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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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256

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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257

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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258

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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259

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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260

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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261

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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262

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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263

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

Appendix A

264

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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265

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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266

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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267

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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268

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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269

/* 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]

Appendix A

270

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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271

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)

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272