0.5 The scaling function and scaling coefficients, wavelet and  (Page 7/13)

Alternate normalizations

An alternate normalization of the scaling coefficients is used by some authors. In some ways, it is a cleaner form than that used here, but itdoes not state the basic recursion as a normalized expansion, and it does not result in a unity norm for $h\left(n\right)$ . The alternate normalization uses the basic multiresolution recursive equation with no $\sqrt{2}$

Some of the relationships and results using this normalization are:

A still different normalization occasionally used has a factor of 2 in [link] rather than $\sqrt{2}$ or unity, giving ${\sum }_{n}h\left(n\right)=1$ . Other obvious modificationsof the results in other places in this book can be worked out. Take care in using scaling coefficients $h\left(n\right)$ from the literature as some must be multiplied or divided by $\sqrt{2}$ to be consistent with this book.

Example scaling functions and wavelets

Several of the modern wavelets had never been seen or described before the 1980's. This section looks at some of the most common wavelet systems.

Haar wavelets

The oldest and most basic of the wavelet systems that has most of our desired properties is constructed from the Haar basis functions. If onechooses a length $N=2$ scaling coefficient set, after satisfying the necessary conditions in [link] and [link] , there are no remaining degrees of freedom. The unique (within normalization) coefficients are

and the resulting normalized scaling function is

The wavelet is, therefore,

Their satisfying the multiresolution equation [link] is illustrated in Figure: Haar and Triangle Scaling Functions . Haar showed that translates and scalings of these functions form anorthonormal basis for $L^2\left(R\right)$ . We can easily see that the Haar functions are also a compact support orthonormal wavelet system thatsatisfy Daubechies' conditions [link] . Although they are as regular as can be achieved for $N=2$ , they are not even continuous. The orthogonality and nesting of spanned subspaces are easily seen because the translates haveno overlap in the time domain. It is instructive to apply the various properties of  [link] and [link] to these functions and see how they are satisfied. They are illustrated in the example in  Figure: Haar Scaling Functions and Wavelets that Span $V_j$ through Figure: Haar Function Approximation in $V_j$ .

Sinc wavelets

The next best known (perhaps the best known) basis set is that formed by the sinc functions. The sinc functions are usually presented in thecontext of the Shannon sampling theorem, but we can look at translates of the sinc function as an orthonormal set of basis functions (or, in somecases, a tight frame). They, likewise, usually form a orthonormal wavelet system satisfying the various required conditions of a multiresolutionsystem.

The sinc function is defined as

where $\text{sinc}\left(0\right)=1$ . This is a very versatile and useful function because its Fourier transform is a simple rectangle function and theFourier transform of a rectangle function is a sinc function. In order to be a scaling function, the sinc must satisfy [link] as