Wavelets - Caltech Multi-Res Modeling Group

and the function can be reconstructed as:
f (t) =
1X
i=0
wi Wi(t)
Figures I.7 and I.8 show the coefficients of the Walsh basis for both of the above signals.
5
4
3
2
1
0
-1
-1
-2
-3
0123456
INTRODUCTION 9
0 5 10 15 20 25 30
Figure I.7: Walsh coefficients for signal 1.
0 5 10 15 20 25 30
Figure I.8: Walsh coefficients for signal 2.
Note that since the original signals have discontinuities only at integral values, the signals are exactly
represented by the first 32 Walsh bases at most. But we should also note that in this example, as well as
would be the case for a Fourier transform, the presence of a single discontinuity at 21 for signal 1 introduces
the highest “frequency” basis, and it has to be added globally for all t. In general cases the coefficients for
each basis function decrease rapidly as the order increases, and that usually allows for a simplification (or
compression) of the representation of the original signal by dropping the basis functions whose coefficients
are small (obviously with loss of information if the coefficients are not 0).
Siggraph ’95 Course Notes: #26 Wavelets