05-12-2012, 11:45 AM
Wavelet Theory and Applications
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Introduction
Most signals are represented in the time domain. More information about the time signals can
be obtained by applying signal analysis, i.e. the time signals are transformed using an analysis
function. The Fourier transform is the most commonly known method to analyze a time signal
for its frequency content. A relatively new analysis method is the wavelet analysis. The wavelet
analysis differs from the Fourier analysis by using short wavelets instead of long waves for the
analysis function. The wavelet analysis has some major advantages over Fourier transform which
makes it an interesting alternative for many applications. The use and fields of application of
wavelet analysis have grown rapidly in the last years.
Historical overview
In 1807, Joseph Fourier developed a method for representing a signal with a series of coefficients
based on an analysis function. He laid the mathematical basis from which the wavelet theory is
developed. The first to mention wavelets was Alfred Haar in 1909 in his PhD thesis. In the 1930’s,
Paul Levy found the scale-varying Haar basis function superior to Fourier basis functions. The
transformation method of decomposing a signal into wavelet coefficients and reconstructing the
original signal again is derived in 1981 by Jean Morlet and Alex Grossman. In 1986, Stephane
Mallat and Yves Meyer developed a multiresolution analysis using wavelets. They mentioned
the scaling function of wavelets for the first time, it allowed researchers and mathematicians to
construct their own family of wavelets using the derived criteria. Around 1998, Ingrid Daubechies
used the theory of multiresolution wavelet analysis to construct her own family of wavelets. Her
set of wavelet orthonormal basis functions have become the cornerstone of wavelet applications
today. With her work the theoretical treatment of wavelet analysis is as much as covered.
Objective
The Fourier transform only retrieves the global frequency content of a signal. Therefore, the
Fourier transform is only useful for stationary and pseudo-stationary signals. The Fourier transform
does not give satisfactory results for signals that are highly non-stationary, noisy, a-periodic,
etc. These types of signals can be analyzed using local analysis methods. These methods include
the short time Fourier transform and the wavelet analysis. All analysis methods are based on the
principle of computing the correlation between the signal and an analysis function.
Approach
The Fourier transform (FT) is probably the most widely used signal analysis method. Understanding
the Fourier transform is necessary to understand the wavelet transform. The transition from
the Fourier transform to the wavelet transform is best explained through the short time Fourier
transform (STFT). The STFT calculates the Fourier transform of a windowed part of the signal
and shifts the window over the signal.
Wavelet analysis can be performed in several ways, a continuous wavelet transform, a discretized
continuous wavelet transform and a true discrete wavelet transform. The application of
wavelet analysis becomes more widely spread as the analysis technique becomes more generally
known. The fields of application vary from science, engineering, medicine to finance.
This report gives an introduction into wavelet analysis. The basics of the wavelet theory are
treated, making it easier to understand the available literature. More detailed information about
wavelet analysis can be obtained using the references mentioned in this report. The applications
described are thought to be of most interest to mechanical engineering.
The various analysis methods presented in this report will be compared using the time signal
x(t), shown in Fig. 1.1. From 0.1 s up to 0.3 s the signal consists of a sine with a frequency of
45 Hz, at 0.2 s the signal has a pulse. At 0.4 s the signal shows a sinusoid with a frequency of
250 Hz which changes to 75 Hz at 0.5 s. The time interval from 0.7 s up to 0.9 s consists of two
superposed sinusoids with frequencies of 30 Hz and 110 Hz. The signal is sampled at a frequency
of 1 kHz.
Outline
This report is organized as follows. The Fourier transform will be shortly addressed in Chapter 2.
The chapter discusses the continuous, discrete, fast and short time Fourier transforms. From
the short time Fourier transform the link to the continuous wavelet transform will be made in
Chapter 3. The wavelet functions and the continuous wavelet analysis method will be explained
together with a discretized version of the continuous wavelet transform. The true discrete wavelet
transform uses filter banks for the analysis and reconstruction of the time signal. Filter banks
and the discrete wavelet transform are the subject of Chapter 4. Wavelet analysis can be applied
for many different purposes. It is not possible to mention all different applications, the most
important application fields will be presented in Chapter 5. Finally conclusions will be drawn in
Chapter 6.
Fourier analysis
The Fourier transform (FT) is probably the most widely used signal analysis method. In 1807, a
French mathematician Joseph Fourier discovered that a periodic function can be represented by
an infinite sum of complex exponentials. Many years later his idea was extended to non-periodic
functions and then to discrete time signals. In 1965 the FT became even more popular by the
development of the Fast Fourier transform (FFT).
The Fourier transform retrieves the global information of the frequency content of a signal
and will be discussed in Section 2.1. A computationally more effective method is the fast Fourier
transform (FFT) which is the subject of Section 2.2. For stationary and pseudo-stationary signals
the Fourier transform gives a good description. However, for highly non-stationary signals some
limitations occur. These limitations are overcome by the short time Fourier transform (STFT),
presented in Section 2.3. The STFT is a time-frequency analysis method which is able to reveil
the local frequency information of a signal.
Fourier transform
The Fourier transform decomposes a signal into orthogonal trigonometric basis functions. The
Fourier transform of a continuous signal x(t) is defined in (2.1). The Fourier transformed signal
XFT (f) gives the global frequency distribution of the time signal x(t) [8, 16]. The original signal
can be reconstructed using the inverse Fourier transform (2.2).
Fast Fourier transform
The calculation of the DFT can become very time-consuming for large signals (large N). The fast
Fourier transform (FFT) algorithm does not take an arbitrary number of intervals N, but only
the intervals N = 2m, m 2 N. The reduction in the number of intervals makes the FFT very fast,
as the name implies. A drawback compared to the ordinary DFT is that the signal must have 2m
samples, this is however in general no problem.
In practice the calculation of the FFT can suffer from two problems. First since only a small
part of the signal x(t) on the interval 0 t T is used, leakage can occur. Leakage is caused
by the discontinuities introduced by periodically extending the signal. Leakage causes energy of
fundamental frequencies to leak out to neighboring frequencies. A solution to prevent signal leakage
is by applying a window to the signal which makes the signal more periodic in the time interval.
A disadvantage is that the window itself has a contribution in the frequency spectrum. The
second problem is the limited number of discrete signal values, this can lead to aliasing. Aliasing
causes fundamental frequencies to appear as different frequencies in the frequency spectrum and
is closely related to the sampling rate of the original signal. Aliasing can be prevented if the
sampling theorem of Shannon is fulfilled. The theorem of Shannon states that no information is
lost by the discretization if the sample time T equals or is smaller than T = 2/fmax. For more
detailed information regarding both problems the reader is referred to [8, 16
Wavelet analysis
The analysis of a non-stationary signal using the FT or the STFT does not give satisfactory results.
Better results can be obtained using wavelet analysis. One advantage of wavelet analysis is the
ability to perform local analysis [17]. Wavelet analysis is able to reveal signal aspects that other
analysis techniques miss, such as trends, breakdown points, discontinuities, etc. In comparison to
the STFT, wavelet analysis makes it possible to perform a multiresolution analysis.
The general idea of multiresolution analysis will be discussed in Section
The wavelet
functions and their properties are the subject of Section 3.2. The continuous wavelet transform
(CWT) will be treated in Section 3.3 together with the discretized version of the CWT.
Multiresolution analysis
The time-frequency resolution problem is caused by the Heisenberg uncertainty principle and exists
regardless of the used analysis technique. For the STFT, a fixed time-frequency resolution is used.
By using an approach called multiresolution analysis (MRA) it is possible to analyze a signal at
different frequencies with different resolutions. The change in resolution is schematically displayed
in Fig. 3.1.