)\right)$ , similar to the FFT, in contrast to the classical wavelet transform which is $O\left(N\right)$ .