0.13 Appendix b

In this appendix we develop most of the results on scaling functions, wavelets and scaling and wavelet coefficients presented in [link] and elsewhere. For convenience, we repeat [link] , [link] , [link] , and [link] here

If normalized

The results in this appendix refer to equations in the text writtenin bold face fonts.

Equation [link] is the normalization of [link] and part of the orthonormal conditions required by [link] for $k=0$ and $E=1$ .

Equation [link] If the $\phi (x-k)$ are orthogonal, [link] states

Summing both sides over $m$ gives

which after reordering is

Using [link] , [link] , and [link] gives

but $\int \phi \left(x\right)dx=A_0$ from [link] , therefore

If the scaling function is not normalized to unity, one can show the more general result of [link] . This is done by noting that a more general form of [link] is

if one does not normalize $A_0=1$ in [link] through [link] .

Equation [link] follows from summing [link] over $m$ as

which after reordering gives

and using [link] gives [link] .

Equation [link] is derived by applying the basic recursion equation to its own right hand side to give

which, with a change of variables of $\ell =2n+k$ and reordering of operation, becomes

Applying this $j$ times gives the result in [link] . A similar result can be derived for the wavelet.

Equation [link] is derived by defining the sum

and using the basic recursive equation [link] to give

Interchanging the order of summation gives

but the summation over $\ell$ is independent of an integer shift so that using [link] and [link] gives

This is the linear difference equation

which has as a solution the geometric sequence

If the limit exists, equation [link] divided by $2^J$ is the Riemann sum whose limit is the definition of the Riemann integral of $\phi \left(x\right)$

It is stated in [link] and shown in [link] that if $\phi \left(x\right)$ is normalized, then $A_0=1$ and [link] becomes

which gives [link] .

Equation [link] shows another remarkable property of $\phi \left(x\right)$ in that the bracketed term is exactly equal to the integral, independent of $J$ . No limit need be taken!

Equation [link] is the “partitioning of unity" by $\phi \left(x\right)$ . It follows from [link] by setting $J=0$ .

Equation [link] is generalization of [link] by noting that the sum in [link] is independent of a shift of the form

for any integers $M\ge J$ and $L$ . In the limit as $M\to \infty$ , $\frac{L}{2^M}$ can be made arbitrarily close to any $x$ , therefore, if $\phi \left(x\right)$ is continuous,

This gives [link] and becomes [link] for $J=0$ . Equation [link] is called a “partitioning of unity" for obvious reasons.

The first four relationships for the scaling function hold in a generalized form for the more general defining equation [link] . Only [link] is different. It becomes

for $M$ an integer. It may be possible to show that certain rational $M$ are allowed.

Equations [link] , [link] , and [link] are the recursive relationship for the Fourier transform of the scaling function and areobtained by simply taking the transform [link] of both sides of [link] giving

which after the change of variables $y=2t-n$ becomes

and using [link] gives

which is [link] and [link] . Applying this recursively gives the infinite product [link] which holds for any normalization.

Equation [link] states that the sum of the squares of samples of the Fourier transform of the scaling function is one if the samples areuniform every $2\pi$ . An alternative derivation to that in Appendix A is shown here by taking the definition of the Fourier transform of $\phi \left(x\right)$ , sampling it every $2\pi k$ points and multiplying it times its complex conjugate.

Summing over $k$ gives

but

therefore

which becomes

Because of the orthogonality of integer translates of $\phi \left(x\right)$ , this is not a function of $\omega$ but is ${\int \left|\phi \left(x\right)\right|}^{2}dx$ which, if normalized, is unity as stated in [link] . This is the frequency domain equivalent of [link] .

Equations [link] and [link] show how the scaling function determines the equation coefficients. This is derived by multiplying bothsides of [link] by $\phi (2x-m)$ and integrating to give

Using the orthogonality condition [link] gives

which gives [link] . A similar argument gives [link] .