Introduction to WaveletOutline of TalkOVERVIEWHistorical DevelopmentMathematical TransformationFREQUENCY ANALYSISSTATIONARITY OF SIGNALSTATIONARITY OF SIGNALCHIRP SIGNALSNOTHING MORE, NOTHING LESSSFORT TIME FOURIER TRANSFORM (STFT)DRAWBACKS OF STFTMULTIRESOLUTION ANALYSIS (MRA)PRINCIPLES OF WAELET TRANSFORMDEFINITION OF CONTINUOUS WAVELET TRANSFORMSCALECOMPUTATION OF CWTRESOLUTION OF TIME & FREQUENCYCOMPARSION OF TRANSFORMATIONSDISCRETIZATION OF CWTMulti Resolution AnalysisSUBBABD CODING ALGORITHMRECONSTRUCTIONWAVELET APPLICATIONSREFERENCESSlide Number 26Introduction to WaveletSSA1D1A2D2A3D3Bhushan D PatilPhD Research Scholar Department of Electrical EngineeringIndian Institute of Technology, BombayPowai, Mumbai. 400076Outline of Talk Overview Historical Development Time vs Frequency Domain Analysis  Fourier Analysis  Fourier vs Wavelet Transforms Wavelet Analysis  Typical Applications ReferencesOVERVIEW 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 smallHistorical 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 todayMathematical 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 transformationFREQUENCY 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() ()knNNnWnxkX ⋅+=+∑−=1011() ()knNNkWkXNnx−−=∑⋅+=+10111⎟⎠⎞⎜⎝⎛−=NjNewπ2() ()dtetxfXftjπ2−∞∞−⋅=∫() ( )dfefXtxftjπ2⋅=∫∞∞−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 SIGNAL0 0.2 0.4 0.6 0.8 1-3-2-101230 5 10 15 20 250100200300400500600TimeMagnitudeMagnitudeFrequency (Hz)2 Hz + 10 Hz + 20HzStationary0 0.5 1-1-0 .8-0 .6-0 .4-0 .200.20.40.60.810 5 10 15 20 25050100150200250TimeMagnitudeMagnitudeFrequency (Hz)Non- Stationary0.0-0.4: 2 Hz + 0.4-0.7: 10 Hz + 0.7-1.0: 20HzCHIRP SIGNALS0 0.5 1-1-0.8-0.6-0.4-0.200.20.40.60.810 5 10 15 20 25050100150TimeMagnitudeMagnitudeFrequency (Hz)0 0.5 1-1-0.8-0.6-0.4-0.200.20.40.60.810 5 10 15 20 25050100150TimeMagnitudeMagnitudeFrequency (Hz)Different in Time DomainFrequency: 2 Hz to 20 HzFrequency: 20 Hz to 2 HzSame in Frequency DomainAt what time the frequency components occur? FT can not tell!At what time the frequency components occur? FT can not tell!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 NeededMost 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) 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()()()()[]dtetttxftftjtπ−ω•′−ω•=′∫2*X,STFT()functionwindowthe:tωA function of time and frequencyDRAWBACKS 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 intervalsVia Narrow WindowVia Wide WindowMULTIRESOLUTION 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 componentsPRINCIPLES 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 IntervalsDEFINITION 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() () ()dtsttxsssxx⎟⎠⎞⎜⎝⎛τ−ψ•=τΨ=τ∫ψψ *1 , ,CWTTranslation(The location of the window)ScaleMother WaveletSCALE 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 NecessaryCOMPUTATION OF CWT() () ()dtsttxsssxx⎟⎠⎞⎜⎝⎛τ−ψ•=τΨ=τ∫ψψ *1 , ,CWTStep 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