0.9 Calculation of the discrete wavelet transform

Although when using the wavelet expansion as a tool in an abstract mathematical analysis, the infinite sum and the continuous description of $t\in \mathbf{R}$ are appropriate, as a practical signal processing or numerical analysis tool, the function or signal $f\left(t\right)$ in [link] is available only in terms of its samples, perhaps with additionalinformation such as its being band-limited. In this chapter, we examine the practical problem of numerically calculating the discrete wavelettransform.

Finite wavelet expansions and transforms

The wavelet expansion of a signal $f\left(t\right)$ as first formulated in [link] is repeated here by

where the $\left\{{\psi }_{j,k}\left(t\right)\right\}$ form a basis or tight frame for the signal space of interest (e.g., $L_2$ ). At first glance, this infinite series expansion seems to have the same practical problems in calculation that aninfinite Fourier series or the Shannon sampling formula has. In a practical situation, this wavelet expansion, where the coefficients arecalled the discrete wavelet transform (DWT), is often more easily calculated. Both the time summation over the index $k$ and the scale summation over the index $j$ can be made finite with little or no error.

The Shannon sampling expansion [link] , [link] of a signal with infinite support in terms of sinc $\left(t\right)=\frac{sin\left(t\right)}{t}$ expansion functions

requires an infinite sum to evaluate $f\left(t\right)$ at one point because the sinc basis functions have infinite support. This is not necessarily true for awavelet expansion where it is possible for the wavelet basis functions tohave finite support and, therefore, only require a finite summation over $k$ in [link] to evaluate $f\left(t\right)$ at any point.

The lower limit on scale $j$ in [link] can be made finite by adding the scaling function to the basis set as was done in [link] . By using the scaling function, the expansion in [link] becomes

where $j=J_0$ is the coarsest scale that is separately represented. The level of resolution or coarseness to start the expansion with is arbitrary,as was shown in  Chapter: A multiresolution formulation of Wavelet Systems in [link] , [link] , and [link] . The space spanned by the scaling function contains all the spaces spanned by the lower resolution wavelets from $j=-\infty$ up to the arbitrary starting point $j=J_0$ . This means ${\mathcal{V}}_{J_0}={\mathcal{W}}_{-\infty }\oplus \cdots \oplus {\mathcal{W}}_{J_0-1}$ . In a practical case, this would be the scale where separating detail becomes important. For asignal with finite support (or one with very concentrated energy), the scaling function might be chosen so that the support of the scalingfunction and the size of the features of interest in the signal being analyzed were approximately the same.

This choice is similar to the choice of period for the basis sinusoids in a Fourier series expansion. If the period of the basis functions ischosen much larger than the signal, much of the transform is used to describe the zero extensions of the signal or the edge effects.

The choice of a finite upper limit for the scale $j$ in [link] is more complicated and usually involves some approximation. Indeed, forsamples of $f\left(t\right)$ to be an accurate description of the signal, the signal should be essentially bandlimited and the samples taken at least at theNyquist rate (two times the highest frequency in the signal's Fourier transform).