# Wavelets An Introduction by flwifnVD

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```									          Wavelets

An Introductory Exposition
Features, Analysis Structures, and
Selected Applications
Entry of Wavelets into the
Domain of SP
 Most significant event in SP after FT.
 Reference to Wavelet Transform:
“Fourier Transform of 20th Century”
 Wavelets fills the missing link in signal
processing: link between time and
frequency studies
    “Little wave with big future”
Groups Active in Wavelet Studies

Wavelets

Mathematics: Function      Engineering, Applied Scientists     Applications
Analysis and Approximation       In Signal Processing          Industrial Groups
Key Idea
Alternative Signal Representations
and Transformation
   Signal representation in a suitable
domain for information extraction,….
Examples:
 Fourier Transform for spectral analysis

 Hilbert Transform in envelop detection

 KLT(PCA) for optimal function approx.

 Laguerre basis function

 Numerous other transforms( DCT,Radon,…)
Two Domains for Signal
Representation

Projection space,
Basis functions

Signal Domain                                      Transformed Domain
Transformation

Different projection spaces        different signal representa-
tions
Fourier Transform

Representation in Time     Representation in Freq
f(t)                      F(ω)

Time Domain Studies        Frequency Domain Studies
Shortcomings of Fourier Transform
signal f(t) with localized changes
0.04

0.02
f(t)

0

-0.02
0       50            100             150           200       time t
250
1
sin(t)

0

-1
0      200            400             600           800          1000
1
cos(t)

0

-1
0      200            400            600             800         1000
Sine and cosine as basis functions of Fourier Transform

- Basis function vary within ± ∞, no localization
- Only one Basis function: Sine and Cosine or its complex form
- Fourier Coefficients: Projection of function f(t) onto sine and
cosine bases functions
- Information about the entire range of a function is contained
in the coefficients, no localization
Fourier Transform
Loss of Local Information
An Illustration

f(x)

x

F(ω)

freq ω

   Information about singularities and sharp changes spread across many
frequencies and many basic functions
   Cost of computations for transients is high
Starting Point of Wavelets
Real-world Signals
   Dominance of transient and non-
stationary signals
   Information often reside in transients,
changes
   The need to develop tools for T-F
analysis
   Shortcomings of STFT: Inability to
model most of the nonstationary real-
world signals
What Are Wavelets
   Wavelets are wavelike oscilatory
signals of finite bandwidth both
in Time and in Frequency
   Wavelets are basis functions of
spaces with certain properties
Examples of Wavelets

Db4                           Db10

Db 4, Db 10 are from Daubechies family of wavelets.
Db wavelets have no analytical expression, they are
constructed numerically
Examples of Wavelet Functions
Coiflet
2
WP1,5
1

0

-1
0   20    40        60   80        100       120     140    160   180
2
WP2,5

0

-2

-4
0    20        40        60             80     100         120    140
2
WP3,5

0

-2

-4
0   20    40        60   80        100       120     140    160   180
Wavelet Functions Generated from
Bior 3.9
bior3.9, wavelet packet W 30
10

5

0

-5

-10
0   50    100      150       200       250   300   350
Bior3.9, wavelet packet W 10
4

2

0

-2

-4
0   100   200      300       400       500   600   700
Shortcomings of Fourier Transform

Representation in Time   Representation in Freq
f(t)                    F(ω)

Time Domain Studies      Frequency Domain Studies
Loss of Time Information in FT
SALAAM with switching the 1st 5000 samples with the tail segment
1

0.5
Original

0

-0.5

-1
0        0.5        1       1.5        2        2.5        3           3.5
4
x 10
1

0.5
shifted

0

-0.5

-1
0        0.5        1       1.5        2        2.5        3           3.5
4
x 10
Fourier Transform
Loss of time information
abs(fft) of SALAAM with shifting the 1st 5000 samples to the tail
4000

3000

2000

1000

0
0          1000           2000          3000           4000             5000

4000

3000

2000

1000

0
0          1000           2000          3000           4000             5000
An Important Property of Wavelets:
Wavelets Filling the Gap
Wavelets
Representation in Time              Representation in Freq
f(t)                               F(ω)

Time Domain Studies                Frequency Domain Studies
Transient Nature of Signals
Most    of the signals we deal are
   Nonstationary:
Non-stationary refers  to time variancy and
spectral variation of the signals with time
   Unpredictable
changes  in statistics of the signals including
changes in pdf function or statistical parameters
The degree of unpredictability varies for
different signals and applications
An Example of a Transient Signal
Speech Signal
SALAAM
1

0

-1
0   0.5    1          1.5          2      2.5     3       3.5
0.5                                                                    4
x 10
sAlaam

0

-0.5
0    500       1000         1500       2000     2500     3000
1
saLaam

0

-1
0              500                     1000              1500
1
saLaam

0

-1
0    500       1000      1500       2000        2500     3000
Three segments of SALAAM
EEG Signals, Awake and Asleep
EEG (EP)
Examples of Transient Signals
Engine Vibrations

2

Healthy
0

-2
500   1000   1500    2000    2500   3000

2
Faulty

0

-2
500   1000    1500    2000   2500   3000
Time (Samplings)
An Example of Real-world Signals
Engine Vibrations
Wavelet Functions, WP
Bior3.1, wavelet packet W2
40

20

0

-20

-40
0   20       40            60         80   100
db4, wavelet packet W 15
4

2

0

-2

-4
0   20       40            60         80   100
Main Stages in Signal Analysis
Signal domain     basis functions     Transformed domain, Coeffs

Signals
Transformation

Information
Analysis      extraction

basis functions

Recon                             Modified
Signal        Reconstruction      Coefficients
Examples of Processing in
Coefficient Domain
 Coding for communication, transmission
 Compression and Data Storage
 Detection and Pattern Recognition
 Modification for Enhancement purposes
 Noise Reduction
 Watermarking
 Signal Separation
 Modeling
Two Main Tasks of Wavelet
Analysis
 Decomposition:
   Information Reside in Signal Constituents /
Components
 Time-Frequency      Representation
 Transformation of a signal into time-
frequency representation
 Different basis and transformations result in

different constituents and T-f information
Key Concern in SP
Resolution/Localization in Signal
Processing
 Resolution: Information Extraction         at a
Narrow Band of Signal Span.
 Localization both in Time and in Frequency
freq
High freq

Low freq
Time/space
Resolution/Localization
in Time

3         3
3
Resolution/Localization in
Frequency
Power Spectrum, Vibration data at combustion zone, original and denoised
150
2                       2

100
Original

50

0
0    2000         4000         6000         8000        10000        12000     14000

150
denoised, soft

100

50

0
0    2000         4000         6000         8000        10000        12000     14000
Freq. Hz
Need for Joint Time-Frequency
Analysis of High Resolution
Why   Joint T-F Analysis:
 Transient signal information reside in
different bands of time and frequency
domain
 The need for the study of signal at the

limits of resolution determined by
Heisenberg Uncertainty Limits
Heisenberg Uncertainty Principle
 Heisenberg uncertainty principle defined in
quantum physics and is applicable to signal
processing problems as well.
 Position and momentum of a particle can not
be determined simultaneously
 Sets a limit given as follows:
       Δt. Δf ≥ 1/4π
     Δx2=⌠{(x-μm )2 |f(x)|2 dx /⌠|f(x)|2 dx
μm   is center of mass of the function f(x):

μm= ⌠{x| f(x)|2 dx /⌠|f(x)|2 dx
Second Concern
Alternative Time–Frequency Tiling
   Information about Different signal behavior reside at
different T-F Bands
    Need to have alternative T-F tiling and cell structures
Freq/scale

Time/scale
Third Concern
   Different information reside at different resolutions.
   Redundancy in Signal Representation
Freq/scale
High Frequency

Redundancy
in tiling

Low Frequency

Time/scale
Wavelets
Tools for T-F Analysis
Wavelets allow:
 High resolution and focused study of signals in time
and in frequency
 Low resolution, Coarse and more general picture
and trend analysis
Comment:
Wavelet analysis resembles human mental activity i.e
capability for a detail focusing as well as general and
a broadly-based data analysis
Fourier vs Wavelet Transform
Fourier Transform              Wavelet Transform
Stationary signal Analysis       Nonstationary Transient signal Analysis
Frequency Information only,      Joint Time and Frequency Information
Time/space information is lost

Single Basis Function            Many Basis Functions
Computational Cost High          Low computational costs

Analysis Structures:             Numerous Analysis structures:
- FS( periodic functions only)    CWT,

- FT and DFT                     DWT(2 Band and M Band),

WP

DDWT, SWT,
 Adaptive Signal Transform, Numerous Best

basis selection algorithms
   Frame structure
Examples of Wavelets

Coiflet 4
coif4 : phi                            coif4 : psi
1.2                                     1.5

1

1

0.8

0.5
0.6

0.4
0

0.2

-0.5
0

-0.2                                     -1
0    5   10       15   20   25          0   5   10       15   20   25

Scaling function               Wavelet function
Wavelet Functions Generated from
HAAR function
Haar, wavelet packet W 13
2

1

0

-1

-2
0   5   10         15        20          25   30   35
dmey, wavelet packet W 10
2

1

0

-1

-2
0   5   10         15        20          25   30   35
Gabor 2D Wavelet
Gabor Wavelet at different scale
and Orientation
Laguerre Gaussian Wavelet
Laguerre Gaussian Wavelet
Wavelet Transform
 WaveletTransform is defined as in other
transforms:
        W = <f(t),ψ(t)> = ∫f(t) ψ*(t)dt
       wjk = <f(t),ψj,k(t)> = ∫f(t) ψj,k*(t)dt
ψj,k(t) is a shifted and scaled wavelet function, j is
scaling and k translation parameters, J,K can be any
scalars. In DWT they assume integer values.

fj,k() 
t w     ,
jk
*
jk
Physical Interpretation of
Wavelet Transform
 Correlation (extent   of matching) of f(t) with
the wavelet ψjk(t) at scale j and location
k.
 It carries signal information at scale j,
and location k. It gives localized
information of a signal.
 Representation of
a signal in a domain
described by wavelet basis functions
Wavelet Transform

Given a function f(t) єL^2
Ψab(t)= Ψ(at-b)

f(t)                                 wab

Function Domain                     Transformed Domain:
single dimensional                    two dimensional,
paramters a,b
f(t)                                 coeffs

Translation b
Time/space

Scale a /freq
Illustration of Translation
by two different Wavelets
f(t)

t

φ0k (t)
s hifte d wave le ts

t

φ0 k (t)   Haar

s hifte d Haar wave le t

t
Illustrative Examples of WT
Signal f(t)

t

φ0k(t)
shifted wavelets

t
t

φ0k(t)     Haar                               φjk(t)
shifted Haar wavelet                      scaled Haar wavelet

t        t                                                 t
Wavelet Transform
Window Function Interpretation
 Different information are extracted form the same
signal using different wavelets. The need to access
multitude of transforms and wavelet bases for
extraction of different information of a given
signal
 Wavelets as window functions may be considered
as lenses having different resolutions. Different
information are extracted by different lenses at
different scales and different locations of a given
signal
 STFT has only one window function at a given
scale
Wavelet Transform,
Translation of Window Function
Illustration of Wavelet
Scaling/Dilation
Common Wavelets
   Old Wavelets:
   Haar function
   Gabor function and wavelet(1D,2D)
   Morlet wavelets
   Shannon function
   1st and 2nd Derivatives of Gaussian function(mexican hat)
   Truncated and lapped Sine or Cosine functions

   Recent Wavelets
   Db Wavelets (orthogonal, biorthogonal),
   Coiflets,
   Symlets
   Biorthgonal wavelets
   Mallat Wavelet (1D,2D)
   Bathlets
   Curvelets
   Ridgelets
   Meyer wavelets
   Banana Wavelets

   Second Generation Wavelets( wavelets constructed by lifting scheme)
Commonly Used Numerical
Wavelets
 Daubechies  Wavelets
 Biorthogonal Wavelets
 Coiflets
 Symmlets
 Bathlets
 Dmeyer
How Wavelets are Generated
A  few number of wavelets are constructed
using known analog functions. Some are
expressed in analog function form, e.g.
Morlet wavelets
 Many others are designed and constructed
numerically. Db wavelets are examples of
these wavelets
 New wavelets are introduced by individuals
each year
Categorization of Wavelets
   Analytical or Numerical
   Real or Complex
   Symmetric, antisymmetric, asymmetric (e.g. db)
   Compactly supported or not compact
   Causal, Non-causal wavelets
   Do they have efficient computation algorithms such as FWT
   Wavelets with special features e.g. complex wavelet with real part
(high freq) and complex part carry low frequency content
   Maximally flat waveletes (Flatness of spectrum, Rate of decay at ω=pi
and origin)
   Have Orthogonal, biorthogonal analysis system
   Wavelets of high Resolution in time-frequency domain: ∆t ∆ω. Gabor:
H=0.5, Haar: H=0.58 in normalized frequency (0,1).
   Balanced Wavelets (Bathlets) are designed under a criteria based on
balancing ∆t and ∆ω.
Two Common Analysis Structures

1- Standard Two Band Discrete Wavelet Transform
High Freq Details

Signal
High Pass         2

High pass

Low Pass          2

Low pass
Low freq Approx

Decimated Wavelet Transform-Analysis Stage
Multiresolution
Pyramidal Signal Analysis Structure

 Multiresolution   as a subband coding of a
signal,
 Signal components at different subbands
are extracted
 Different analyzing wavelets results in
different components at different bands
An Illustration of Multiresolution
Signal Decomposition
Binary Tree of DWT
Common Analysis Structures
Wavelet Packets
High freq details
High pass        2

High Pass               2
V space of the signal

Low pass      2

High pass        2

Low Pass                2

Low pass      2

Wavelet Packets -Analysis Stages
Wavelet Packets Tree
Standard Two Channel DWT and
WP T-F Tiling
Time-Freq. Tiling

Standard DWT               Standard WP
Perfect Reconstruction Filterbank

Filterbank implementation of DWT
Wavelet
Subspace W

High Pass                           High pass

Low Pass                            Low pass

Perfect Reconstrution Filter bank
Alternative Signal Analysis
Architectures
Wavelets are rich in analysis structures and
algorithms
Main Analysis Structures
   ORTHOGONAL
   BIORTHOGONAL
   REDUNDANT WAVELET TRANSFROM
   FRAME-BASED REDUNDANT
TRANSFORM
Beyond Orthogonality: Compression
Requirements
Coeffs 1                   Coeffs 2

Slow rate of decay       Fast rate of decay

Orthogonal Expansion       Non-orthogonal Expansion
Primary Features of Wavelets used
in Signal Processing
1. Time- frequency Localization (useful for transient
data analysis)
2.   Sparsity: Coefficients are often grouped into high
and low amplitude
3.   Function Approximation: near Optimal Function
Approximation
4.    Noise Reduction, No wavelet can model white noise
leading to small amplitude coefficients
5.   Decorrelation: Signal components( coefficients) in
wavelet domain have a lower degree of correlation
Categorization of
Wavelet Applications
Applications: Methodology/Algorithm Intensive
   Change Detection ( of all kinds)
   Compression
   Feature Extraction and Pattern Recognition with Numerous Applications
   Image Enhancement
   Noise reduction with Numerous Applications
   Diagnostics (such as NDT, NDE)
   Signal Separation, ICA using Wavelets
   Solution of Differential and Partial Differential Equations
   Analysis of Fractals
   System Identification in Control systems
   Data and Image Fusion
   Watermarking
Applications, Industry
Categorization
Application Areas: Industry Intensive
   Power Systems, Transient data Analysis, Component Diagnosis, Signal
detection
   NDT, Ultra sonic Flaw Detection, Diagnosis
   Communications, Coding, Transmission
   Medical Applications: Compression, Noise Reduction, signal separation,
signal classification, image enhancement
   Machine Vision, Robotics, geometry and spatial edge detection
   Industrial Machinery and Machine Diagnosis,
   Remote Sensing, Satellite Image Analysis, Image segmentation, fusion
   Biometrics: Fingerprint, Iris and Facial Identification
   Nuclear Reactors, Boiler Pipe fault analysis,
   Oil Industry, Oil Exploration
   Transportation Industry (railroad engine diagnosis, Aero jet engine and
Helicopter drive and gear diagnosis
   Seismic Data analysis
    Genetics
Applications, Fields of Work

Applications in different fields
 Physics, Geophysics,
 Mathematics, function approximation, signal
processing, Harmonic analysis, Differential
Equation, Frames, Function Space Analysis
 Electrical Engineering, numerous aplications
 Biomedical Engineering, EEG, ECG, MRI
 Finance and Financial Data Analysis
 Mechanical, Civil, Ocean Science and Engineering
 Remote Sensing, Forestry
 Nontechnical: Historial document retrieval
Details of Applications

   To Be Presented in a Separate
   Slide Presentation

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