0.6 Regularity, moments, and wavelet system design  (Page 4/13)

which gives a complete parameterization of Daubechies' maximum zero wavelet moment design. It also gives a very straightforward procedurefor the calculation of the $h\left(n\right)$ that satisfy these conditions. Herrmann derived this expression for the design of Butterworth or maximally flatFIR digital filters [link] .

If the regularity is $K<N/2$ , $P\left(y\right)$ must be of higher degree and the form of the solution is

where $R\left(y\right)$ is chosen to give the desired filter length $N$ , to achieve some other desired property, and to give $P\left(y\right)\ge 0$ .

The steps in calculating the actual values of $h\left(n\right)$ are to first choose the length $N$ (or the desired regularity) for $h\left(n\right)$ , then factor ${\left|H\left(\omega \right)\right|}^2$ where there will be freedom in choosing which roots to use for $H\left(\omega \right)$ . The calculations are more easily carried out using the z-transform form of the transferfunction and using convolution in the time domain rather than multiplication (raising to a power) in the frequency domain. That is donein the Matlab program [hn,h1n] = daub(N) in Appendix C where the polynomial coefficients in [link] are calculated from the binomial coefficient formula. This polynomial is factored with the roots command in Matlab and the roots are mapped from the polynomial variable $y_{\ell }$ to the variable $z_{\ell }$ in [link] using first $cos\left(\omega \right)=1-2y_{\ell }$ , then with $isin\left(\omega \right)=\sqrt{{cos}^2\left(\omega \right)-1}$ and $e^{i\omega }=cos\left(\omega \right)\pm isin\left(\omega \right)$ we use $z=e^{i\omega }$ . These changes of variables are used by Herrmann [link] and Daubechies [link] .

Examine the Matlab program to see the details of just how this is carried out. The program uses the sort command to order the roots of $H\left(z\right)H(1/z)$ after which it chooses the $N-1$ smallest ones to give a minimum phase $H\left(z\right)$ factorization. You could choose a different set of $N-1$ roots in an effort to get a more linear phase or even maximum phase. This choice allows some variation in Daubechieswavelets of the same length. The $M$ -band generalization of this is developed by Heller in [link] , [link] . In [link] , Daubechies also considers an alternation of zeros inside and outside the unit circle whichgives a more symmetric $h\left(n\right)$ . A completely symmetric real $h\left(n\right)$ that has compact support and supports orthogonal wavelets is not possible;however, symmetry is possible for complex $h\left(n\right)$ , biorthogonal systems, infinitely long $h\left(n\right)$ , and multiwavelets. Use of this zero moment design approach will also assure the resulting wavelets system is an orthonormalbasis.

If all the degrees of freedom are used to set moments to zero, one uses $K=N/2$ in [link] and the above procedure is followed. It is possible to explicitly set a particular pair of zeros somewhere other than at $\omega =\pi$ . In that case, one would use $K=(N/2)-2$ in [link] . Other constraints are developed later in this chapter and in later chapters.

To illustrate some of the characteristics of a Daubechies wavelet system, [link] shows the scaling function and wavelet coefficients, $h\left(n\right)$ and $h_1\left(n\right)$ , and the corresponding discrete scaling coefficient moments and wavelet coefficient moments for a length-8 Daubechies system. Note the $N/2=4$ zero moments of the wavelet coefficients and the zero ^th scaling coefficient moment of $\mu \left(0\right)=\sqrt{2}$ .