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Appendix A (Wavelets: Evolution, Types and Properties)
A. History of Wavelets and its Evolution
The development of wavelets can be linked to several separate trains of
thought, starting with Haar's work in the early 20th century. Later work by Dennis
Gabor yielded Gabor atoms (1946), which are constructed similarly and applied to
similar purposes as wavelets. Notable contributions to wavelet theory can be
attributed to Zweig’s discovery of the continuous wavelet transform in 1975
(originally called the cochlear transform and discovered while studying the
reaction of the ear to sound), Pierre Goupillaud, Grossmann and Morlet's
formulation of what is now known as the CWT (1982), Jan-OlovStrömberg's early
work on discrete wavelets (1983), Daubechies' orthogonal wavelets with compact
support (1988), Mallat’s multiresolution framework (1989), Nathalie Delprat's
time-frequency interpretation of the CWT (1991), Newland's harmonic wavelet
transform (1993) and many others since.
Timeline
First wavelet (Haar wavelet) by Alfred Haar (1909)
Since the 1970s: George Zweig, Jean Morlet, Alex Grossmann
Since the 1980s: Yves Meyer, Stéphane Mallat, Ingrid Daubechies,
Ronald Coifman, Victor Wickerhauser
Since the 1990s: Wavelet Packet Transform: Ronald Coifman, Victor
Wickerhauser, Ridgelelet Transform: E. J. Candès, D. L. Donoho.
Since the 2000s: Digital Curvelet Transform: E. J. Candès, D. L.
Donoho, FDCT: Laurent Demanety, Lexing Yingz, Denoising: J.
Starck
Prior to wavelet analysis, Fourier transform and Cosine transform were in use for
solution of majority of the problems. The need of simultaneous representation and
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localization of both time and frequency for non-stationary signals (e.g. music,
speech, images) led to the evolution of wavelet transform from the popular
Fourier transform and Cosine transform.
A.1 Fourier Analysis
Signal analysts have at their disposal an impressive arsenal of tools.
Perhaps the most well-known of these is Fourier analysis, which breaks down a
signal into constituent sinusoids of different frequencies. Another way to think of
Fourier analysis is as a mathematical technique for transforming our view of the
signal from time-based to frequency-based as can be seen in figure A.1.
Fig. A.1: Fourier transform representation
For many signals, Fourier analysis is extremely useful because the signal's
frequency content is of great importance. So why are the other techniques like
wavelet analysis required?
Fourier analysis has a serious drawback. In transforming to the frequency
domain, time information is lost. When looking at a Fourier transform of a signal, it
is impossible to tell when a particular event took place. If the signal properties do
not change much over time (for example stationary signals), this drawback is not
very important. However, most interesting signals contain numerous non-
stationary or transitory characteristics such as drift, trends, abrupt changes and
beginnings and ends of events. These characteristics are often the most important
part of the signal and Fourier analysis is not suited to detect them [Doc2013].
A.2 Short Time Fourier Transform (STFT)
In an effort to correct the deficiency of Fourier analysis, Dennis Gabor
(1946) adapted the Fourier transform to analyze only a small section of the signal
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at a time. This technique is called windowing the signal and Gabor's adaptation is
called the Short Time Fourier Transform (STFT) which maps a signal into a two
dimensional function of time and frequency as shown in figure A.2.
Fig. A.2: Short Time Fourier Transform representations
The STFT represents a sort of compromise between the time and frequency based
representations of a signal. It provides some information about that when and also
at what frequencies a signal event occurs. However, it can only be obtained with
limited precision and that precision is determined by the size of the window.
While the STFT compromise between time and frequency information can be
useful, the drawback is that once a particular size for the time window is chosen, it
remains same for all the frequencies. Many signals require a more flexible
approach, one where the window size can be varied to determine more accurately
either time or frequency [Doc2013].
A.3 Wavelet Analysis
Wavelet analysis represents the next logical step: a windowing technique
with variable sized regions. Wavelet analysis allows the use of long time intervals
where more precise low-frequency information is required and shorter regions
where high frequency information is required. The representation is shown in
figure A.3.
Fig. A.3: Wavelet transform representation
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In contrast with the time, frequency and Gabor wavelet based analysis, wavelet
analysis does not use a time-frequency region, but rather a time-scale region as
shown in figure A.4 [Doc2013].
Fig. A.4: Time-based, frequency-based, STFT and Wavelet views of a signal
A wavelet is a waveform of effectively limited time duration that has an
average value of zero. Comparing wavelets with sine waves, which are the basis of
Fourier analysis, sinusoids do not have limited duration. They extend from minus
infinity to plus infinity. The sinusoids are smooth and predictable while wavelets
tend to be irregular and asymmetric.
Fourier analysis consists of breaking up a signal into sine waves of various
frequencies. Similarly, wavelet analysis is the breaking up of a signal into shifted
and scaled versions of the original (or mother) wavelet. A sinusoid (basis of
Fourier) and a wavelet (basis of wavelet transform) are shown in figure A.5.
Sine wave Wavelet (Db10)
Fig. A.5: Comparison of wavelets with sine wave
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From figure A.5, it can be observed intuitively that signals with sharp changes
might be better analyzed with an irregular wavelet than with a smooth sinusoid.
Similarly, local features can be described better with wavelets that have local
extent.
A.2 Wavelet Analysis Vs. Fourier Analysis
One major advantage afforded by wavelets is the ability to perform local
analysis i.e. analysis of a localized area of a larger signal. Consider a sinusoidal
signal with a small discontinuity. This discontinuity could be as tiny as to be barely
visible. Such a signal with a tiny discontinuity is shown in figure A.6. Such a signal
easily could be generated in the real world, perhaps by a power fluctuation or a
noisy switch.
Fig. A.6: Signal representing discontinuity
Plots of the Fourier coefficients and wavelet coefficients of the signal of figure A.6
are shown in figure A.7 (a) and A.7 (b).
(a) Fourier Coefficients (b) Wavelet Coefficients
Fig. A.7: Discontinuity represented by Fourier and Wavelet coefficients
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The plot of Fourier coefficients shows nothing particularly interesting, rather a flat
spectrum with two peaks representing a single frequency. However, a plot of
wavelet coefficients clearly shows the exact location in time of the discontinuity.
Therefore, wavelet analysis is capable of revealing aspects of data that other signal
analysis techniques generally miss such as trends, breakdown points,
discontinuities in higher derivatives and self-similarity [Zho2010]. Some
similarities and differences in Fourier and wavelet analysis are explained in
following sections.
A.2.1 Similarities between Wavelet and Fourier analysis
Following are some similarities between Fourier and Wavelet analysis,
The Fast Fourier Transform (FFT) and the Discrete Wavelet Transform (DWT)
are both linear operations that generate a data structure that contains log2 𝑛
segments of various lengths, usually filling and transforming it into a different
data vector of length 2𝑛 .
The mathematical properties of the matrices involved in the transforms are
similar as well. The inverse transform matrix for both the FFT and the DWT is
the transpose of the original. As a result, both transforms can be viewed as a
rotation in function space to a different domain.
For the FFT, the transformed domain contains basis functions that are sine and
cosines. For the wavelet transform, this new domain contains more
complicated basis functions called wavelets, mother wavelets, or analyzing
wavelets.
Both transforms have another similarity. The basis functions are localized in
frequency, making mathematical tools such as power spectra (how much
power is contained in a frequency interval) and Scalogram useful at picking out
frequencies and calculating power distributions [Doc2013].
A.2.2 Dissimilarities between Wavelet and Fourier analysis
Following are some dissimilarities between Fourier and Wavelet analysis.
The most interesting dissimilarity between these two kinds of transforms is
that individual wavelet functions are localized in space while Fourier sine and
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cosine functions are not. This localization feature, along with wavelets'
localization of frequency, makes many functions and operators using wavelets
"sparse" when transformed into the wavelet domain. This sparseness, in turn,
results in a number of useful applications such as data compression, detecting
features in images, and removing noise from time series.
One way to see the time-frequency resolution differences between the Fourier
transform and the wavelet transform is to look at the basis function coverage
of the time-frequency plane. Figure A.8 shows a windowed Fourier transform,
where the window is simply a square wave. The square wave window
truncates the sine or cosine function to fit a window of a particular width.
Because a single window is used for all frequencies in the WFT, the resolution
of the analysis is the same at all locations in the time-frequency plane
[Doc2013].
Fig. A.8: Time-frequency tiles and coverage of the time-frequency plane for Fourier basis
An advantage of wavelet transforms is that the windows vary. In order to
isolate signal discontinuities, one would like to have some very short basis
functions. At the same time, in order to obtain detailed frequency analysis, one
would like to have some very long basis functions. A way to achieve this is to
have short high-frequency basis functions and long low-frequency ones. Figure
A.9 shows the coverage in the time-frequency plane with one wavelet function
for the Daubechies wavelet (Db2) [Doc2013].
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Fig. A.9: Time-frequency tiles and coverage of the time-frequency plane for Db2 wavelet
Wavelet transforms do not have a single set of basis functions like the Fourier
transform, which utilizes just the sine and cosine functions. Instead, wavelet
transforms have an infinite set of possible basis functions. Thus wavelet
analysis provides immediate access to information that can be obscured by
other time-frequency methods such as Fourier analysis [Bov2009].
A.3 Types of Wavelets and their properties
Since their evolution several new wavelet functions have been developed
and used in various applications depending on their suitability for that particular
application. The wavelets can be classified on the basis of their the following basic
properties as follows [Mis2007]. ,
i) Real or Complex
ii) Orthogonal or Bi-Orthogonal
iii) Compactly supported or not
iv) Arbitrary or Infinite Regularity
v) Symmetric or Asymmetric
vi) Number of Vanishing Moments
vii) Existence of scaling function 𝜑 or not.
There are several wavelets inwavelets in use such as Haar, Db, Bior, Rbior,
Morlett, Coiflet, Symlet, Meaxican Hat, Shannon, B-Spline, Gaussian, Meyer etc.
These wavelets along with their abbreviations are shown in table A.1. Various
wavelets and their properties are summarized in the following table A.2.
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Table A.1: Various types of wavelets and their abbreviations
Wavelets Abbreviations
Haar Wavelet Haar
Daubechies Wavelet Db
Symlets Sym
Coiflets Coif
Bi-Orthogonal Wavelet Bior
Meyer Wavelet Meyr
Discrete Meyer Wavelet Dmey
Battle and Lemarié Wavelets Btlm
Gaussian Wavelet Gaus
Meaxican Hat Wavelets Mexh
Morlet Wavelet Morl
Complex Gaussian Wavelets Cgau
Complex Shannon Wavelets Shan
Complex B-spline frequency Wavelets Fbsp
Complex Morlet Wavelets Cmor
Table A.2: Wavelets and their properties
Properties
Wavelet Families
Mo
rlet
Mex
ican
Hat
Mey
er
Bat
tle
-Lem
a
Haa
r
Db
N
Sym
N
Co
ifN
Bio
rN-R
bio
rN
Gau
ssia
n
D M
eye
r
C G
auss
ian
C M
orl
et
C B
-Sp
line
C S
han
no
n
Admissible
Infinite Regularity
Arbitrary Regularity
Orthogonal with Compact Support
Biorthogonal with Compact
Support
Symmetry
Asymmetry
Near Symmetry
Arbitrary number of zero moments
Existence of 𝝋
Zero moments for 𝝋
Orthogonal Analysis
Biorthogonal Analysis
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Exact Reconstruction # # #
FIR Filters
Continuous Transformation
Discrete Transformation
Fast Algorithm
Explicit Expression *
Complex Wavelet
Complex Continuous
Transformation
Approximation with FIR
# - Nearly exact reconstruction, * - Explicit expression for Splines
On the basis of these properties wavelets are grouped in various families as shown
in table A.3.
Table A.3: Various wavelet families based on their properties
Wavelets with filters Wavelets without filters
With compact support With non-compact
support Real Complex
Orthogonal Biorthogonal Orthogonal
Gaus, Mexh, Morl Cgau, Shan, Fbsp, Cmor Db, Haar, Sym, Coif Bior Meyr, Dmey, Btlm
/* Better to show in the form of a tree as is normally done*/
Some common wavelets along with their main properties are explained as
follows.,
A.3.1 Haar Wavelet
Haar wavelet is the first and simplest wavelet. Haar wavelet is
discontinuous and resembles a step function. It represents the same wavelet as
Daubechies Db1. Haar scaling function (𝜑(𝑡)) and wavelet function (𝜓(𝑡)) are
shown in figure A.10. Following are sSome basic properties of the Haar wavelet
are as follows. ,
Orthogonal : Yes
Formatted: Left, Line spacing: single
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Biorthogonal : Yes
Compact support : Yes
DWT : Possible
CWT : Possible
Support width : 1
Filters length : 2
Regularity : Discontinuous
Symmetry : Yes
No. of vanishing moments for 𝜑 : 1
Scaling Function (𝜑(𝑡)) Wavelet Function (𝜓(𝑡))
Fig. A.10: Haar wavelet and Haar Scaling function
A.3.2 Daubechies Wavelet
Daubechies are compactly supported orthonormal wavelets. , which
Development of these made discrete wavelet analysis practicable.
The names of the Daubechies family wavelets are written as DbN, where N
is the order and Db the "surname" of the wavelet. The Db1 wavelet, as mentioned
above, is the same as Haar wavelet. Wavelet functions 𝜓(𝑡) of the next nine
members of the Daubechies family are named as Db2, Db3, Db4, Db5, Db6, db7,
Db8, Db9 and Db10 wavelet. Their representation is shown in figure A.11. Some
basic properties are as follows.,
Order : N (2, 3, 4, ……)
Orthogonal : Yes
Biorthogonal : Yes
Compact support : Yes
DWT : Possible
CWT : Possible
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Support width : 2N-1
Filters length : 2N
Regularity : About 0.2N for large N
Symmetry : Far from
No. of vanishing moments for 𝜑 : N
Fig. A.11: Daubechies wavelets
A.3.3 Coiflet Wavelet
The Coiflet wavelet function has 2N moments equal to 0 and the scaling
function has 2N-1 moments equal to 0. The two functions have a support of length
6N-1. Some Coiflets are shown in figure A.12. Following are sSome basic
properties of the Coiflets are as follows. ,
Order : N (1, 2, 3, ……)
Orthogonal : Yes
Biorthogonal : Yes
Compact support : Yes
DWT : Possible
CWT : Possible
Support width : 6N-1
Filters length : 6N
Regularity : -
Symmetry : Near from
No. of vanishing moments for 𝜓 : 2N
No. of vanishing moments for 𝜑 : 2N-1
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Fig. A.12: Coiflet wavelets
A.3.4 Bi-Orthogonal Wavelet
This family of f waveletswavelets exhibitss the property of linear phase,
which is needed for signal and image reconstruction. By using These use two
wavelets, one for decomposition and the other for reconstruction instead of the
same single one. Some interesting properties are also derived. These wavelets are
represented by ‘BiorNr.Nd’,. wWhere, Nr and Nd represent orders of
reconstruction and decomposition filters respectively.
These bi-orthogonal wavelets are shown in figure A.13. In this figure, left
side wavelets on the left side are used for decomposition and right side the
wavelets on the right side are used for reconstruction. The bi-orthogonal wavelets
are better known for image reconstruction with least possible errors.
Some basic properties of BiorNr.Nd are given as follows.
,
Order : Nr, Nd (1, 2, 3, ……)
Orthogonal : No
Biorthogonal : Yes
Compact support : Yes
DWT : Possible
CWT : Possible
Support width : 2Nr+1, 2Nd+1
Filters length : Max(2Nr,2Nd)+2
Regularity : Nr-1 and Nr-2 at the knots
Symmetry : Yes
No. of vanishing moments for 𝜓 : Nr
Formatted: No Spacing, Left, Line spacing: single
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Fig. A.13: Bi-orthogonal wavelets
Few
A.3.5 Symlet Wavelet
The Symlets are nearly symmetrical wavelets proposed by Daubechies as
modifications to the Db family. The properties of the two wavelet families are
similar. Some Symlet wavelets functions are shown in figure A.14. The basic
properties of Symlets are as follows.
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/* What are the modifications? It is not clear how these are different*/
Order : N (2, 3, ……)
Orthogonal : Yes
Biorthogonal : Yes
Compact support : Yes
DWT : Possible
CWT : Possible
Support width : 2N-1
Filters length : 2N
Regularity : -
Symmetry : Near from
No. of vanishing moments for 𝜑 : N
Fig. A.14: Symlet wavelets
A.3.6 Morlet Wavelet
This wavelet has no scaling function, but it is explicit. /* The statement is
not clear*/ The simplest form of Morlet wavelet is represented in figure A.15.
Some Few properties are as follows.,
Orthogonal : No
Biorthogonal : No
Compact support : No
DWT : No
CWT : Possible
Support width : Infinity
Effective Support : [-4, 4]
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Symmetry : Yes
Fig. A.15: Morlet wavelet
A.3.7 Mexican Hat Wavelet
This wavelet has no scaling function and is derived from a function that is
proportional to the second derivative function of the Gaussian probability density
function. The wavelet is shown in figure A.16 and its basic properties are as
follows.,
Orthogonal : No
Biorthogonal : No
Compact support : No
DWT : No
CWT : Possible
Support width : Infinity
Effective Support : [-5, 5]
Symmetry : Yes
Fig. A.16: Mexican Hat Wavelet
A.3.8 Meyer Wavelet
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The Meyer wavelet and scaling function are defined in the frequency
domain. An eExample Meyer wavelet and scaling functions are shown in figure
A.17. Some properties are as follows.,
Orthogonal : Yes
Biorthogonal : Yes
Compact support : No
DWT : Possible but without FWT
CWT : Possible
Support width : Infinity
Effective Support : [-8, 8]
Symmetry : Yes
Regularity : Indefinitely derivable
Scaling Function (𝜑) Wavelet Function (𝜓)
Fig. A.17: Meyer wavelet
A.3.9 Other Real Wavelets
Some other real wavelets are also in use such as,
Reverse Biorthogonal Wavelet Pairs (RbioNr.Nd)
Gaussian derivatives family (Gauss)
FIR based approximation of the Meyer wavelet (Dmey)
A.3.10 Complex Wavelets
There are some complex wavelet families also such as,
Complex Gaussian Wavelet (Cgau)
Complex Morlet Wavelet (Cmor)
Complex Frequency B-Spline Wavelet (Fbsp)
Complex Shannon Wavelet (Shan)