0.5 The scaling function and scaling coefficients, wavelet and  (Page 8/13)

for the appropriate scaling coefficients $h\left(n\right)$ and some $K$ . If we construct the scaling function from the generalized sampling function aspresented in [link] , the sinc function becomes

In order for these two equations to be true, the sampling period must be $T=1/2$ and the parameter

which gives the scaling coefficients as

We see that $\phi \left(t\right)=\text{sinc}\left(Kt\right)$ is a scaling function with infinite support and its corresponding scaling coefficients are samples ofa sinc function. If $R=1$ , then $K=\pi$ and the scaling function generates an orthogonal wavelet system. For $R>1$ , the wavelet system is a tight frame, the expansion set is not orthogonal or a basis, and $R$ is the amount of redundancy in the system as discussed in this chapter.For the orthogonal sinc scaling function, the wavelet is simply expressed by

The sinc scaling function and wavelet do not have compact support, but they do illustrate an infinitely differentiable set of functions that resultfrom an infinitely long $h\left(n\right)$ . The orthogonality and multiresolution characteristics of the orthogonal sinc basis is best seen in thefrequency domain where there is no overlap of the spectra. Indeed, the Haar and sinc systems are Fourier duals of each other. The sinc generatingscaling function and wavelet are shown in [link] .

Spline and battle-lemarié wavelet systems

The triangle scaling function illustrated in Figure: Haar and Triangle Scaling Functions is a special case of a more general family of spline scaling functions. The scaling coefficient system $h\left(n\right)=\{\frac{1}{2\sqrt{2}},\frac{1}{\sqrt{2}},\frac{1}{2\sqrt{2}},0\}$ gives rise to the piecewise linear, continuous triangle scaling function. Thisfunction is a first-order spline, being a concatenation of two first order polynomials to be continuous at the junctions or “knots". A quadraticspline is generated from $h=\{1/4,3/4,3/4,1/4\}/\sqrt{2}$ as three sections of second order polynomials connected to give continuous first orderderivatives at the junctions. The cubic spline is generated from $h\left(n\right)=\{1/16,1/4,3/8,1/4,1/16\}/\sqrt{2}$ . This is generalized to an arbitrary $N$ th order spline with continuous $(N-1)$ th order derivatives and with compact support of $N+1$ . These functions have excellent mathematical properties, but they are not orthogonal over integer translation. Iforthogonalized, their support becomes infinite (but rapidly decaying) and they generate the “Battle-Lemarié wavelet system" [link] , [link] , [link] , [link] . [link] illustrates the first-order spline scaling function which is the triangle functionalong with the second-, third-, and fourth-order spline scaling functions.

Further properties of the scaling function and wavelet

The scaling function and wavelet have some remarkable properties thatshould be examined in order to understand wavelet analysis and to gain some intuition for these systems. Likewise, the scaling and waveletcoefficients have important properties that should be considered.

We now look further at the properties of the scaling function and the wavelet in terms of the basic defining equations and restrictions. Wealso consider the relationship of the scaling function and wavelet to the equation coefficients. A multiplicity or rank of two is used here but themore general multiplicity-M case is easily derived from these (See Section: Multiplicity-M (M-Band) Scaling Functions and Wavelets and Appendix B ). Derivations or proofs for some of these properties are included in Appendix B .