28-08-2012, 12:36 PM
A Tutorial of the Wavelet Transform
WaveletTutorial.pdf (Size: 1.71 MB / Downloads: 348)
Introduction
The Fourier transform is an useful tool to analyze the frequency components
of the signal. However, if we take the Fourier transform over the whole time
axis, we cannot tell at what instant a particular frequency rises. Short-time
Fourier transform (STFT) uses a sliding window to nd spectrogram, which
gives the information of both time and frequency. But still another problem
exists: The length of window limits the resolution in frequency. Wavelet
transform seems to be a solution to the problem above. Wavelet transforms
are based on small wavelets with limited duration. The translated-version
wavelets locate where we concern. Whereas the scaled-version wavelets allow
us to analyze the signal in dierent scale.
History
The rst literature that relates to the wavelet transform is Haar wavelet. It
was proposed by the mathematician Alfrd Haar in 1909. However, the concept
of the wavelet did not exist at that time. Until 1981, the concept was
proposed by the geophysicist Jean Morlet. Afterward, Morlet and the physicist
Alex Grossman invented the term wavelet in 1984. Before 1985, Haar
wavelet was the only orthogonal wavelet people know. A lot of researchers
even thought that there was no orthogonal wavelet except Haar wavelet. Fortunately,
the mathematician Yves Meyer constructed the second orthogonal
wavelet called Meyer wavelet in 1985. As more and more scholars joined in
this eld, the 1st international conference was held in France in 1987.
Continuous Wavelet Transform
Introduction
In this chapter, we talk about continuous wavelet transform. This transform
works when we use a continuous wavelet function to nd the detailed coef-
cients of a continuous signal. Like the concept in chapter 2, we have to
establish a basis to do such analysis. First, we give the denition of continuous
wavelet transform and do some comparison between that and the Fourier
transform.
Conclusions
In this tutorial, we explore the world of wavelets. From the abstract idea
in approximation, multiresolition theory, we generalize the Fourier transform
and start the journey of wavelets. The discrete wavelet is more useful in realization.
We often use 2D wavelets to do image compression. The continuous
wavelet transform analyze the continuous-time signal in a dierent perspective.
By the advantage of multiresolution, we can locate time and frequency
more accurately. The wavelet design is more complicated in mathematics but
the design procedure completes the existence of the wavelets. The application
chapter mentions the nowadays JPEG and JPEG2000 standard.
[8] and [9] give a thorough approach to the wavelets but in purely mathematically
words. [1] and [15] illustrate some examples on the wavelet and
then the abstract concepts. Other references mainly focus on some parts of
the materials.
The wavelets bring us to a new vision of signal processing. It tactically
avoids the problem that Fourier analysis encounters. Its implementation
is simple. We need some designed lters to do the task. Although the
implementation is quite easy, the lter design includes lots of mathematical
originality and this is the research topic.