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The case where $\phi \left(t\right)$ and $h\left(n\right)$ have compact support is very important. It aids in the time localization properties of the DWT andoften reduces the computational requirements of calculating the DWT. If $h\left(n\right)$ has compact support, then the filters described in Chapter: Filter Banks and the Discrete Wavelet Transform are simple FIR filters. We have stated that $N$ , the length of the sequence $h\left(n\right)$ , must be even and $h\left(n\right)$ must satisfy the linear constraint of [link] and the $\frac{N}{2}$ bilinear constraints of [link] . This leaves $\frac{N}{2}-1$ degrees of freedom in choosing $h\left(n\right)$ that will still guarantee the existence of $\phi \left(t\right)$ and a set of essentially orthogonal basis functions generated from $\phi \left(t\right)$ .
For a length-2 $h\left(n\right)$ , there are no degrees of freedom left after satisfying the required conditions in [link] and [link] . These requirements are
and
which are uniquely satisfied by
These are the Haar scaling functions coefficients which are also the length-2 Daubechies coefficients [link] used as an example in Chapter: A multiresolution formulation of Wavelet Systems and discussed later in this book.
For the length-4 coefficient sequence, there is one degree of freedom or one parameter that gives all the coefficients that satisfy the requiredconditions:
and
Letting the parameter be the angle $\alpha $ , the coefficients become
These equations also give the length-2 Haar coefficients [link] for $\alpha =0,\pi /2,3\pi /2$ and a degenerate condition for $\alpha =\pi $ . We get the Daubechies coefficients (discussed later in this book) for $\alpha =\pi /3$ . These Daubechies-4 coefficients have a particularly clean form,
For a length-6 coefficient sequence $h\left(n\right)$ , the two parameters are defined as $\alpha $ and $\beta $ and the resulting coefficients are
Here the Haar coefficients are generated for any $\alpha =\beta $ and the length-4 coefficients [link] result if $\beta =0$ with $\alpha $ being the free parameter. The length-4 Daubechies coefficients are calculatedfor $\alpha =\pi /3$ and $\beta =0$ . The length-6 Daubechies coefficients result from $\alpha =1.35980373244182\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}\beta =-0.78210638474440$ .
The inverse of these formulas which will give $\alpha $ and $\beta $ from the allowed $h\left(n\right)$ are
As $\alpha $ and $\beta $ range over $-\pi $ to $\pi $ all possible $h\left(n\right)$ are generated. This allows informative experimentation to better see whatthese compactly supported wavelets look like. This parameterization is implemented in the Matlab programs in Appendix C and in the Aware, Inc. software, UltraWave [link] .
Since the scaling functions and wavelets are used with integer translations, the location of their support is not important, only thesize of the support. Some authors shift $h\left(n\right)$ , ${h}_{1}\left(n\right)$ , $\phi \left(t\right)$ , and $\psi \left(t\right)$ to be approximately centered around the origin. This is achieved by having the initial nonzero scaling coefficient start at $n=-\frac{N}{2}+1$ rather than zero. We prefer to have the origin at $n=t=0$ .
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