Here we start by defining a unitary scaling filter to be an FIR filter with coefficients h(n) from the basic recursive Equation 6.1 satisfying the admissibility conditions from Equation 6.10 and orthogonality conditions from Equation 6.14 as

(7.1)

The term “scaling filter" comes from Mallat's algorithm, and the relation to filter banks discussed in Chapter: Filter Banks and the Discrete Wavelet Transform. The term “unitary" comes from the orthogonality conditions expressed in filter bank language, which is explained in Chapter: Filter Banks and Transmultiplexers.

A unitary scaling filter is said to be K-regular if its z-transform has K zeros at z=e^iπ. This looks like

(7.2)

where is the z-transform of the scaling coefficients h(n) and Q(z) has no poles or zeros at z=e^iπ. Note that we are presenting a definition of regularity of h(n), not of the scaling function φ(t) or wavelet ψ(t). They are related but not the same. Note also from Equation 6.20 that any unitary scaling filter is at least K=1 regular.

The length of the scaling filter is N which means H(z) is an N–1 degree polynomial. Since the multiple zero at z=–1 is order K, the polynomial Q(z) is degree N–1–K. The existence of φ(t) requires the zero^th moment be which is the result of the linear condition in Equation 7.1. Satisfying the conditions for orthogonality requires N/2 conditions which are the quadratic equations in Equation 7.1. This means the degree of regularity is limited by

(7.3)

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 freedom for 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 the scaling coefficients. This is related to the fact that the differentiability of a function is tied to how fast its Fourier series coefficients drop off as the index goes to infinity or how fast the Fourier transform magnitude drops off as frequency goes to infinity. The relation of the Fourier transform of the scaling function to the frequency response of the FIR filter with coefficients h(n) is given by the infinite product Equation 6.74. From these connections, we reason that since H(z) is lowpass and, if it has a high order zero at z=–1 (i.e., ω=π), the Fourier transform of φ(t) should drop off rapidly and, therefore, φ(t) should be smooth. This turns out to be true.

We next define the k^th moments of φ(t) and ψ(t) as

(7.4)

and

(7.5)

and the discrete k^th moments of h(n) and h₁(n) as

(7.6)

and

(7.7)

The partial moments of h(n) (moments of samples) are defined as

(7.8)

Note that μ(k)=ν(k,0)+ν(k,1).

From these equations and the basic recursion Equation 6.1 we obtain 10

(7.9)

which can be derived by substituting Equation 6.1 into Equation 7.4, changing variables, and using Equation 7.6. Similarly, we obtain

(7.10)

These equations exactly calculate the moments defined by the integrals in Equation 7.4 and Equation 7.5 with simple finite convolutions of the discrete moments with the lower order continuous moments. A similar equation also holds for the multiplier-M case described in Section: Multiplicity-M (M-Band) Scaling Functions and Wavelets 10. A Matlab program that calculates the continuous moments from the discrete moments using Equation 7.9 and Equation 7.10 is given in Appendix C.

Requiring the moments of ψ(t) to be zero has several interesting consequences. The following three theorems show a variety of equivalent characteristics for the K-regular scaling filter, which relate both to our desire for smooth scaling functions and wavelets as well as polynomial representation.

Theorem 20 (Equivalent Characterizations of K-Regular Filters) A unitary scaling filter is K-regular if and only if the following equivalent statements are true:

This is a very powerful result 29, 15. It not only ties the number of zero moments to the regularity but also to the degree of polynomials that can be exactly represented by a sum of weighted and shifted scaling functions.

Theorem 21 If ψ(t) is K-times differentiable and decays fast enough, then the first K–1 wavelet moments vanish 7; i.e.,

(7.11)

implies

(7.12)

Unfortunately, the converse of this theorem is not true. However, we can relate the differentiability of ψ(t) to vanishing moments by

Theorem 22 There exists a finite positive integer L such that if m₁(k)=0 for 0≤k≤K–1 then

(7.13)

for .

For example, a three-times differentiable ψ(t) must have three vanishing moments, but three vanishing moments results in only one-time differentiability.

These theorems show the close relationship among the moments of h₁(n), ψ(t), the smoothness of H(ω) at ω=0 and π and to polynomial representation. It also states a loose relationship with the smoothness of φ(t) and ψ(t) themselves.

Daubechies used the above relationships to show the following important result which constructs orthonormal wavelets with compact support with the maximum number of vanishing moments.

Theorem 23 The discrete-time Fourier transform of h(n) having K zeros at ω=π of the form

(7.14)

satisfies

if and only if L(ω)=|L(ω)|² can be written

(7.16)

with K≤N/2 where

(7.17)

and R(y) is an odd polynomial chosen so that P(y)≥0 for 0≤y≤1.

If R=0, the length N is minimum for a given regularity K=N/2. If , the second term containing R has terms with higher powers of y whose coefficients can be used for purposes other than regularity.

The proof and a discussion are found in Daubechies 5, 7. Recall from Equation 6.20 that H(ω) always has at least one zero at ω=π as a result of h(n) satisfying the necessary conditions for φ(t) to exist and have orthogonal integer translates. We are now placing restrictions on h(n) to have as high an order zero at ω=π as possible. That accounts for the form of Equation 7.14. Requiring orthogonality in Equation 6.22 gives Equation 7.15.

Because the frequency domain requirements in Equation 7.15 are in terms of the square of the magnitudes of the frequency response, spectral factorization is used to determine H(