0.6 Regularity, moments, and wavelet system design  (Page 5/13)

[link] gives the same information for the length-6, 4, and 2 Daubechies scaling coefficients, wavelet coefficients, scaling coefficient moments,and wavelet coefficient moments. Again notice how many discrete wavelet moments are zero.

[link] shows the continuous moments of the scaling function $\phi \left(t\right)$ and wavelet $\psi \left(t\right)$ for the Daubechies systems with lengths six and four. The discrete moments are the momentsof the coefficients defined by [link] and [link] with the continuous moments defined by [link] and [link] calculated using [link] and [link] with the programs listed in Appendix C.

These tables are very informative about the characteristics of wavelet systems in general as well as particularities of the Daubechies system.We see the $\mu \left(0\right)=\sqrt{2}$ of [link] and [link] that is necessary for the existence of a scaling function solution to [link] and the ${\mu }_1\left(0\right)=m_1\left(0\right)=0$ of [link] and [link] that is necessary for the orthogonality of the basis functions. Ortho normality requires [link] which is seen in comparison of the $h\left(n\right)$ and $h_1\left(n\right)$ , and it requires $m\left(0\right)=1$ from [link] and [link] . After those conditions are satisfied, there are $N/2-1$ degrees of freedom left which Daubechies uses to set wavelet moments $m_1\left(k\right)$ equal zero. For length-6 we have two zero wavelet moments and for length-4, one. For all longer Daubechies systems we have exactly $N/2-1$ zero wavelet moments in addition to the one $m_1\left(0\right)=0$ for a total of $N/2$ zero wavelet moments. Note $m\left(2\right)=m{\left(1\right)}^2$ as will be explained in [link] and there exist relationships among some of the values of the even-ordered scaling functionmoments, which will be explained in [link] through [link] .

As stated earlier, these systems have a maximum number of zero moments of the wavelets which results in a high degree ofsmoothness for the scaling and wavelet functions.  [link] and  [link] show the Daubechies scaling functions and wavelets for $N=4,6,8,10,12,16,20,40$ . The coefficients were generated by the techniques described in Section: Parameterization of the Scaling Coefficients and Chapter: Regularity, Moments, and Wavelet System Design . The Matlab programs are listed in Appendix C and values of $h\left(n\right)$ can be found in [link] or generated by the programs. Note the increasing smoothness as $N$ is increased. For $N=2$ , the scaling function is not continuous; for $N=4$ , it is continuous but not differentiable; for $N=6$ , it is barely differentiable once; for $N=14$ , it is twice differentiable, and similarly for longer $h\left(n\right)$ . One can obtain any degree of differentiability for sufficiently long $h\left(n\right)$ .