Digital Signal Processing Reference
In-Depth Information
Thus, the wavelet transformation is used not only for frequency analysis but also for a ver-
satile analysis of patterns. The wavelet transformation is not directly a “multi-frequency”
analysis (FOURIER analysis) but rather a multi-scale analysis. The whole wavelet trans-
formation could take place entirely in the time domain but would then require a great deal
of computing.
The point of departure is the “mother wavelet”, the actual basic pattern. This is now com-
pressed (or stretched) by a scaling function. In the case of a high scaling value a wide time
window is present in which the wavelet changes in a slow rhythm. The “rhythmic simi-
larities” between the signal and the wavelet are to be determined by multiplication. If the
momentary median value (“sliding median value”) of this signal window is large, the
wavelet and the signal agree rhythmically in this time domain! The smaller the scaling
value becomes, the more the window is compressed and the more rapid (high-frequency)
rhythmic changes are registered.
The scaling works like a zoom lens which can be changed almost as with a slide control
from a wide-angle lens to a telephoto lens. The scaling is thus inversely proportional to
the frequency. A small scaling value thus implies “high frequency”. Instead of the fre-
quency axis in the FOURIER transformation, in the wavelet transformation the scaling
axis is used. Usually the scaling values are entered on the axis logarithmically, i.e. small
values appear disproportionately large. This is shown very clearly by Illustration 64 in
contrast to Illustration 65.
By means of the Wavelet Transformation it is possible to extend
the uncertainty phenomenon not only to the time-frequency prob-
lem but also to other patterns which are contained in the mother
wavelet, e.g. jumps and discontinuities. It is specialised in
analysing, filtering out and saving any number of kinds of change
more efficiently.
The Uncertainty Principle is also applicable without modification
to wavelet transformation. A suitable choice of pattern for the
mother wavelet and skilful scaling makes a more precise analysis
of measurements possible,
when
or
where
certain frequency
bands - which always point to a temporal or local change (e.g. in
the case of pictures) - are present in a signal.
The wavelet transformation is much more efficient in many practical applications than the
FOURIER transformation. This can be explained in a particularly straightforward way in
the case of a sudden change in a signal: in such a sudden change the FOURIER transfor-
mation results in “an infinite number” of frequencies (“FOURIER coefficients”). Illustra-
tion 27 to Illustration 29 show most clearly how difficult it is to “reproduce” a sudden
change even with a large number of frequencies.
Sudden changes of this kind can be reproduced using far fewer wavelet coefficients by the
choice of wavelets with a very small scaling - i.e. short time windows. The wavelet trans-
formation makes it possible to get rid of redundant information (signal compression), i.e.
reduce the volume of the data. The most efficient processes at present for the compression
of pictures are therefore based on wavelet transformation.
