0.5 The scaling function and scaling coefficients, wavelet and  (Page 2/13)

Fourier transforms

We will need the Fourier transform of $\phi \left(t\right)$ which, if it exists, is defined to be $$\text{Φ}$$

and the discrete-time Fourier transform (DTFT) [link] of $h\left(n\right)$ defined to be

where $i=\sqrt{-1}$ and $n$ is an integer ( $n\in \mathbf{Z}$ ). If convolution with $h\left(n\right)$ is viewed as a digital filter, as defined in Section: Analysis - From Fine Scale to Coarse Scale , then the DTFT of $h\left(n\right)$ is the filter's frequency response, [link] , [link] which is $2\pi$ periodic.

If $\text{Φ}\left(\omega \right)$ exists, the defining recursive equation [link] becomes

which after iteration becomes

if ${\sum }_{n}h\left(n\right)=\sqrt{2}$ and $\text{Φ}\left(0\right)$ is well defined. This may be a distribution or it may be asmooth function depending on $H\left(\omega \right)$ and, therefore, $h\left(n\right)$ [link] , [link] . This makessense only if $\text{Φ}\left(0\right)$ is well defined. Although [link] and [link] are equivalent term-by-term, the requirement of $\text{Φ}\left(0\right)$ being well defined and the nature of the limits in the appropriate function spacesmay make one preferable over the other. Notice how the zeros of $H\left(\omega \right)$ determine the zeros of $\text{Φ}\left(\omega \right)$ .

Refinement and transition matrices

There are two matrices that are particularly important to determining the properties of wavelet systems. The first is the refinement matrix $M$ , which is obtained from the basic recursion equation [link] by evaluating $\phi \left(t\right)$ at integers [link] , [link] , [link] , [link] , [link] . This looks like a convolution matrix with the even (or odd) rows removed.Two particular submatrices that are used later in [link] to evaluate $\phi \left(t\right)$ on the dyadic rationals are illustrated for $N=6$ by

which we write in matrix form as

with ${\mathbf{M}}_0$ being the $6\times 6$ matrix of the $h\left(n\right)$ and $\underline{\phi }$ being $6\times 1$ vectors of integer samples of $\phi \left(t\right)$ . In other words, the vector $\underline{\phi }$ with entries $\phi \left(k\right)$ is the eigenvector of ${\mathbf{M}}_0$ for an eigenvalue of unity.

The second submatrix is a shifted version illustrated by

with the matrix being denoted ${\mathbf{M}}_1$ . The general refinement matrix $\mathbf{M}$ is the infinite matrix of which ${\mathbf{M}}_0$ and ${\mathbf{M}}_1$ are partitions. If the matrix $\mathbf{H}$ is the convolution matrix for $h\left(n\right)$ , we can denote the $\mathbf{M}$ matrix by $[\downarrow 2]\mathbf{H}$ to indicate the down-sampled convolution matrix $\mathbf{H}$ . Clearly, for $\phi \left(t\right)$ to be defined on the dyadic rationals, ${\mathbf{M}}_0$ must have a unity eigenvalue.

A third, less obvious but perhaps more important, matrix is called the transition matrix $\mathbf{T}$ and it is built up from the autocorrelation matrix of $h\left(n\right)$ . The transition matrix is constructed by

This matrix (sometimes called the Lawton matrix) was used by Lawton (who originally called it the Wavelet-Galerkin matrix) [link] to derive necessary and sufficient conditions for an orthogonal wavelet basis.As we will see later in this chapter, its eigenvalues are also important in determining the properties of $\phi \left(t\right)$ and the associated wavelet system.

Necessary conditions

Theorem 1 If $\phi \left(t\right)\in L^1$ is a solution to the basic recursion equation [link] and if $\int \phi \left(t\right)dt\ne 0$ , then

The proof of this theorem requires only an interchange in the order of a summation and integration (allowed in $L^1$ ) but no assumption of orthogonality of the basis functions or any otherproperties of $\phi \left(t\right)$ other than a nonzero integral. The proof of this theorem and several of the others stated here are contained inAppendix A.