Introduction to Wavelet Bhushan D Patil
PhD Research Scholar
Department of Electrical Engineering
Indian Institute of Technology, Bombay
Powai, Mumbai. 400076

Outline of Talk :

Outline of Talk Overview
Historical Development
Time vs Frequency Domain Analysis
Fourier Analysis
Fourier vs Wavelet Transforms
Wavelet Analysis
Typical Applications
References

OVERVIEW :

OVERVIEW Wavelet
A small wave
Wavelet Transforms
Convert a signal into a series of wavelets
Provide a way for analyzing waveforms, bounded in both frequency and duration
Allow signals to be stored more efficiently than by Fourier transform
Be able to better approximate real-world signals
Well-suited for approximating data with sharp discontinuities
“The Forest & the Trees”
Notice gross features with a large "window“
Notice small features with a small

Historical Development :

Historical Development Pre-1930
Joseph Fourier (1807) with his theories of frequency analysis
The 1930s
Using scale-varying basis functions; computing the energy of a function
1960-1980
Guido Weiss and Ronald R. Coifman; Grossman and Morlet
Post-1980
Stephane Mallat; Y. Meyer; Ingrid Daubechies; wavelet applications today

Mathematical Transformation :

Mathematical Transformation Why
To obtain a further information from the signal that is not readily available in the raw signal.
Raw Signal
Normally the time-domain signal
Processed Signal
A signal that has been "transformed" by any of the available mathematical transformations
Fourier Transformation
The most popular transformation

FREQUENCY ANALYSIS :

FREQUENCY ANALYSIS Frequency Spectrum
Be basically the frequency components (spectral components) of that signal
Show what frequencies exists in the signal
Fourier Transform (FT)
One way to find the frequency content
Tells how much of each frequency exists in a signal

STATIONARITY OF SIGNAL :

STATIONARITY OF SIGNAL Stationary Signal
Signals with frequency content unchanged in time
All frequency components exist at all times
Non-stationary Signal
Frequency changes in time
One example: the “Chirp Signal”

STATIONARITY OF SIGNAL :

STATIONARITY OF SIGNAL

CHIRP SIGNALS :

CHIRP SIGNALS Frequency: 2 Hz to 20 Hz Frequency: 20 Hz to 2 Hz Same in Frequency Domain At what time the frequency components occur? FT can not tell!

NOTHING MORE, NOTHING LESS :

NOTHING MORE, NOTHING LESS FT Only Gives what Frequency Components Exist in the Signal
The Time and Frequency Information can not be Seen at the Same Time
Time-frequency Representation of the Signal is Needed Most of Transportation Signals are Non-stationary.
(We need to know whether and also when an incident was happened.) ONE EARLIER SOLUTION: SHORT-TIME FOURIER TRANSFORM (STFT)

SFORT TIME FOURIER TRANSFORM (STFT) :

SFORT TIME FOURIER TRANSFORM (STFT) Dennis Gabor (1946) Used STFT
To analyze only a small section of the signal at a time -- a technique called Windowing the Signal.
The Segment of Signal is Assumed Stationary
A 3D transform A function of time and frequency

DRAWBACKS OF STFT :

DRAWBACKS OF STFT Unchanged Window
Dilemma of Resolution
Narrow window -> poor frequency resolution
Wide window -> poor time resolution
Heisenberg Uncertainty Principle
Cannot know what frequency exists at what time intervals Via Narrow Window Via Wide Window

MULTIRESOLUTION ANALYSIS (MRA) :

MULTIRESOLUTION ANALYSIS (MRA) Wavelet Transform
An alternative approach to the short time Fourier transform to overcome the resolution problem
Similar to STFT: signal is multiplied with a function
Multiresolution Analysis
Analyze the signal at different frequencies with different resolutions
Good time resolution and poor frequency resolution at high frequencies
Good frequency resolution and poor time resolution at low frequencies
More suitable for short duration of higher frequency; and longer duration of lower frequency components

PRINCIPLES OF WAELET TRANSFORM :

PRINCIPLES OF WAELET TRANSFORM Split Up the Signal into a Bunch of Signals
Representing the Same Signal, but all Corresponding to Different Frequency Bands
Only Providing What Frequency Bands Exists at What Time Intervals

DEFINITION OF CONTINUOUS WAVELET TRANSFORM :

DEFINITION OF CONTINUOUS WAVELET TRANSFORM Wavelet
Small wave
Means the window function is of finite length
Mother Wavelet
A prototype for generating the other window functions
All the used windows are its dilated or compressed and shifted versions

SCALE :

SCALE Scale
S>1: dilate the signal
S<1: compress the signal
Low Frequency -> High Scale -> Non-detailed Global View of Signal -> Span Entire Signal
High Frequency -> Low Scale -> Detailed View Last in Short Time
Only Limited Interval of Scales is Necessary

COMPUTATION OF CWT :

COMPUTATION OF CWT Step 1: The wavelet is placed at the beginning of the signal, and set s=1 (the most compressed wavelet);
Step 2: The wavelet function at scale “1” is multiplied by the signal, and integrated over all times; then multiplied by ;
Step 3: Shift the wavelet to t= , and get the transform value at t= and s=1;
Step 4: Repeat the procedure until the wavelet reaches the end of the signal;
Step 5: Scale s is increased by a sufficiently small value, the above procedure is repeated for all s;
Step 6: Each computation for a given s fills the single row of the time-scale plane;
Step 7: CWT is obtained if all s are calculated.

RESOLUTION OF TIME & FREQUENCY :

RESOLUTION OF TIME & FREQUENCY

COMPARSION OF TRANSFORMATIONS :

COMPARSION OF TRANSFORMATIONS

DISCRETIZATION OF CWT :

DISCRETIZATION OF CWT It is Necessary to Sample the Time-Frequency (scale) Plane.
At High Scale s (Lower Frequency f ), the Sampling Rate N can be Decreased.
The Scale Parameter s is Normally Discretized on a Logarithmic Grid.
The most Common Value is 2.
The Discretized CWT is not a True Discrete Transform
Discrete Wavelet Transform (DWT)
Provides sufficient information both for analysis and synthesis
Reduce the computation time sufficiently
Easier to implement
Analyze the signal at different frequency bands with different resolutions
Decompose the signal into a coarse approximation and detail information

Multi Resolution Analysis :

Multi Resolution Analysis Analyzing a signal both in time domain and frequency domain is needed many a times
But resolutions in both domains is limited by Heisenberg uncertainty principle
Analysis (MRA) overcomes this , how?
Gives good time resolution and poor frequency resolution at high frequencies and good frequency resolution and poor time resolution at low frequencies
This helps as most natural signals have low frequency content spread over long duration and high frequency content for short durations

SUBBABD CODING ALGORITHM :

SUBBABD CODING ALGORITHM Halves the Time Resolution
Only half number of samples resulted
Doubles the Frequency Resolution
The spanned frequency band halved

RECONSTRUCTION :

RECONSTRUCTION What
How those components can be assembled back into the original signal without loss of information?
A Process After decomposition or analysis.
Also called synthesis
How
Reconstruct the signal from the wavelet coefficients
Where wavelet analysis involves filtering and down sampling, the wavelet reconstruction process consists of up sampling and filtering

WAVELET APPLICATIONS :

WAVELET APPLICATIONS Typical Application Fields
Astronomy, acoustics, nuclear engineering, sub-band coding, signal and image processing, neurophysiology, music, magnetic resonance imaging, speech discrimination, optics, fractals, turbulence, earthquake-prediction, radar, human vision, and pure mathematics applications
Sample Applications
Identifying pure frequencies
De-noising signals
Detecting discontinuities and breakdown points
Detecting self-similarity
Compressing images

REFERENCES :

REFERENCES Mathworks, Inc. Matlab Toolbox http://www.mathworks.com/access/helpdesk/help/toolbox/wavelet/wavelet.html
Robi Polikar, The Wavelet Tutorial, http://users.rowan.edu/~polikar/WAVELETS/WTpart1.html
Robi Polikar, Multiresolution Wavelet Analysis of Event Related Potentials for the Detection of Alzheimer's Disease, Iowa State University, 06/06/1995
Amara Graps, An Introduction to Wavelets, IEEE Computational Sciences and Engineering, Vol. 2, No 2, Summer 1995, pp 50-61.
Resonance Publications, Inc. Wavelets. http://www.resonancepub.com/wavelets.htm
R. Crandall, Projects in Scientific Computation, Springer-Verlag, New York, 1994, pp. 197-198, 211-212.
Y. Meyer, Wavelets: Algorithms and Applications, Society for Industrial and Applied Mathematics, Philadelphia, 1993, pp. 13-31, 101-105.
G. Kaiser, A Friendly Guide to Wavelets, Birkhauser, Boston, 1994, pp. 44-45.
W. Press et al., Numerical Recipes in Fortran, Cambridge University Press, New York, 1992, pp. 498-499, 584-602.
M. Vetterli and C. Herley, "Wavelets and Filter Banks: Theory and Design," IEEE Transactions on Signal Processing, Vol. 40, 1992, pp. 2207-2232.
I. Daubechies, "Orthonormal Bases of Compactly Supported Wavelets," Comm. Pure Appl. Math., Vol 41, 1988, pp. 906-966.
V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software, AK Peters, Boston, 1994, pp. 213-214, 237, 273-274, 387.
M.A. Cody, "The Wavelet Packet Transform," Dr. Dobb's Journal, Vol 19, Apr. 1994, pp. 44-46, 50-54.
J. Bradley, C. Brislawn, and T. Hopper, "The FBI Wavelet/Scalar Quantization Standard for Gray-scale Fingerprint Image Compression," Tech. Report LA-UR-93-1659, Los Alamos Nat'l Lab, Los Alamos, N.M. 1993.
D. Donoho, "Nonlinear Wavelet Methods for Recovery of Signals, Densities, and Spectra from Indirect and Noisy Data," Different Perspectives on Wavelets, Proceeding of Symposia in Applied Mathematics, Vol 47, I. Daubechies ed. Amer. Math. Soc., Providence, R.I., 1993, pp. 173-205.
B. Vidakovic and P. Muller, "Wavelets for Kids," 1994, unpublished. Part One, and Part Two.

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