A fast algorithm for wavelet transformations uses the Mallat tree as seen in Figure . The algorithm involves two wavelets, low-pass and high-pass, with the one derived from the other by changing the sign of the even coefficients.

In Figure , the input data series is represented by
the circle marked `input'. The result referred to as Lo-1 is
from an application of the low-pass filter with a decimation.
The low-pass filter for the Haar transform
is the 2-term low-pass operator in Figure ,
and the low-pass filter for the *D _{4}* transform is
4-term low-pass operator in Figure .
Hi-1 comes from filtering with the corresponding high-pass filters.
These filters
are illustrated by the 2-term high-pass operator in Figure ,
and the 4-term high-pass operator in Figure . Hi-1 is now set aside,
and the data in Lo-1 becomes the input for the next step.
Elsewhere in this report, Schwab expands on how the filters
are calculated (Schwab, 1992).

At each step, the high-pass output is saved and becomes part of the transformed output, and the final lo-pass output makes up the rest of the output.

11/17/1997