1 The wavelet transform - International Computer Science Institute

83
Audio encoding schemes
output signal. The 16 buttons together correspond to the complete transform block. Activating
the first button on the left will set one sixteenth of the transform coefficients to zero. In order to
use this kind of compression in a meaningful way, it is necessary to know how the transform
coefficients are arranged within the transform interval. When a signal is compressed its reconstruction
from the remaining transform coefficients differs from the original signal. In the top
right corner of the user interface the error label (11) gives a coarse measurement of a mean
error history. We calculate the mean error with
e
n – 1
1
= -- u
n ∑ k – ûk k = 0
where n is the number of samples, ui the i -th coefficient of the original sample vector and ûi the i -th coefficient of the reconstructed vector. Since this value varies too quickly with every
other transformation, we rather display an error value that is an average over the history of the
mean error with
eh = 0, 99eh + 0, 01e
The error label (11) shows the value multiplied with a scaling factor of 100.
The two big buttons (12) + (13) open two separate windows. The signal viewer (12) is a small
strip chart for a real-time visualization of either the signal coefficients - button (2) deactivated -
or the transform coefficients - button (2) activated. In figure 80 two screen shots of the signal
a) b)
e h
FIGURE 80. The signal viewer shows the coefficients of a Daubechies8 wavelet transform with 50%
absolute compression (a) without and (b) with activated sort option.
viewer are shown. It should be noted that the strip chart depicts only absolute values. The base
viewer (13) is of relevance for the wave packet transform. On one hand it allows to illustrate the
wave packet bases that are result of the best bases search. On the other hand it is the means for
explicitly selecting a certain basis for a wave packet transform and use it as fixed basis. See
figure 81 on page 84 for an example screen shot. The same illustration with rectangles was
already used when the wavelet theory was introduced, see “The wave packet transform” on
page 58The two top rectangles correspond to the first decomposition level and from the top to
the bottom seven decomposition levels are shown. The coefficients of the eight decomposition
level are used when none of the above levels was selected. With simply clicking on the appropriate
rectangles the user can compose an arbitrary basis for the wave packet transform. The
library of bases of the wave packet transform includes the transform basis of both the wavelet