Digital Signal Processing Reference
In-Depth Information
It is also characteristic of real signals that higher frequency segments are often in the form
of short time bursts (see Illustration 45 and Illustration 46). By contrast lower frequencies
must naturally last longer and are of greater constancy.
This “defect” led from a historical point of view to the so-called Wavelet Transformation,
the point of departure of which will be described in the following.
“Wavelets” meaning short time wave forms may provide a more precise description of the
relationship between frequency and time. Almost all real signals are non-stationary, i.e.
their spectrum changes over time. As can be seen from Illustration 60 and Illustration 61
the most important part of its signal - its information content - is hidden in its time-fre-
quency signature. It is here immaterial whether it is the analysis of an ECG time segment
by a cardiologist or the identification of a whale by a marine biologist by means of its
song.
At the beginning of the 80s the geophysicist Jean MORLET developed an alternative to
the FOURIER Transformation STFT, which took up precisely the idea suggested here. He
wanted to analyze seismographic data, which had properties with very different frequen-
cies at different positions in the time domain, in order to locate new oilfields more pre-
cisely than the STFT permitted. For this purpose he used an adjustable window function,
which he could condense and stretch. He slid this over the entire “signal” for each window
width.
The aim of this procedure is constant relative uncertainty, a process in which the product
of the frequency to be analysed and the frequency uncertainty of the window used is con-
stant. Put simply, this means:
For all frequencies to be analysed the window is made sufficiently
wide by scaling that in each case the same number of periods fills
the window.
Put even more simply: wide window in the case of lower frequen-
cies and a narrow window with high frequencies so that near- pe-
riodic signals with comparable uncertainty are always analysed
in the time and frequency domain.
In this connection the word “wavelet” occurred for the first time because the window
looked like a short wave which begins gently and directly ends gently. MORLET was in
touch with the theoretical physicist Alex GROSSMAN and later with the mathematician
Yves MEYER. MEYER demonstrated among other things that numerous mathematical
solutions for the problems of pattern-recognition and the reduction of complex patterns to
simple basic patterns already existed.
The formal development of these insights led to the Wavelet Transformation which makes
possible a unified way of looking at pattern-recognition of many individual developments
in the field of signal analysis, signal compression and signal transmission.
In the course of this development other basic patterns (“mother-wavelets”), which can
“dissect” complex signals - e.g. images - more efficiently, were used instead of a sinu-
soidal basic pattern. These merely had to fulfil certain mathematical criteria which,
however, hold good more or less for all wave forms.
