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Looking at the roots of sequences of polynomials is always interesting,
and our sequence of filters just described is no exception. Again, Mallat's
Tree can be used with these three rules:
- Convolving two polynomials is equivalent to forming the union of
their roots.
- Filtering decimated data is equivalent to filtering the undecimated
data with a zero-padded filter followed by decimating the output.
- The roots of a filter zero-padded between every coefficient are the
square roots of the roots of the same unpadded filter.

With these rules in place, the roots of the new low-pass filter are the roots
of the old low-pass filter in union with the roots of the basic low-pass
wavelet zero-padded by the number of stages.
This is illustrated in Figures and .
**tree
**

Figure 11 Mallat's tree.

**haarwave
**

Figure 12 The wavelets and the zeros
for the Haar wavelets.
In each plot, the top row shows the high-pass Haar wavelet on the left
and its zeros on the complex plane on the right. The bottom
row shows the low-pass Haar wavelet and its zeros. Multiple zeros have
multiple circles drawn around them. In the electronic version of
this paper, pressing the button starts a movie of these figures.

**wave
**

Figure 13 The wavelets and the zeros
for various forms of *D*_{4}.
In each plot, the top row shows the high-pass *D*_{4} wavelet on the left
and its zeros on the complex plane on the right. The bottom
row shows the low-pass *D*_{4} wavelet and its zeros. Multiple zeros have
multiple circles drawn around them. In the electronic version of
this paper, pressing the button starts a movie of these figures.

** Next:** CONCLUSIONS
** Up:** MALLAT'S TREE
** Previous:** The tree as a
Stanford Exploration Project

11/17/1997