0.6 Regularity, moments, and wavelet system design  (Page 10/13)

with the first formula [link] giving the same result as Daubechies [link] , [link] (corrected) and that of Odegard [link] and the third giving the same result as Wickerhauser [link] . The results from [link] are included in [link] along with the discrete moments of the scaling function and wavelet, $\mu \left(k\right)$ and ${\mu }_1\left(k\right)$ for $k=0,1,2,3$ . The design of a length-6 Coifman system specifies one zero scaling function moment and one zero wavelet moment (in addition to ${\mu }_1\left(0\right)=0$ ), but we, in fact, obtain one extra zero scaling function moment. That is the result of $m\left(2\right)=m{\left(1\right)}^2$ from [link] . In other words, we get one more zero scaling function moment than the two degreesof freedom would seem to indicate. This is true for all lengths $N=6\ell$ for $\ell =1,2,3,\cdots$ and is a result of the interaction between the scaling function moments and the wavelet moments described later.

The property of zero wavelet moments is shift invariant, but the zero scaling function moments are shift dependent [link] . Therefore, a particular shift for the scaling function must be used. This shift istwo for the length-6 example in [link] , but is different for the solutions in [link] and [link] . Compare this table to the corresponding one for Daubechies length-6 scaling functions and waveletsgiven in [link] where there are two zero discrete wavelet moments – just as many as the degrees of freedom in that design.

The scaling function from [link] is fairly symmetric, but not around its center and the other three designs in [link] , [link] , and [link] are not symmetric at all. The scaling function from [link] is also fairly smooth, and from [link] only slightly less so but the scaling function from [link] is very rough and from [link] seems to be fractal. Examination of the frequency response $H\left(\omega \right)$ and the zero location of the FIR filters $h\left(n\right)$ shows very similar frequency responses for [link] and [link] with [link] having a somewhat irregular but monotonic frequency response and [link] having a zero on the unit circle at $\omega =\pi /3$ , i.e., not satisfying Cohen's condition [link] for an orthognal basis. It is also worth noticing that the design in [link] has the largest Hölder smoothness. These four designs, all satisfying the same necessary conditions, have very different characteristics. This tells usto be very careful in using zero moment methods to design wavelet systems. The designs are not unique and some are much better than others.

[link] contains the scaling function and wavelet coefficients for the length-6 and 12 designed by Daubechies and length-8 designed byTian together with their discrete moments. We see the extra zero scaling function moments for lengths 6 and 12 and also the extra zero forlengths 8 and 12 that occurs after a nonzero one.

The continuous moments can be calculated from the discrete moments and lower order continuous moments [link] , [link] , [link] using [link] and [link] . An important relationship of the discrete moments for a system with $K-1$ zero wavelet moments is found by calculating the derivatives of the magnitude squared of the discrete time Fouriertransform of $h\left(n\right)$ which is $H\left(\omega \right)={\sum }_{n}h\left(n\right)e^{-i\omega n}$ and has $2K-1$ zero derivatives of the magnitude squared at $\omega =0$ . This gives [link] the $k^{th}$ derivative for $k$ even and $1<k<2K-1$

Solving for $\mu \left(k\right)$ in terms of lower order discrete moments and using $\mu \left(0\right)=\sqrt{2}$ gives for $k$ even