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An Introductory Exposition Features, Analysis Structures, and Selected Applications. Wavelets. Entry of Wavelets into the Domain of SP. Most significant event in SP after FT. Reference to Wavelet Transform: “Fourier Transform of 20 th Century”
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An Introductory Exposition Features, Analysis Structures, and Selected Applications Wavelets
Entry of Wavelets into theDomain 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 Analysis and Approximation Engineering, Applied Scientists In Signal Processing Applications 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 Different projection spaces different signal representa-tions Projection space, Basis functions Transformed Domain Signal Domain Transformation
Fourier Transform Representation in Time f(t) Representation in Freq F(ω) Time Domain Studies Frequency Domain Studies
Shortcomings 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 TransformLoss of Local Information An Illustration Information about singularities and sharp changes spread across many frequencies and many basic functions Cost of computations for transients is high f(x) x F(ω) freq ω
Starting Point of WaveletsReal-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
Shortcomings of Fourier Transform Representation in Time f(t) Representation in Freq F(ω) Time Domain Studies Frequency Domain Studies
An Important Property of Wavelets: Wavelets Filling the Gap Representation in Time f(t) Wavelets Representation in Freq 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
Main Stages in Signal Analysis Signal domain basis functionsTransformed domain, Coeffs Signals Transformation Analysis Information extraction basis functions Recon Signal Reconstruction Modified 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
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 ConcernAdaptive Multi-Resolution Study 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 TransformFourierTransformWaveletTransform Stationary signal Analysis Nonstationary Transient signal Analysis Frequency Information only, Time/space information is lost Joint Time and Frequency Information Single Basis Function Many Basis Functions Computational Cost High Low computational costs Analysis Structures: - FS( periodic functions only) - FT and DFT Numerous Analysis structures: • CWT, • DWT(2 Band and M Band), • WP • DDWT, SWT, • Adaptive Signal Transform, Numerous Best basis selection algorithms • Frame structure
Examples of Wavelets Coiflet 4 Scaling function Wavelet function
Wavelet Transform • Wavelet Transform 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.
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) wab f(t) Transformed Domain: two dimensional, paramters a,b Function Domain single dimensional f(t) coeffs Translation b Time/space Scale a /freq
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
