0.6 Regularity, moments, and wavelet system design  (Page 2/13)

Daubechies used the degrees of freedom to obtain maximum regularity for a given $N$ , or to obtain the minimum $N$ for a given regularity. Others have allowed a smaller regularity and used the resulting extra degrees of freedomfor other design purposes.

Regularity is defined in terms of zeros of the transfer function or frequency response function of an FIR filter made up from thescaling coefficients. This is related to the fact that the differentiability of a function is tied to how fast its Fourier series coefficients drop off asthe index goes to infinity or how fast the Fourier transform magnitude drops off as frequency goes to infinity. The relation of the Fouriertransform of the scaling function to the frequency response of the FIR filter with coefficients $h\left(n\right)$ is given by the infinite product [link] . From these connections, we reason that since $H\left(z\right)$ is lowpass and, if it has a high order zero at $z=-1$ (i.e., $\omega =\pi$ ), the Fourier transform of $\phi \left(t\right)$ should drop off rapidly and, therefore, $\phi \left(t\right)$ should be smooth. This turns out to be true.

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

and

and the discrete $k^{th}$ moments of $h\left(n\right)$ and $h_1\left(n\right)$ as

and

The partial moments of $h\left(n\right)$ (moments of samples) are defined as

Note that $\mu \left(k\right)=\nu (k,0)+\nu (k,1)$ .

From these equations and the basic recursion [link] we obtain [link]

which can be derived by substituting [link] into [link] , changing variables, and using [link] . Similarly, we obtain

These equations exactly calculate the moments defined by the integrals in [link] and [link] with simple finite convolutions of the discrete moments with the lower order continuous moments. A similar equationalso holds for the multiplier- $M$ case described in Section: Multiplicity-M (M-Band) Scaling Functions and Wavelets [link] . A Matlab program that calculates the continuous moments from the discrete moments using [link] and [link] is given in Appendix C.

Vanishing wavelet moments

Requiring the moments of $\psi \left(t\right)$ to be zero has several interesting consequences. The following three theorems show a variety of equivalentcharacteristics for the $K$ -regular scaling filter, which relate both to our desire for smooth scaling functions and wavelets as well as polynomialrepresentation.

Theorem 20 (Equivalent Characterizations of K-Regular 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, ${\mu }_1\left(k\right)=0$ , for $k=0,1,\cdots ,(K-1)$
2. All moments of the wavelets are zero, $m_1\left(k\right)=0$ , for $k=0,1,\cdots ,(K-1)$
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 a zero of order K at $\omega =\pi$ , i.e. [link] .
5. The $k^{th}$ derivative of the magnitude-squared frequency response of the scaling filter is zero at $\omega =0$ for $k=1,2,\cdots ,2K-1$ .
6. All polynomial sequences up to degree $(K-1)$ can be expressed as a linear combination of shifted scaling filters.
7. All polynomials of degree up to $(K-1)$ can be expressed as a linear combination of shifted scaling functions at any scale.