0.5 The scaling function and scaling coefficients, wavelet and  (Page 6/13)

Theorems [link] , [link] , and [link] show that $h\left(n\right)$ has the characteristics of a lowpass FIR digital filter. We will later see that the FIR filter made up of the wavelet coefficients is ahigh pass filter and the filter bank view developed in  Chapter: Filter Banks and the Discrete Wavelet Transform and Section: Multiplicity-M (M-Band) Scaling Functions and Wavelets further explains this view.

Theorem 16 If $h\left(n\right)$ has finite support and if $\phi \left(t\right)\in L^1$ , then $\phi \left(t\right)$ has finite support [link] .

If $\phi \left(t\right)$ is not restricted to $L^1$ , it may have infinite support even if $h\left(n\right)$ has finite support.

These theorems give a good picture of the relationship between the recursive equation coefficients $h\left(n\right)$ and the scaling function $\phi \left(t\right)$ as a solution of [link] . More properties and characteristics are presented in [link] .

Wavelet system design

One of the main purposes for presenting the rather theoretical results of this chapter is to set up the conditions for designing wavelet systems.One approach is to require the minimum sufficient conditions as constraints in an optimization or approximation,then use the remaining degrees of freedom to choose $h\left(n\right)$ that will give the best signalrepresentation, decomposition, or compression. In some cases, the sufficient conditions are overly restrictive and it is worthwhile to usethe necessary conditions and then check the design to see if it is satisfactory. In many cases, wavelet systems are designed by a frequencydomain design of $H\left(\omega \right)$ using digital filter design techniques with wavelet based constraints.

The wavelet

Although this chapter is primarily about the scaling function, some basic wavelet properties are included here.

Theorem 17 If the scaling coefficients $h\left(n\right)$ satisfy the conditions for existence and orthogonality of the scaling function and the wavelet is defined by [link] , then the integer translates of this wavelet span ${\mathcal{W}}_0$ , the orthogonal compliment of ${\mathcal{V}}_0$ , both being in ${\mathcal{V}}_1$ , i.e., the wavelet is orthogonal to the scaling function at the samescale,

if and only if the coefficients $h_1\left(n\right)$ are given by

where $N$ is an arbitrary odd integer chosen to conveniently position $h_1\left(n\right)$ .

An outline proof is in Appendix A.

Theorem 18 If the scaling coefficients $h\left(n\right)$ satisfy the conditions for existence and orthogonality of the scaling function and the wavelet is defined by [link] , then the integer translates of this wavelet span ${\mathcal{W}}_0$ , the orthogonal compliment of ${\mathcal{V}}_0$ , both being in ${\mathcal{V}}_1$ ; i.e., the wavelet is orthogonal to the scaling function at the samescale. If

then

which is derived in Appendix A, [link] .

The translation orthogonality and scaling function-wavelet orthogonality conditions in [link] and [link] can be combined to give

if $h_0\left(n\right)$ is defined as $h\left(n\right)$ .

Theorem 19 If $h\left(n\right)$ satisfies the linear and quadratic admissibility conditions of [link] and [link] , then

and

The wavelet is usually scaled so that its norm is unity.

The results in this section have not included the effects of integer shifts of the scaling function or wavelet coefficients $h\left(n\right)$ or $h_1\left(n\right)$ . In a particular situation, these sequences may be shifted to make thecorresponding FIR filter causal.