0.5 The scaling function and scaling coefficients, wavelet and  (Page 4/13)

Frequency domain necessary conditions

We turn next to frequency domain versions of the necessary conditions for the existence of $\phi \left(t\right)$ . Some care must be taken in specifying the space offunctions that the Fourier transform operates on and the space that the transform resides in. We do not go into those details in this book butthe reader can consult [link] .

Theorem 5 If $\phi \left(t\right)$ is a $L^1$ solution of the basic recursion equation [link] , then the following equivalent conditions must be true:

This follows directly from [link] and states that the basic existence requirement [link] is equivalent to requiring that the FIR filter's frequency response at DC ( $\omega =0$ ) be $\sqrt{2}$ .

Theorem 6 For $h\left(n\right)\in {\ell }^1$ , then

which says the frequency response of the FIR filter with impulse response $h\left(n\right)$ is zero at the so-called Nyquist frequency ( $\omega =\pi$ ). This follows from [link] and [link] , and supports the fact that $h\left(n\right)$ is a lowpass digital filter. This is also equivalent to the $\mathbf{M}$ and $\mathbf{T}$ matrices having a unity eigenvalue.

Theorem 7 If $\phi \left(t\right)$ is a solution to [link] in $L^2\cap L^1$ and $\text{Φ}\left(\omega \right)$ is a solution of [link] such that $\text{Φ}\left(0\right)\ne 0$ , then

This is a frequency domain equivalent to the time domain definition of orthogonality of the scaling function [link] , [link] , [link] . It allows applying the orthonormal conditions to frequency domain arguments.It also gives insight into just what time domain orthogonality requires in the frequency domain.

Theorem 8 For any $h\left(n\right)\in {\ell }^1$ ,

This theorem [link] , [link] , [link] gives equivalent time and frequency domain conditions on the scaling coefficients and states thatthe orthogonality requirement [link] is equivalent to the FIR filter with $h\left(n\right)$ as coefficients being what is called a Quadrature Mirror Filter (QMF) [link] . Note that [link] , [link] , and [link] require $\left|H\right(\pi /2\left)\right|=1$ and that the filter is a “half band" filter.

Sufficient conditions

The above are necessary conditions for $\phi \left(t\right)$ to exist and the following are sufficient. There are many forms these could and do takebut we present the following as examples and give references for more detail [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] .

Theorem 9 If ${\sum }_{n}h\left(n\right)=\sqrt{2}$ and $h\left(n\right)$ has finite support or decays fast enough so that $\sum \left|h\left(n\right)\right|(1+{\left|n\right|)}^{ϵ}<\infty$ for some $ϵ>0$ , then a unique (within a scalar multiple) $\phi \left(t\right)$ (perhaps a distribution) exists that satisfies [link] and whose distributional Fourier transform satisfies [link] .

This [link] , [link] , [link] can be obtained in the frequency domain by considering the convergence of [link] . It has recently been obtained using a much more powerful approach in the time domain by Lawton [link] .

Because this theorem uses the weakest possible condition, the results are weak. The scaling function obtained from only requiring ${\sum }_{n}h\left(n\right)=\sqrt{2}$ may be so poorly behaved as to be impossible to calculate or use. The worst cases will not support a multiresolutionanalysis or provide a useful expansion system.

Theorem 10 If ${\sum }_{n}h\left(2n\right)={\sum }_{n}h(2n+1)=1/\sqrt{2}$ and $h\left(n\right)$ has finite support or decays fast enough so that $\sum \left|h\left(n\right)\right|(1+{\left|n\right|)}^{ϵ}<\infty$ for some $ϵ>0$ , then a $\phi \left(t\right)$ (perhaps a distribution) that satisfies [link] exists, is unique, and is well-defined on the dyadic rationals. In addition, the distributional sum