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Wavelet Toolbox User's Guide

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1. i A f l J 1 1 0 1 2 3 4 0 1 2 3 4 0 2 4 6 8 0 2 4 6 8 bior1 3 bior1 5 ayy Rip i I 1 p 2 3 4 0 1 2 3 4 0 2 4 6 8 0 2 4 6 8 bior2 2 bior2 4 i 1 1 1 0 o 0 0 4 1 0 5 10 0 5 10 0 5 10 r 5 10 15 bior2 6 bior2 8 100 i 4 1 4 a 0 1 2 0 1 2 0 2 4 6 i 0 2 4 6 bior3 1 bior3 3 1 1 05 0 0 0 0 2 a 2 l 0 S 10 i 0 5 10 0 5 10 bior3 5 bior3 7 1 0 5 f 10 15 0 10 15 n 2 4 6 0 2 4 6 8 bior3 9 bior4 4 1 0 0 0 0 0 5 1 i 0 5 10 0 5 10 0 5 10 0 5 10 15 bior5 5 bior6 8 You can obtain a survey of the main properties of this family by typing waveinfo bior from the MATLAB command line See Biorthogonal Wavelet Pairs biorNr Nd on page 6 76 for more detail Coiflets Introduction to the Wavelet Families Built by I Daubechies at the request of R Coifman The 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 You can obtain a survey of the main properties of this family by typing waveinfo coif from the MATLAB command line See Coiflet Wavelets coifN on page 6 75 for more detail 0 2 4 coif1 Symlets coif3 10 coif4 15 20 0 5 10 15 20 25 coif5 The symlets are nearly symmetrical wavelets proposed by Daubechies as modifications to the db family
2. Displayed statistics include measures of tendency mean mode median and dispersion range standard deviation In addition the tool provides frequency distribution diagrams histograms and cumulative histograms as well as time series diagrams autocorrelation function and spectrum The same feature exists for the Wavelet 1 D De noising tool 2400 2600 2800 3000 3200 3400 Dismiss the Wavelet 1 D Compression window click the Close button l When the Update Synthesized Si l dialog b click No You can see that the compression process removed most of the noise but P e E E eee Pana preserved 99 74 of the energy of the signal The automatic thresholding 11 Show statistics was very efficient zeroing out all but 3 2 of the wavelet coefficients You can view a variety of statistics about your signal and its components 10 Show the residuals From the Wavelet 1 D tool click the Statisties button From the Wavelet 1 D Compression tool click the Residuals button The More on Residuals for Wavelet 1 D Compression window appears 2 54 2 55 One Dimensional Discrete Wavelet Analysis 2 Using Wavelets The Wavelet 1 D Statistics window appears displaying by default statistics on the original signal Displayed statistics include measures of tendency mean mode median and dispersion range standard deviation 22933 82 8 8 In addition the tool provides frequency distribution diagrams histograms and cumul
3. Compress the signal The graphical interface tools feature a compression option with automatic or manual thresholding One Dimensional Discrete Wavelet Analysis met Analyze Statistics Compress e Histograms De noise Bring up the Compression window click the Compress button located in the middle right of the window underneath the Analyze button The Compression window appears While you always have the option of choosing by level thresholding here we ll take advantage of the global thresholding feature for quick and easy compression 2 53 One Dimensional Discrete Wavelet Analysis 2 Using Wavelets Note If you want to experiment with manual thresholding choose the By Level thresholding option from the menu located at the top right of the Wavelet 1 D Compression window The sliders located below this menu then control the level dependent thresholds indicated by yellow dotted lines running horizontally through the graphs on the left of the window The yellow dotted lines can also be dragged directly using the left mouse button Curruistive histagrmarn te re te eed t 100 mo FFT Spectrum Click the Compress button located at the center right After a pause for computation the electrical consumption signal is redisplayed in red with the compressed version superimposed in yellow Below we ve zoomed in to get a closer look at the noisy part of the signal Original and corrpressed signals
4. Show and Scroll App Det 2 x More Display Options These options include the ability to suppress the display of various E components and to choose whether or not to display the original signal Tree Mode along with the details and approximations lt 3 3 7 3 8 8 7 Remove noise from a signal The graphical interface tools feature a de noising option with a predefined thresholding strategy This makes it very easy to remove noise from a signal Bring up the de noising tool click the De noise button located in the middle right of the window underneath the Analyze button ad sation _comees You can change the default display mode on a per session basis Select the Histograms De noise desired mode from the View gt Default Display Mode submenu Show amp Scroll Mode Show amp Scroll Mode Stem Cfs Note The Compression and De noising windows opened from the Wavelet 1 D tool will inherit the current coefficient visualization attribute stems or colored blocks Depending on which display mode you select you may have access to additional display options through the More Display Options button for more information see More Display Options on page A 19 2 48 2 49 2 Using Wavelets The Wavelet 1 D De noising window appears Wavelet 10 Denoirmg p p a 1p aoe 20 Original cneltie emis While a number of options are available for fine tuning the de noising algorithm we ll
5. The properties of the two wavelet families are similar Here are the wavelet functions psi 5 10 sym6 sym3 5 sym7 10 sym4 sym8 You can obtain a survey of the main properties of this family by typing waveinfo sym from the MATLAB command line See Symlet Wavelets symN on page 6 74 for more detail 1 45 T Wavelets A New Tool for Signal Analysis Morlet This wavelet has no scaling function but is explicit 0 5 Wavelet function psi 0 5 8 6 4 2 0 2 4 6 8 You can obtain a survey of the main properties of this family by typing waveinfo morl from the MATLAB command line See Morlet Wavelet morl on page 6 81 for more detail Mexican Hat 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 0 8 0 6 0 4 0 2 Wavelet function psi 0 0 2 8 6 4 2 0 2 4 6 8 You can obtain a survey of the main properties of this family by typing waveinfo mexh from the MATLAB command line See Mexican Hat Wavelet mexh on page 6 80 for more information 1 46 Introduction to the Wavelet Families loes Meyer The Meyer wavelet and scaling function are defined in the frequency domain 0 5 Wavelet function psi 05 29 0 5 You can obtain a survey of the mai
6. setting each element to some fraction of the vectors peak or average value Then we could reconstruct new detail signals D1 D2 and D3 from the thresholded coefficients To denoise the signal use the ddencmp command to calculate the default parameters and the wdencmp command to perform the actual de noising type thr sorh keepapp ddencmp den wv s clean wdencmp gbl C L db1 3 thr sorh keepapp 2 38 One Dimensional Discrete Wavelet Analysis Note that wdencmp uses the results of the decomposition C and L that we calculated in step 6 on page 2 33 We also specify that we used the db1 wavelet to perform the original analysis and we specify the global thresholding option gb1 See ddencmp and wdencmp in the reference pages for more information about the use of these commands To display both the original and denoised signals type subplot 2 1 1 plot s 2000 3920 title Original subplot 2 1 2 plot clean 2000 3920 title De noised Original 500 fh 400 300 200 hes 200 400 600 800 1000 1200 1400 1600 1800 De noised 500 F 400 300 200 200 400 600 800 1000 1200 1400 1600 1800 We ve plotted here only the noisy latter part of the signal Notice how we ve removed the noise without compromising the sharp detail of the original signal This is a strength of wavelet analysis While using command line functions to remove the noise from a signal can be cumberso
7. WILT TLL AA S A EA ANN Small 3950 4000 4050 4100 4150 4200 4250 Coefficients ime 1 18 1 19 Continuous Wavelet Transform T Wavelets A New Tool for Signal Analysis e High scale a gt Stretched wavelet gt Slowly changing coarse features gt Low These coefficient plots resemble a bumpy surface viewed from above If you frequency could look at the same surface from the side you might see something like this Scale of Nature It s important to understand that the fact that wavelet analysis does not produce a time frequency view of a signal is not a weakness but a strength of the technique Not only is time scale a different way to view data it is a very natural way to view data deriving from a great number of natural phenomena Consider a lunar landscape whose ragged surface simulated below is a result of centuries of bombardment by meteorites whose sizes range from gigantic boulders to dust specks If we think of this surface in cross section as a one dimensional signal then it is reasonable to think of the signal as having components of different scales large features carved by the impacts of large meteorites and finer features abraded by small meteorites The continuous wavelet transform coefficient plots are precisely the time scale view of the signal we referred to earlier It is a different view of signal data from the time frequency Fourier view but it is not unrelated Scale and Frequency N
8. Wavelet a Select cgav4 Sampling Period Scale Settings Step by Step Mode z Min gt 0 oaio Local Maxirra Lines Step gt 0 2 Select scales from 1 to 64 in steps of 2 Max lt 512 40 8 1000 200 400 600 800 1000 AWD VRSSG REGGE LDN ND NDO hl Scale of colors fromMIN to MAX Modulus Ca b fora 31 frq 0 016 Angle Ca b fora 31 frq 0 016 1 5 20 Local Maxirra Lines ab OND O N LOT O a i gt CONDON LOW ON LS 400 600 800 1000 200 400 600 0 600 00 Choe Aie Analyze julian Each side has exactly the same representation that we found in Continuous After a pause for computation the tool displays the usual plots associated to Analysis Using the Graphical Interface on page 2 8 the modulus of the coefficients on the left side and the angle of the sochiciontaomiheriahtside Select the plots related to the modulus of the coefficients using the Modulus option button in the Selected Axes frame 2 26 2 27 2 Using Wavelets 2 28 Analyzed Signal length 1024 a 100 200 300 400 500 600 700 800 900 1000 Modulus of Ca b Coefficients Coloration mode init by scale AoD RASS REONE a Scale of colors fromMIN to MAX Coefficients Line Modulus Ca b forscale a 31 frequency 0 016 L J J L J fl 400 500 600 700 800 900 1000 Local Maxima Lines i AOON ATIO ON h The figure now looks like the one in the real Continuous Wavelet 1 D tool Importi
9. adding your own M files 1 2 1 3 T Wavelets A New Tool for Signal Analysis 1 4 Background Reading Wavelet Toolbox software provides a complete introduction to wavelets and assumes no previous knowledge of the area The toolbox allows you to use wavelet techniques on your own data immediately and develop new insights It is our hope that through the use of these practical tools you may want to explore the beautiful underlying mathematics and theory Excellent supplementary texts provide complementary treatments of wavelet theory and practice see References on page 6 155 For instance e Burke Hubbard Bur96 is an historical and up to date text presenting the concepts using everyday words e Daubechies Dau92 is a classic for the mathematics e Kaiser Kai94 is a mathematical tutorial and a physics oriented book e Mallat Mal98 is a 1998 book which includes recent developments and consequently is one of the most complete e Meyer Mey93 is the father of the wavelet books e Strang Nguyen StrN96 is especially useful for signal processing engineers It offers a clear and easy to understand introduction to two central ideas filter banks for discrete signals and for wavelets It fully explains the connection between the two Many exercises in the book are drawn from Wavelet Toolbox software The Wavelet Digest Internet site http Wwww wavelet org provides much useful and practical informatio
10. introduced when a defect developed in the monitoring equipment as the measurements were being made Wavelet analysis effectively removes the noise 1 Load a signal From the MATLAB prompt type load leleccum Set the variables Type s leleccum 1 3920 l s length s 2 Perform a single level wavelet decomposition of a signal Perform a single level decomposition of the signal using the db1 wavelet Type cA1 cD1 dwt s db1 This generates the coefficients of the level 1 approximation cA1 and detail cD1 2 31 2 Using Wavelets 2 32 3 Construct approximations and details from the coefficients To construct the level 1 approximation and detail A1 and D1 from the coefficients cA1 and cD1 type A1 upcoef a cA1 db1 1 1 s D1 upcoef d cD1 db1 1 1 s or A1 idwt cAi db1 1_s D1 idwt cD1 dbi 1_s Display the approximation and detail To display the results of the level one decomposition type subplot 1 2 1 plot A1 title Approximation A1 subplot 1 2 2 plot D1 title Detail D1 Approximation A1 Detail D1 550 25 500 20 450 400 350 300 250 200 150 20 100 25 0 1000 2000 3000 4000 0 1000 2000 3000 4000 5 Regenerate a signal by using the Inverse Wavelet Transform To find the inverse transform type AO idwt cAi cD1 db1 1 s err max abs s A0O One Dimensional Discrete Wavelet Analysis err
11. 2 27 378 013 Perform a multilevel wavelet decomposition of a signal To perform a level 3 decomposition of the signal again using the db1 wavelet type C L wavedec s 3 db1 The coefficients of all the components of a third level decomposition that is the third level approximation and the first three levels of detail are returned concatenated into one vector C Vector L gives the lengths of each component CA cD C 3 3 Extract approximation and detail coefficients To extract the level 3 approximation coefficients from C type cA3 appcoef C L db1 3 To extract the levels 3 2 and 1 detail coefficients from C type cD3 detcoef C L 3 cD2 detcoef C L 2 cD1 detcoef C L 1 or cD1 cD2 cD3 detcoef C L 1 2 3 2 33 2 Using Wavelets 2 34 Results are displayed in the figure below which contains the signal s the approximation coefficients at level 3 cA3 and the details coefficients from level 3 to 1 cD3 cD2 and cD1 from the top to the bottom Original signal s and coefficients 1000 AA TT A ee l 0 i i i ji ji ji Pr 500 1000 1500 2000 2500 3000 3500 4000 tooo UN O 500 100 10000 50 f 500 1000 50 l 50 1 1 i 0 500 1000 1500 2000 Reconstruct the Level 3 approximation and the Level 1 2 and 3 details To reconstruct the level 3 approximation from C type A3 wrcoef a C L db1 3 To reconstr
12. ATLAB software really allows one to write complex and powerful applications in a very short amount of time The Graphic User Interface is both user friendly and intuitive It provides an excellent interface to explore the various aspects and applications of wavelets it takes away the tedium of typing and remembering the various function calls Ingrid C Daubechies Professor Princeton University Department of Mathematics and Program in Applied and Computational Mathematics Product Overview T Wavelets A New Tool for Signal Analysis Taaa E The second category of tools is a collection of graphical interface tools that Pred uct Overview afford access to extensive functionality Access these tools from the command line by typing Everywhere around us are signals that can be analyzed For example there are seismic tremors human speech engine vibrations medical images financial data music and many other types of signals Wavelet analysis is a new and promising set of tools and techniques for analyzing these signals wavemenu Note The examples in this guide are generated using Wavelet Toolbox Wavelet Toolbox software is a collection of functions built on the MATLAB software with the DWT extension mode set to zpd for zero padding except technical computing environment It provides tools for the analy sis and l when it is explicitly mentioned So if you want to obtain exactly the same synthesis of signals and images and t
13. Cantor curve 2 18 One Dimensional Continuous Wavelet Analysis ERI After saving the continuous wavelet coefficients to the file cantor wc1 load the variables into your workspace load cantor wc1 mat whos Name Size Bytes Class coeff 64x2188 1120256 double array scales 1x64 512 double array wname 1x4 8 char array Variables coefs and scales contain the continuous wavelet coefficients and the associated scales More precisely in the above example coefs is a 64 by 2188 matrix one row for each scale and scales is the 1 by 64 vector 1 64 Variable wname contains the wavelet name 2 19 2 Using Wavelets 2 20 One Dimensional Complex Continuous Wavelet Analysis This section takes you through the features of complex continuous wavelet analysis using the Wavelet Toolbox software and focuses on the differences between the real and complex continuous analysis You can refer to the section One Dimensional Continuous Wavelet Analysis on page 2 4 if you want to learn how to e Zoom in on detail e Display coefficients in normal or absolute mode e Choose the scales at which the analysis is performed e Switch from scale to pseudo frequency information e Exchange signal and coefficient information between the disk and the graphical tools Wavelet Toolbox software requires only one function for complex continuous wavelet analysis of a real valued signal cwt You ll find full information about this function
14. H2 importance of choosing the right filters In fact the choice of filters not only determines whether perfect reconstruction is possible it also determines the shape of the wavelet we use to perform the analysis To construct a wavelet of some practical utility you seldom start by drawing a waveform Instead it usually makes more sense to design the appropriate quadrature mirror filters and then use them to create the waveform Let s see how this is done by focusing on an example Consider the low pass reconstruction filter L for the db2 wavelet If we iterate this process several more times repeatedly upsampling and convolving the resultant vector with the four element filter vector Lprime a pattern begins to emerge 0 db2 wavelet 1 0 1 2 3 The filter coefficients can be obtained from the dbaux command Lprime dbaux 2 1 32 1 33 T Wavelets A New Tool for Signal Analysis 1 34 Second Iteration Third Iteration 0 1 0 15 0 1 0 05 0 05 0 0 0 05 0 1 0 05 5 10 15 20 10 20 30 40 Fourth Iteration Fifth Iteration 0 04 0 02 0 02 0 01 0 0 0 02 0 01 0 04 0 02 20 40 60 80 50 100 150 The curve begins to look progressively more like the db2 wavelet This means that the wavelet s shape is determined entirely by the coefficients of the reconstruction filters This relationship has profound implications It means that you cannot choose just any shape call it a wavelet and p
15. Let s switch to time aspects The main goals are e Rupture and edges detection e Study of short time phenomena as transient processes 1 7 T Wavelets A New Tool for Signal Analysis Fourier Analysis Fourier Analysis As domain applications we get e Industrial supervision of gear wheel e Checking undue noises in craned or dented wheels and more generally in nondestructive control quality processes e Detection of short pathological events as epileptic crises or normal ones as evoked potentials in EEG medicine e SAR imagery e Automatic target recognition e Intermittence in physics Wavelet Decomposition as a Whole Many applications use the wavelet decomposition taken as a whole The common goals concern the signal or image clearance and simplification which are parts of de noising or compression We find many published papers in oceanography and earth studies One of the most popular successes of the wavelets is the compression of FBI fingerprints When trying to classify the applications by domain it is almost impossible to sum up several thousand papers written within the last 15 years Moreover it is difficult to get information on real world industrial applications from companies They understandably protect their own information Some domains are very productive Medicine is one of them We can find studies on micro potential extraction in EKGs on time localization of His bundle electrical heart ac
16. Wavelet Toolbox 4 User s Guide Michel Misiti Yes Misiti Georges Oppenheim Jean Michel Poggi MATLAB 4 The MathWorks Accelerating the pace of engineering and science X C9 How to Contact The MathWorks www mathworks com Web comp soft sys matlab Newsgroup www mathworks com contact_TS html Technical support Product enhancement suggestions Bug reports Documentation error reports Order status license renewals passcodes Sales pricing and general information suggest mathworks com bugs mathworks com doc mathworks com service mathworks com info mathworks com 508 647 7000 Phone 508 647 7001 Fax The MathWorks Inc 3 Apple Hill Drive Natick MA 01760 2098 For contact information about worldwide offices see the MathWorks Web site Wavelet Toolbox User s Guide COPYRIGHT 1997 2009 by The MathWorks Inc The software described in this document is furnished under a license agreement The software may be used or copied only under the terms of the license agreement No part of this manual may be photocopied or repro duced in any form without prior written consent from The MathWorks Inc FEDERAL ACQUISITION This provision applies to all acquisitions of the Program and Documentation by for or through the federal government of the United States By accepting delivery of the Program or Documentation the government hereby agrees that this software or documentation qualifies as commercial comp
17. accept the defaults of soft fixed form thresholding and unscaled white noise Continue by clicking the De noise button The de noised signal appears superimposed on the original The tool also plots the wavelet coefficients of both signals 2 50 One Dimensional Discrete Wavelet Analysis Wavnlet 1 H Demm Ee Wer mai Tou Window Heo Zoom in on the plot of the original and de noised signals for a closer look Drag a rubber band box around the pertinent area and then click the XY button The De noise window magnifies your view By default the original signal is shown in red and the de noised signal in yellow 2 51 2 Using Wavelets 2 52 Original and de noised signals 2000 2500 3000 3500 Dismiss the Wavelet 1 D De noising window click the Close button You cannot have the De noise and Compression windows open simultaneously so close the Wavelet 1 D De noising window to continue When the Update Synthesized Signal dialog box appears click No If you click Yes the Synthesized Signal is then available in the Wavelet 1 D main window Refine the analysis The graphical tools make it easy to refine an analysis any time you want to Up to now we ve looked at a level 3 analysis using db1 Let s refine our analysis of the electrical consumption signal using the db3 wavelet at level 5 Select 5 from the Level menu at the upper right and select the db3 from the Wavelet menu Click the Analyze button
18. al scheme known in the signal processing community as a two channel subband coder see page 1 of the book Wavelets and Filter Banks by Strang and Nguyen StrN96 This very practical filtering algorithm yields a fast wavelet transform a box into which a signal passes and out of which wavelet coefficients quickly emerge Let s examine this in more depth One Stage Filtering Approximations and Details For many signals the low frequency content is the most important part It is what gives the signal its identity The high frequency content on the other hand imparts flavor or nuance Consider the human voice If you remove the high frequency components the voice sounds different but you can still tell what s being said However if you remove enough of the low frequency components you hear gibberish In wavelet analysis we often speak of approximations and details The approximations are the high scale low frequency components of the signal The details are the low scale high frequency components The filtering process at its most basic level looks like this Discrete Wavelet Transform OT ue a Filters a ee low pass high pass The original signal S passes through two complementary filters and emerges as two signals Unfortunately if we actually perform this operation on a real digital signal we wind up with twice as much data as we started with Suppose for instance that the original signal S consis
19. alled decomposition or analysis The other half of the story is how those components can be assembled back into the original signal without loss of information This process is called reconstruction or synthesis The mathematical manipulation that effects synthesis is called the inverse discrete wavelet transform IDWT To synthesize a signal using Wavelet Toolbox software we reconstruct it from the wavelet coefficients H Where wavelet analysis involves filtering and downsampling the wavelet reconstruction process consists of upsampling and filtering Upsampling is the process of lengthening a signal component by inserting zeros between samples Il 2 3 4 5 1 2 3 4 5 6 7 8 9 10 Signal component Upsampled signal component The toolbox includes commands like idwt and waverec that perform single level or multilevel reconstruction respectively on the components of one dimensional signals These commands have their two dimensional analogs idwt2 and waverec2 1 28 1 29 T Wavelets A New Tool for Signal Analysis 1 30 Reconstruction Filters The filtering part of the reconstruction process also bears some discussion because it is the choice of filters that is crucial in achieving perfect reconstruction of the original signal The downsampling of the signal components performed during the decomposition phase introduces a distortion called aliasing It turns out that by carefully choosing filters for the decomposition
20. and reconstruction phases that are closely related but not identical we can cancel out the effects of aliasing A technical discussion of how to design these filters is available on page 347 of the book Wavelets and Filter Banks by Strang and Nguyen The low and high pass decomposition filters L and H together with their associated reconstruction filters L and H form a system of what is called quadrature mirror filters Decomposition Reconstruction Reconstructing Approximations and Details We have seen that it is possible to reconstruct our original signal from the coefficients of the approximations and details H cD 500 coefs 1000 samples a OE 500 coefs L Wavelet Reconstruction II I UOU It is also possible to reconstruct the approximations and details themselves from their coefficient vectors As an example let s consider how we would reconstruct the first level approximation A1 from the coefficient vector cA1 We pass the coefficient vector cA1 through the same process we used to reconstruct the original signal However instead of combining it with the level one detail cD1 we feed in a vector of zeros in place of the detail coefficients vector 0 500 zeros 1000 samples cA1 D 500 coefs L The process yields a reconstructed approximation A1 which has the same length as the original signal S and which is a real approximation of it Similarly we can reconstruct the first level
21. asuring average fluctuations at different scales might prove less sensitive to noise Biorthogonal on page 1 43 Coiflets on page 1 45 Symlets on page 1 45 The first recorded mention of what we now call a wavelet seems to be in 1909 in a thesis by Alfred Haar Morlet on page 1 46 Mexican Hat on page 1 46 The concept of wavelets in its present theoretical form was first proposed by Jean Morlet and the team at the Marseille Theoretical Physics Center working under Alex Grossmann in France Meyer on page 1 47 Other Real Wavelets on page 1 47 Complex Wavelets on page 1 47 The methods of wavelet analysis have been developed mainly by Y Meyer and his colleagues who have ensured the methods dissemination The main To explore all wavelet families on your own check out the Wavelet Display algorithm dates back to the work of Stephane Mallat in 1988 Since then taol research on wavelets has become international Such research is particularly active in the United States where it is spearheaded by the work of scientists 1 Type wavemenu at the MATLAB command line The Wavelet Toolbox such as Ingrid Daubechies Ronald Coifman and Victor Wickerhauser Main Menu appears Barbara Burke Hubbard describes the birth the history and the seminal concepts in a very clear text See The World According to Wavelets A K Peters Wellesley 1996 The wavelet domain is growing up very quick
22. ation A3 is quite clean as a comparison between it and the original signal To compare the approximation to the original signal type subplot 2 1 1 plot s title Original axis off subplot 2 1 2 plot A3 title Level 3 Approximation axis off Of course in discarding all the high frequency information we ve also lost many of the original signal s sharpest features Optimal de noising requires a more subtle approach called thresholding This involves discarding only the portion of the details that exceeds a certain limit 12 Remove noise by thresholding Let s look again at the details of our level 3 analysis To display the details D1 D2 and D3 type subplot 3 1 1 plot D1i title Detail Level 1 axis off subplot 3 1 2 plot D2 title Detail Level 2 axis off subplot 3 1 3 plot D3 title Detail Level 3 axis off 2 36 2 37 2 Using Wavelets Detail Level 1 lt q Setting a threshold Detail Level 2 sot fpr Detail Level 3 E tina ht bro Most of the noise occurs in the latter part of the signal where the details show their greatest activity What if we limited the strength of the details by restricting their maximum values This would have the effect of cutting back the noise while leaving the details unaffected through most of their durations But there s a better way Note that cD1 cD2 and cD3 are just MATLAB vectors so we could directly manipulate each vector
23. ative histograms You can plot these histograms separately using the Histograms button from the Wavelets 1 D window Click the Approximation option button A menu appears from which you choose the level of the approximation you want to examine Select the synthesized signal or signal component whose statistics you want to examine Click the appropriate option button and then click the Show Statistics button Here we ve chosen to examine the compressed signal using more 100 bins instead of 30 which is the default Select Level 1 and again click the Show Statistics button Statistics appear for the level 1 approximation 2 56 2 57
24. cal tools to emulate what we did in the previous wavelet decomposition section using command line functions To perform a level 3 decomposition of the signal using the db1 wavelet The default display mode is called Full Decomposition Mode Other alternatives include Select 3 from the Level menu at the upper right and then click the Analyze button again Separate Mode which shows the details and the approximations in separate columns Superimpose Mode which shows the details on a single plot superimposed in different colors The approximations are plotted similarly Tree Mode which shows the decomposition tree the original signal and one additional component of your choice Click on the decomposition tree to select the signal component you d like to view After the decomposition is performed you ll see a new analysis appear in the Show and Scroll Mode which displays three windows The first shows Wavelet 1 D tool the original signal superimposed on an approximation you select The second window shows a detail you select The third window shows the wavelet coefficients Show and Scroll Mode Stem Cfs is very similar to the Show and Scroll Mode except that it displays in the third window the wavelet coefficients as stem plots instead of colored blocks Select a view 2 46 2 47 One Dimensional Discrete Wavelet Analysis 2 Using Wavelets ae Display mode
25. clarify them we try to untangle the aspects somewhat arbitrarily For scale aspects we present one idea around the notion of local regularity For time aspects we present a list of domains When the decomposition is taken as a whole the de noising and compression processes are center points Scale Aspects As a complement to the spectral signal analysis new signal forms appear They are less regular signals than the usual ones The cusp signal presents a very quick local variation Its equation is t with t close to 0 and 0 lt r lt 1 The lower r the sharper the signal To illustrate this notion physically imagine you take a piece of aluminum foil The surface is very smooth very regular You first crush it into a ball and then you spread it out so that it looks like a surface The asperities are clearly visible Each one represents a two dimension cusp and analog of the one dimensional cusp If you crush again the foil more tightly in a more compact ball when you spread it out the roughness increases and the regularity decreases Several domains use the wavelet techniques of regularity study e Biology for cell membrane recognition to distinguish the normal from the pathological membranes e Metallurgy for the characterization of rough surfaces e Finance which is more surprising for detecting the properties of quick variation of values e In Internet traffic description for designing the services size Time Aspects
26. d by a complex exponential Recall that a complex exponential can be broken down into real and imaginary sinusoidal components The results of the transform are the Fourier coefficients F which when multiplied by a sinusoid of frequency yield the constituent sinusoidal components of the original signal Graphically the process looks like Fourier Transform Signal Constituent sinusoids of different frequencies Similarly the continuous wavelet transform CWT is defined as the sum over all time of the signal multiplied by scaled shifted versions of the wavelet function y C scale position f w scale position t dt The results of the CWT are many wavelet coefficients C which are a function of scale and position 1 15 T Wavelets A New Tool for Signal Analysis 1 16 Multiplying each coefficient by the appropriately scaled and shifted wavelet yields the constituent wavelets of the original signal Wavelet p t Ve Transform hi j Signal Constituent wavelets of different scales and positions Scaling We ve already alluded to the fact that wavelet analysis produces a time scale view of a signal and now we re talking about scaling and shifting wavelets What exactly do we mean by scale in this context Scaling a wavelet simply means stretching or compressing it To go beyond colloquial descriptions such as stretching we introduce the scale factor often denoted by the letter a If we r
27. detail D1 using the analogous process m cD1 w 500 coefs 1000 samples o 500 zeros L The reconstructed details and approximations are true constituents of the original signal In fact we find when we combine them that A D S Note that the coefficient vectors cA1 and cD1 because they were produced by downsampling and are only half the length of the original signal cannot directly be combined to reproduce the signal It is necessary to reconstruct the approximations and details before combining them 1 31 Wavelet Reconstruction T Wavelets A New Tool for Signal Analysis Taaa E Lprime Extending this technique to the components of a multilevel analysis we find 0 3415 0 5915 0 1585 0 0915 that similar relationships hold for all the reconstructed signal constituents That is there are several ways to reassemble the original signal If we reverse the order of this vector see wrev and then multiply every even sample by 1 we obtain the high pass filter H Hprime 0 0915 0 1585 0 5915 0 3415 S A Dj f Next upsample Hprime by two see dyadup inserting zeros in alternate Reconstructed positions Signal D D Components HU A r DaD D 0 0915 O 0 1585 0 0 8915 0 0 345 Finally convolve the upsampled vector with the original low pass filter Relationship of Filters to Wavelet Shapes H2 conv HU Lprime In the section Reconstruction Filters on page 1 30 we spoke of the plot
28. e talking about sinusoids for example the effect of the scale factor is very easy to see 0 f t sin t a 1 4 0 1 2 3 4 5 6 1 a saal P f t sin 2t a 5 4 0 1 2 3 4 5 6 _ a gt 1 0 f t sin 4t a 4 1 0 1 2 3 4 5 6 Continuous Wavelet Transform Ten The scale factor works exactly the same with wavelets The smaller the scale factor the more compressed the wavelet 0 ft wt a 1 0 02 200 400 600 800 1000 1200 1400 1600 1800 2000 0 04 0 02 1 f t w 2t es 0 02 200 400 600 800 1000 1200 1400 1600 1800 2000 0 04 0 02 2 Z 1 f t w 4t a ri 0 200 400 600 800 1000 1200 1400 1600 1800 2000 It is clear from the diagrams that for a sinusoid sin t the scale factor a is related inversely to the radian frequency Similarly with wavelet analysis the scale is related to the frequency of the signal We ll return to this topic later Shifting Shifting a wavelet simply means delaying or hastening its onset Mathematically delaying a function f t by k is represented by f t k H Wavelet function Shifted wavelet function w t w t k Five Easy Steps to a Continuous Wavelet Transform The continuous wavelet transform is the sum over all time of the signal multiplied by scaled shifted versions of the wavelet This process produces wavelet coefficients that are a function of scale and position 1 17 Continuous Wavelet Transform T Wavelets A New Tool fo
29. entally What is a wavelet A wavelet is a waveform of effectively limited duration that has an average value of zero Compare wavelets with sine waves which are the basis of Fourier analysis Sinusoids do not have limited duration they extend from minus to plus infinity And where sinusoids are smooth and predictable wavelets tend to be irregular and asymmetric Sine Wave Wavelet db10 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 Just looking at pictures of wavelets and sine waves you can see intuitively that signals with sharp changes might be better analyzed with an irregular wavelet than with a smooth sinusoid just as some foods are better handled with a fork than a spoon It also makes sense that local features can be described better with wavelets that have local extent Number of Dimensions Thus far we ve discussed only one dimensional data which encompasses most ordinary signals However wavelet analysis can be applied to two dimensional data images and in principle to higher dimensional data This toolbox uses only one and two dimensional analysis techniques Mathematically the process of Fourier analysis is represented by the Fourier transform F o f f t e dt which is the sum over all time of the signal f t multiplie
30. enting the methods settled the issue by pointing us in the right direction For wavelets the period of growth and intuition is becoming a time of consolidation and implementation In this context a toolbox is not only possible but valuable It provides a working environment that permits experimentation and enables implementation Since the field still grows it has to be vast and open The Wavelet Toolbox product addresses this need offering an array of tools that can be organized according to several criteria e Synthesis and analysis tools e Wavelet and wavelet packets approaches e Signal and image processing e Discrete and continuous analyses e Orthogonal and redundant approaches e Coding de noising and compression approaches What can we anticipate for the future at least in the short term It is difficult to make an accurate forecast Nonetheless it is reasonable to think that the pace of development and experimentation will carry on in many different fields Numerical analysis constantly uses new bases of functions to encode its operators or to simplify its calculations to solve partial differential equations The analysis and synthesis of complex transient signals touches musical instruments by studying the striking up when the bow meets the cello string The analysis and synthesis of multifractal signals whose regularity or rather irregularity varies with time localizes information of interest at its geographic loca
31. erform an analysis At least you can t choose an arbitrary wavelet waveform if you want to be able to reconstruct the original signal accurately You are compelled to choose a shape determined by quadrature mirror decomposition filters Scaling Function We ve seen the interrelation of wavelets and quadrature mirror filters The wavelet function y is determined by the high pass filter which also produces the details of the wavelet decomposition There is an additional function associated with some but not all wavelets This is the so called scaling function The scaling function is very similar to the wavelet function It is determined by the low pass quadrature mirror filters and thus is associated with the approximations of the wavelet decomposition In the same way that iteratively upsampling and convolving the high pass filter produces a shape approximating the wavelet function iteratively upsampling and convolving the low pass filter produces a shape approximating the scaling function Wavelet Reconstruction Multistep Decomposition and Reconstruction A multistep analysis synthesis process can be represented as ee 7 D E Tog Analysis Synthesis Decomposition Wavelet Reconstruction DWT Coefficients IDWT This process involves two aspects breaking up a signal to obtain the wavelet coefficients and reassembling the signal from the coefficients We ve already discussed decomposition and reconstruction at some len
32. es one of the brightest stars in the world of wavelet research for signal and image reconstruction By using two wavelets one for invented what are called compactly supported orthonormal wavelets thus decomposition on the left side and the other for reconstruction on the right making discrete wavelet analysis practicable side instead of the same single one interesting properties are derived The names of the Daubechies family wavelets are written 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 Here are the wavelet functions psi of the next nine members of the family db2 db3 db4 db5 db6 0 5 10 0 5 10 15 0 5 10 15 0 5 10 15 db7 db8 db9 db10 You can obtain a survey of the main properties of this family by typing waveinfo db from the MATLAB command line See Daubechies Wavelets dbN on page 6 72 for more detail 1 42 1 43 T Wavelets A New Tool for Signal Analysis 1 44
33. eudo frequency information 1 Load a signal e Zoom in on detail e Display coefficients in normal or absolute mode From the MATLAB prompt type e Choose the scales at which analysis is performed load noissin Since you can perform analyses either from the command line or using the You now have the signal noissin in your workspace graphical interface tools this section has subsections covering each method 7 p whos The final subsection discusses how to exchange signal and coefficient information between the disk and the graphical tools Name Size Bytes Class noissin 1x1000 8000 double array 2 Perform a Continuous Wavelet Transform Use the cwt command Type c cwt noissin 1 48 db4 One Dimensional Continuous Wavelet Analysis 2 Using Wavelets CN 4 Choose scales for the analysis The arguments to cwt specify the signal to be analyzed the scales of the analysis and the wavelet to be used The returned argument c contains the The second argument to cwt gives you fine control over the scale levels on coefficients at various scales In this case c is a 48 by 1000 matrix with each which the continuous analysis is performed In the previous example we row corresponding to a single scale used all scales from 1 to 48 but you can construct any scale vector subject to these constraints 3 Plot the coefficients o All scales must be real positive numbers The cwt command accepts a fourth argument This is a flag that w
34. g 1 D Wavelet Packet 1 D Density Estimation 1 D Continuous Wavelet 1 D Regression Estimation 1 D Complex Continuous Wavelet 1 D Wavelet Coefficients Selection 1 D Fractional Brownian Generation 1 D Wavelet 2 D j Specialized Tools 2 D Wavelet Packet 2 D True Compression 2 D msa iano C er oe WawktDisay Choose the File gt Load Signal menu option WaeletPacketoisniay When the Load Signal dialog box appears select the demo MAT file cuspamax mat which should reside in the MATLAB directory Newwaveiettorcwt toolbox wavelet wavedemo Click the OK button The cusp signal is loaded into the Complex Continuous Wavelet 1 D tool 2 Load a signal The default value for the sampling period is equal to 1 second 3 Perform a Complex Continuous Wavelet Transform Click the Complex Continuous Wavelet 1 D menu item To start our analysis let s perform an analysis using the cgau4 wavelet at scales 1 through 64 in steps of 2 just as we did using command line 2 24 2 25 One Dimensional Complex Continuous Wavelet Analysis 2 Using Wavelets functions in Complex Continuous Analysis Using the Command Line on _ Analyzed Signal length 1024 Analyzed Signal length 1024 page 2 21 1 In the upper right portion of the Complex Continuous Wavelet 1 D tool o o r ed select the cgau4 wavelet and scales 1 64 in steps of 2 Modulus of Ca b Coefficients Angle of Ca b Coefficients Dats ce e EAT
35. g command line functions in the previous section i Wi Eesi oi eolian freee pe E EETA Coto eti Line Cab oracle do Sd ieQuency OOF In the upper right portion of the Continuous Wavelet 1 D tool select the db4 wavelet and scales 1 48 Das ie ae mT Wavelet es Te Select db4 5 View Wavelet Coefficients Line Scale Settings Step by Step Mode Select another scale a 40 by clicking in the coefficients plot with the right Min gt 0 mouse button See step 9 to know more precisely how to select the desired scale Step gt 0 oa A Select scales 1 to 48 in steps of 1 Max lt 255 48 Click the New Coefficients Line button The tool updates the plot 2 10 2 11 9 One Dimensional Continuous Wavelet Analysis Using Wavelets Anslyed digaal length 1000 Coefficients Line Ca b for scale a 40 frequency 0 018 6 View Maxima Line Click the Refresh Maxima Line button The local maxima plot displays the re reece ee chaining across scales of the coefficients local maxima from a 40 down to a 1 Local Maxirra Lines 46 ri 37 34 31 28 33 19 16 13 9 4 1 E i ee es ee ee Se e S eee ees ae ss eee eee ee eee 100 200 300 400 500 600 700 800 900 Hold down the right mouse button over the coefficients plot The position of the mouse is given by the Info frame located at the bottom of the screen in 7 itch fi l P F Infi ion terme of locaton OO and stale Sca Switch fr
36. gth Of course there is no point breaking up a signal merely to have the satisfaction of immediately reconstructing it We may modify the wavelet coefficients before performing the reconstruction step We perform wavelet analysis because the coefficients thus obtained have many known uses de noising and compression being foremost among them But wavelet analysis is still a new and emerging field No doubt many uncharted uses of the wavelet coefficients lie in wait The toolbox can be a means of exploring possible uses and hitherto unknown applications of wavelet analysis Explore the toolbox functions and see what you discover 1 35 Introduction to the Wavelet Families T Wavelets A New Tool for Signal Analysis nas Introduction to the Wavelet Families History of Wavelets Several families of wavelets that have proven to be especially useful are From an historical point of view wavelet analysis is a new method though its included in this toolbox What follows is an introduction to some wavelet mathematical underpinnings date back to the work of Joseph Fourier in the families nineteenth century Fourier laid the foundations with his theories of frequency analysis which proved to be enormously important and influential e Haar on page 1 41 The attention of researchers gradually turned from frequency based analysis e Daubechies on page 1 42 to scale based analysis when it started to become clear that an approach me
37. he drawback is that once you choose a particular size for the time window that window is the same for all frequencies Many signals require a more flexible approach one where we can vary the window size to determine more accurately either time or frequency Wavelet Analysis 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 we want more precise low frequency information and shorter regions where we want high frequency information Amplitude Scale Wavelet A Transform Time Wavelet Analysis Here s what this looks like in contrast with the time based frequency based and STFT views of a signal gt 5 5 D lt Ta Time Amplitude Time Domain Shannon Frequency Domain Fourier ao S 8 D 09 Le Time Time STFT Gabor Wavelet Analysis You may have noticed that wavelet analysis does not use a time frequency region but rather a time scale region For more information about the concept of scale and the link between scale and frequency see How to Connect Scale to Frequency on page 6 66 Wavelet Analysis Wavelets A New Tool for Signal Analysis Indeed in their brief history within the signal processing field wavelets have What Can Wavelet Analysis Do already proven themselves to be an indispensable addition to the analyst s One major advantage aff
38. hematics professors at Ecole Centrale de Lyon University of Marne La Vall e and Paris 5 University Yves Misitiis a research engineer specializing in Computer Sciences at Paris 11 University The authors are members of the Laboratoire de Math matique at Orsay Paris 11 University France Their fields of interest are statistical signal processing stochastic processes adaptive control and wavelets The authors group established more than 15 years ago has published numerous theoretical papers and carried out applications in close collaboration with industrial teams For instance e Robustness of the piloting law for a civilian space launcher for which an expert system was developed e Forecasting of the electricity consumption by nonlinear methods e Forecasting of air pollution Notes by Yves Meyer The history of wavelets is not very old at most 10 to 15 years The field experienced a fast and impressive start characterized by a close knit international community of researchers who freely circulated scientific information and were driven by the researchers youthful enthusiasm Even as the commercial rewards promised to be significant the ideas were shared the trials were pooled together and the successes were shared by the community There are lots of successes for the community to share Why Probably because the time is ripe Fourier techniques were liberated by the appearance of windowed Fourier methods that operate l
39. hen The scale increment must be positive present causes cwt to produce a plot of the absolute values of the continuous The high l j l j h D A E cack oaks e highest scale cannot exceed a maximum value depending on the signal The cwt command can accept more arguments to define the different characteristics of the produced plot For more information see the cwt reference page c cwt noissin 2 2 128 db4 plot Let s repeat the analysis using every other scale from 2 to 128 Type c cwt noissin 1 48 db4 plot A new plot appears A plot appears Absolute Values of Ca b Coefficients fora 246810 Absolute Values of Ca b Coefficients fora 12345 46 i l 43 i scales a scales a 100 200 300 400 500 600 700 800 900 1000 time or space b 100 200 300 400 500 600 700 800 900 1000 time or space b This plot gives a clearer picture of what s happening with the signal Of course coefficient plots generated from the command line can be highlighting the periodicity manipulated using ordinary MATLAB graphics commands PE One Dimensional Continuous Wavelet Analysis Using Wavelets Continuous Analysis Using the Graphical Interface We now use the Continuous Wavelet 1 D tool to analyze the same noisy sinusoidal signal we examined earlier using the command line interface in Continuous Analysis Using the Command Line on page 2 5 Click the Continuous Wavelet 1 D menu item The conti
40. in its reference page In this section you ll learn how to e Load a signal e Perform a complex continuous wavelet transform of a signal e Produce plots of the coefficients Since you can perform analyses either from the command line or using the graphical interface tools this section has subsections covering each method One Dimensional Complex Continuous Wavelet Analysis SEER Complex Continuous Analysis Using the Command Line This example involves a cusp signal 2 5 0 5 0 200 400 600 800 1000 1200 1 Load a signal From the MATLAB prompt type load cuspamax You now have the signal cuspamax in your workspace whos Name Size Bytes Class Caption 1x71 142 char array Cuspamax 1x1024 8192 double array caption Caption x linspace 0 1 1024 y exp 128 x 0 3 2 3 abs x 0 7 0 4 caption is a string that contains the signal definition 2 21 2 Using Wavelets 2 22 2 Perform a Continuous Wavelet Transform Use the cwt command Type c cwt cuspamax 1 2 64 cgau4 The arguments to cwt specify the signal to be analyzed the scales of the analysis and the wavelet to be used The returned argument c contains the coefficients at various scales In this case c is a complex 32 by 1024 matrix each row of which corresponds to a single scale Plot the coefficients The cwt command accepts a fourth argument This is a flag that when present causes cwt to produce fo
41. let wavedemo Click the OK button After a pause for computation the tool displays the decomposition 2 42 2 43 One Dimensional Discrete Wavelet Analysis 2 Using Wavelets Wavelet 1 D File View Insert Tools Window Help Decomposition at level 1 za mi eat Decomposition at level 1 s ai di Click the X button located at the bottom of the screen to zoom horizontally The Wavelet 1 D tool zooms all the displayed signals Decorposition at evel ice at edi 20 500 1000 1500 10 10 x Y XY se 5 Zoom in on relevant detail One advantage of using the graphical interface tools is that you can zoom in easily on any part of the signal and examine it in greater detail 0 9 e 350 eo 250 0 180 2 Drag a rubber band box by holding down the left mouse button over the portion of the signal you want to magnify Here we ve selected the noisy part of the original signal The other zoom controls do more or less what you d expect them to The X button for example zooms out horizontally The history function keeps track of all your views of the signal Return to a previous zoom level by clicking the left arrow button 2 44 2 45 One Dimensional Discrete Wavelet Analysis 2 Using Wavelets CE Selecting Different Views of the Decomposition 6 Perform a multilevel decomposition The Display mode menu middle right lets you choose different views of the Again we ll use the graphi
42. ly A lot of mathematical papers and practical trials are published every month 1 38 1 39 T Wavelets A New Tool for Signal Analysis 1 40 SVYT Denoising 1 D Wavelet 2 D Wavelet Packet 2 D Density Estimation 1 D Regression Estimation 1 1 Wavelet Coefficients Selectior Fractional Brownian Generatio True Compression 2 D SYVT Denoising 2 D Multisignal Analysis 1 D Wavelet Coefficients Selectior Multivariate Denoising Image Fusion 7 Multiscale Princ Comp Analysis Signal Extension Wavelet Display Imana Evtancinn 2 Click the Wavelet Display menu item The Wavelet Display tool appears 3 Select a family from the Wavelet menu at the top right of the tool 4 Click the Display button Pictures of the wavelets and their associated filters appear 5 Obtain more information by clicking the information buttons located at the right Introduction to the Wavelet Families Haar Any discussion of wavelets begins with Haar wavelet the first and simplest Haar wavelet is discontinuous and resembles a step function It represents the same wavelet as Daubechies db1 See Haar on page 6 73 for more detail 0 Wavelet function psi 1 41 Introduction to the Wavelet Families T Wavelets A New Tool for Signal Analysis nw bechi Biorthogonal Daubechies This family of wavelets exhibits the property of linear phase which is needed Ingrid Daubechi
43. many valuable ideas Colleagues and friends who have helped us steadily are Patrice Abry ENS Lyon Samir Akkouche Ecole Centrale de Lyon Mark Asch Paris 11 Patrice Assouad Paris 11 Roger Astier Paris 11 Jean Coursol Paris 11 Didier Dacunha Castelle Paris 11 Claude Deniau Marseille Patrick Flandrin Ecole Normale de Lyon Eric Galin Ecole Centrale de Lyon Christine Graffigne Paris 5 Anatoli Juditsky Grenoble G rard Kerkyacharian Paris 10 G rard Malgouyres Paris 11 Olivier Nowak Ecole Centrale de Lyon Dominique Picard Paris 7 and Franck Tarpin Bernard Ecole Centrale de Lyon Several student groups have tested preliminary versions One of our first opportunities to apply the ideas of wavelets connected with signal analysis and its modeling occurred during a close and pleasant cooperation with the team Analysis and Forecast of the Electrical Consumption of Electricit de France Clamart Paris directed first by Jean Pierre Desbrosses and then by Herv Laffaye and which included Xavier Brossat Yves Deville and Marie Madeleine Martin Many thanks to those who tested and helped to refine the software and the printed matter and at last to The MathWorks group and specially to Roy Lurie Jim Tung Bruce Sesnovich Jad Succari Jane Carmody and Paul Costa And finally apologies to those we may have omitted About the Authors Michel Misiti Georges Oppenheim and Jean Michel Poggi are mat
44. me the software s graphical interface tools include an easy to use de noising feature that includes automatic thresholding 2 39 2 Using Wavelets 2 40 More information on the de noising process can be found in the following sections Remove noise from a signal on page 2 49 De Noising on page 6 97 One Dimensional Variance Adaptive Thresholding of Wavelet Coefficients on page 2 158 One Dimensional Variance Adaptive Thresholding of Wavelet Coefficients on page 6 107 One Dimensional Analysis Using the Graphical Interface In this section we explore the same electrical consumption signal as in the previous section but we use the graphical interface tools to analyze the signal 1 Start the 1 D Wavelet Analysis Tool From the MATLAB prompt type wavemenu One Dimensional Discrete Wavelet Analysis erent The Wavelet Toolbox Main Menu appears Wavelet Toolbox Main Menu File Window Help One Dimensional Wavelet 1 D Wavelet Packet 1 D Continuous Wavelet 1 D Complex Continuous Wavelet 1 D Two Dimensional Wavelet 2 D Wavelet Packet 2 D Multiple 1 D Multisignal Analysis 1 D Multivariate Denoising Multiscale Princ Comp Analysis Display Wavelet Display Wavelet Packet Display Wavelet Design _ New Wavelet for CVT Specialized Tools 1 D SVT Denoising 1 D Density Estimation 1 D Regression Estimation 1 D Wavelet C
45. n Installing Wavelet Toolbox Software SEE Installing Wavelet Toolbox Software To install this toolbox on your computer see the appropriate platform specific MATLAB installation guide To determine if the Wavelet Toolbox software is already installed on your system check for a subdirectory named wavelet within the main toolbox directory or folder Wavelet Toolbox software can perform signal or image analysis For indexed images or truecolor images represented by m by n by 3 arrays of uint8 all wavelet functions use floating point operations To avoid Out of Memory errors be sure to allocate enough memory to process various image sizes The memory can be real RAM or can be a combination of RAM and virtual memory See your operating system documentation for how to configure virtual memory System Recommendations While not a requirement we recommend you obtain Signal Processing Toolbox and Image Processing Toolbox software to use in conjunction with the Wavelet Toolbox software These toolboxes provide complementary functionality that give you maximum flexibility in analyzing and processing signals and images This manual makes no assumption that your computer is running any other MATLAB toolboxes Platform Specific Details Some details of the use of the Wavelet Toolbox software may depend on your hardware or operating system Windows Fonts We recommend you set your operating system to use Small Font
46. n 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 that is if it is what is called a stationary signal this drawback isn t very important However most interesting signals contain numerous nonstationary or transitory characteristics 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 detecting them 1 9 T Wavelets A New Tool for Signal Analysis 1 10 Short Time Fourier Analysis In an effort to correct this deficiency Dennis Gabor 1946 adapted the Fourier transform to analyze only a small section of the signal at a time a technique called windowing the signal Gabor s adaptation called the Short Time Fourier Transform STFT maps a signal into a two dimensional function of time and frequency window Short Time Amplitude Fourier Frequency Transform Time Time The STFT represents a sort of compromise between the time and frequency based views of a signal It provides some information about both when and at what frequencies a signal event occurs However you can only obtain this information 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 t
47. n properties of this family by typing waveinfo meyer from the MATLAB command line See Meyer Wavelet meyr on page 6 78 for more detail Other Real Wavelets Some other real wavelets are available in the toolbox e Reverse Biorthogonal e Gaussian derivatives family e FIR based approximation of the Meyer wavelet See Additional Real Wavelets on page 6 82 for more information Complex Wavelets Some complex wavelet families are available in the toolbox e Gaussian derivatives e Morlet e Frequency B Spline e Shannon See Complex Wavelets on page 6 84 for more information 1 47 D U5 One Dimensional Continuous Wavelet Analysis Using Wavelets Continuous Analysis Using the Command Line This example involves a noisy sinusoidal signal One Dimensional Continuous Wavelet Analysis This section takes you through the features of continuous wavelet analysis using Wavelet Toolbox software The toolbox requires only one function for continuous wavelet analysis cwt Youll find full information about this function in its reference page In this section you ll learn how to e Load a signal e Perform a continuous wavelet transform of a signal e Produce a plot of the coefficients e Produce a plot of coefficients at a given scale e Produce a plot of local maxima of coefficients across scales e Select the displayed plots 100 200 300 400 500 600 700 800 900 1000 e Switch from scale to ps
48. ng and Exporting Information from the Graphical Interface To know how to import and export information from the Complex Continuous Wavelet Graphical Interface see the corresponding paragraph in One Dimensional Continuous Wavelet Analysis on page 2 4 The only difference is that the variable coefs is a complex matrix see Saving Wavelet Coefficients on page 2 18 One Dimensional Discrete Wavelet Analysis et One Dimensional Discrete Wavelet Analysis This section takes you through the features of one dimensional discrete wavelet analysis using the Wavelet Toolbox software The toolbox provides these functions for one dimensional signal analysis For more information see the reference pages Analysis Decomposition Functions Function Name Purpose dwt Single level decomposition wavedec Decomposition wmaxlev Maximum wavelet decomposition level Synthesis Reconstruction Functions Function Name Purpose idwt Single level reconstruction waverec Full reconstruction wrcoef Selective reconstruction upcoef Single reconstruction Decomposition Structure Utilities Function Name Purpose detcoef Extraction of detail coefficients appcoef Extraction of approximation coefficients upwlev Recomposition of decomposition structure 2 29 2 Using Wavelets 2 30 De noising and Compression Function Name Purpose ddencmp Provide default values for de noising and compression wbmpen Penalized threshold for wavele
49. noise than does the original signal length cA Length cD ans 501 501 You may observe that the actual lengths of the detail and approximation coefficient vectors are slightly more than half the length of the original signal This has to do with the filtering process which is implemented by convolving the signal with a filter The convolution smears the signal introducing several extra samples into the result Discrete Wavelet Transform OE TE Multiple Level Decomposition The decomposition process can be iterated with successive approximations being decomposed in turn so that one signal is broken down into many lower resolution components This is called the wavelet decomposition tree Looking at a signal s wavelet decomposition tree can yield valuable information CA3 Number of Levels Since the analysis process is iterative in theory it can be continued indefinitely In reality the decomposition can proceed only until the individual 1 27 T Wovelets A New Tool for Signal Analysis Wavelet Reconstruction details consist of a single sample or pixel In practice you ll select a suitable Wavelet Reconstruction number of levels based on the nature of the signal or on a suitable criterion such as entropy see Choosing the Optimal Decomposition on page 6 147 We ve learned how the discrete wavelet transform can be used to analyze or decompose signals and images This process is c
50. nuous wavelet analysis tool for one dimensional signal data appears 1 Start the Continuous Wavelet 1 D Tool From the MATLAB prompt type wavemenu Continuous Wavelet 1 D The Wavelet Toolbox Main Menu appears Wavelet Toolbox Main Menu C ommaan rerenmeseann e lt lt Wavelet 2 D Wavelet Packet 2 D ee True Compression 2 D C orome oe nian _ las Signal Extension Wavelet Display Image Extension Wavelet Packet Display New Wavelet for CAT 2 8 2 9 One Dimensional Continuous Wavelet Analysis e s 2 Using Wavelets 4 Click the Analyze button 2 Load a signal After a pause for computation the tool displays the coefficients plot the coefficients line plot corresponding to the scale a 24 and the local maxima plot which displays the chaining across scales from a 48 down to a 1 of the coefficients local maxima Choose the File gt Load Signal menu option When the Load Signal dialog box appears select the demo MAT file noissin mat which should reside in the MATLAB directory toolbox wavelet wavedemo Click the OK button Anata Signin lenge bbi The noisy sinusoidal signal is loaded into the Continuous Wavelet 1 D tool The default value for the sampling period is equal to 1 second Sa eae NASW 3 Perform a Continuous Wavelet Transform To start our analysis let s perform an analysis using the db4 wavelet at scales 1 through 48 just as we did usin
51. ocally on a time frequency approach In another direction Burt Adelson s pyramidal algorithms the quadrature mirror filters and filter banks and subband coding are available The mathematics underlying those algorithms existed earlier but new computing techniques enabled researchers to try out new ideas rapidly The numerical image and signal processing areas are blooming The wavelets bring their own strong benefits to that environment a local outlook a multiscaled outlook cooperation between scales and a time scale analysis They demonstrate that sines and cosines are not the only useful SLL functions and that other bases made of weird functions serve to look at new foreign signals as strange as most fractals or some transient signals Recently wavelets were determined to be the best way to compress a huge library of fingerprints This is not only a milestone that highlights the practical value of wavelets but it has also proven to be an instructive process for the researchers involved in the project Our initial intuition generally was that the proper way to tackle this problem of interweaving lines and textures was to use wavelet packets a flexible technique endowed with quite a subtle sharpness of analysis and a substantial compression capability However it was a biorthogonal wavelet that emerged victorious and at this time represents the best method in terms of cost as well as speed Our intuitions led one way but implem
52. oefficients Selection 1 D Fractional Brownian Generation 1 D Specialized Tools 2 D True Compression 2 D SVT Denoising 2 D Wavelet Coefficients Selection 2 D Image Fusion Extension Signal Extension Image Extension Close 2 Click the Wavelet 1 D menu item The discrete wavelet analysis tool for one dimensional signal data appears 2 41 2 Using Wavelets One Dimensional Discrete Wavelet Analysis Load Signal ry ex2cusp2 mat linchirp mat ex2gauss mat 4 freqbrk mat 4 mandel mat ex2ntix mat 4 geometry mat 4 mitqbrk mat ex2nsto mat 4 heavysin mat 4 mishmash mat ex3nfix mat 4 julia mat 4 nbarb1 mat ex3nsto mat leleccum mat 4 nblocr1 mat Signal mat wav au The electrical consumption signal is loaded into the Wavelet 1 D tool 4 Perform a single level wavelet decomposition To start our analysis let s perform a single level decomposition using the 3 Load a signal db1 wavelet just as we did using the command line functions in One Dimensional Analysis Using the Command Line on page 2 31 From the File menu choose the Load gt Signal option In the upper right portion of the Wavelet 1 D tool select the db1 wavelet and Wavelet 1 D single level decomposition Load Signal om When the Load Signal dialog box appears select the demo MAT file leleccum mat which should reside in the MATLAB directory Click the Analyze button toolbox wave
53. om scale to Pseudo Frequency Information Using the option button on the right part of the screen select Frequencies instead of Scales Again hold down the right mouse button over the coefficients plot the position of the mouse is given in terms of location X and frequency Frq in Hertz 2 12 2 13 2 Using Wavelets Anaid Signal engeh 100 Seale of colon frorn MIH io MAX Contficknte Lint Cab Pereak a 24 irequemcy Sudo Sete eel This facility allows you to interpret scale in terms of an associated pseudo frequency which depends on the wavelet and the sampling period For more information on the connection between scale and frequency see How to Connect Scale to Frequency on page 6 66 8 Deselect the last two plots using the check boxes in the Selected Axes frame 2 14 One Dimensional Continuous Wavelet Analysis Anima Sige agin 10 4 Hall l WANA HA AAA Rt Mt AE Vea Me al PAC ST IGS EAE I peat 1 i i Milt Hl I l ae f Ml Mi i 1 Ij AWONOMO 9 Zoom in on detail Drag a rubber band box by holding down the left mouse button over the portion of the signal you want to magnify Signal length 1000 10 Click the X button located at the bottom of the screen to zoom horizontally only X Y XY aly x AE AL mo T ito EH 2 15 2 Using Wavelets The Continuous Wavelet 1 D tool enlarges the displayed signal and coefficients plot for more inf
54. only Online only Online only Third printing Online only Online only Fourth printing Online only Online only Online only Online only Online only Fifth printing Online only Online only Online only Online only New for Version 1 0 Revised for Version 2 0 Release 12 Revised for Version 2 1 Release 12 1 Revised for Version 2 2 Release 13 Revised for Version 3 0 Release 14 Revised for Version 3 0 Revised for Version 3 0 1 Release 14SP1 Revised for Version 3 0 2 Release 14SP2 Minor revision for Version 3 0 2 Minor revision for Version 3 0 3 Release R14SP3 Minor revision for Version 3 0 4 Release 2006a Revised for Version 3 1 Release 2006b Revised for Version 4 0 Release 2007a Revised for Version 4 1 Release 2007b Revised for Version 4 1 Revised for Version 4 2 Release 2008a Revised for Version 4 3 Release 2008b Revised for Version 4 4 Release 2009a Minor revision for Version 4 4 1 Release 2009b ee Acknowledgments The authors wish to express their gratitude to all the colleagues who directly or indirectly contributed to the making of the Wavelet Toolbox software Specifically e For the wavelet questions to Pierre Gilles Lemari Rieusset Evry and Yves Meyer ENS Cachan e For the statistical questions to Lucien Birg Paris 6 Pascal Massart Paris 11 and Marc Lavielle Paris 5 e To David Donoho Stanford and to Anestis Antoniadis Grenoble who give generously so
55. ools for statistical applications using numerical results type dwtmode zpd before to execute the example code wavelets and wavelet packets within the framework of MATLAB The MathWorks provides several products that are relevant to the kinds of In most of the command line examples figures are displayed To clarify the tasks you can perform with the toolbox For more information about any of presentation the plotting commands are partially or completely omitted To these products see the products section of The MathWorks Web site reproduce the displayed figures exactly you would need to insert some i graphical commands in the example code Wavelet Toolbox software provides two categories of tools e Command line functions e Graphical interactive tools The first category of tools is made up of functions that you can call directly from the command line or from your own applications Most of these functions are M files series of statements that implement specialized wavelet analysis or synthesis algorithms You can view the code for these functions using the following statement type function_name You can view the header of the function the help part using the statement help function_name A summary list of the Wavelet Toolbox functions is available to you by typing help wavelet You can change the way any toolbox function works by copying and renaming the M file then modifying your copy You can also extend the toolbox by
56. orded by wavelets is the ability to perform local collection of tools and continue to enjoy a burgeoning popularity today analysis that is to analyze a localized area of a larger signal Consider a sinusoidal signal with a small discontinuity one so tiny as to be barely visible Such a signal easily could be generated in the real world perhaps by a power fluctuation or a noisy switch Sinusoid with a small discontinuity A plot of the Fourier coefficients as provided by the fft command of this signal shows nothing particularly interesting 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 40 60 Fourier Coefficients Wavelet Coefficients Wavelet analysis is capable of revealing aspects of data that other signal analysis techniques miss aspects like trends breakdown points discontinuities in higher derivatives and self similarity Furthermore because it affords a different view of data than those presented by traditional techniques wavelet analysis can often compress or de noise a signal without appreciable degradation 1 12 1 13 Wavelets A New Tool for Signal Analysis Continuous Wavelet Transform What Is Wavelet Analysis Continuous Wavelet Transform Now that we know some situations when wavelet analysis is useful it is worthwhile asking What is wavelet analysis and even more fundam
57. ormation on zooming see Connection of Plots on page A 3 Analysed Signal lengih 180 i in l i i 4 i i eid Gall i WHATMAN ARTE i As with the command line analysis on the preceding pages you can change the scales or the analyzing wavelet and repeat the analysis To do this just edit the necessary fields and click the Analyze button 11 View normal or absolute coefficients The Continuous Wavelet 1 D tool allows you to plot either the absolute values of the wavelet coefficients or the coefficients themselves More generally the coefficients coloration can be done in several different ways For more details on the Coloration Mode see Controlling the Coloration Mode on page A 7 2 16 One Dimensional Continuous Wavelet Analysis anaemia Choose either one of the absolute modes or normal modes from the Coloration Mode menu In normal modes the colors are scaled between the minimum and maximum of the coefficients In absolute modes the colors are scaled between zero and the maximum absolute value of the coefficients The coefficients plot is redisplayed in the mode you select Oe ee ee peot de MDD Bess oi sota ter ER mE a Absolute Mode Normal Mode Importing and Exporting Information from the Graphical Interface The Continuous Wavelet 1 D graphical interface tool lets you import information from and export information to disk You can e Load signal
58. otice that the scales in the coefficients plot shown as y axis labels run from 1 to 31 Recall that the higher scales correspond to the most stretched wavelets The more stretched the wavelet the longer the portion of the signal with which it is being compared and thus the coarser the signal features being measured by the wavelet coefficients j Wavelet 1 h Low scale High scale Thus there is a correspondence between wavelet scales and frequency as revealed by wavelet analysis e Low scale a gt Compressed wavelet gt Rapidly changing details gt High frequency 1 20 1 21 Continuous Wavelet Transform T Wavelets A New Tool for Signal Analysis OE E The CWT is also continuous in terms of shifting during computation the Here is a case where thinking in terms of scale makes much more sense than analyzing wavelet is shifted smoothly over the full domain of the analyzed thinking in terms of frequency Inspection of the CWT coefficients plot for this function signal reveals patterns among scales and shows the signal s possibly fractal nature Absolute Values of Ca b Coefficients fora 1 1 48 Even though this signal is artificial many natural phenomena from the intricate branching of blood vessels and trees to the jagged surfaces of mountains and fractured metals lend themselves to an analysis of scale What s Continuous About the Continuous Wavelet Transform Any signal processing perfo
59. r Signal Analysis It s really a very simple process In fact here are the five steps of an easy recipe for creating a CWT Signal 1 Take a wavelet and compare it to a section at the start of the original signal Wavelet CC Tee NANNAN AN 2 Calculate a number C that represents how closely correlated the wavelet is with this section of the signal The higher C is the more the similarity More precisely if the signal energy and the wavelet energy are equal to one C may be interpreted as a correlation coefficient C 0 2247 5 Repeat steps 1 through 4 for all scales Note that the results will depend on the shape of the wavelet you choose When yov re done you ll have the coefficients produced at different scales by different sections of the signal The coefficients constitute the results of a regression of the original signal performed on the wavelets Signal How to make sense of all these coefficients You could make a plot on which the x axis represents position along the signal time the y axis represents scale and the color at each x y point represents the magnitude of the wavelet coefficient C These are the coefficient plots generated by the graphical tools Wavelet O MANN Large 3 Shift the wavelet to the right and repeat steps 1 and 2 until you ve covered Coefficients the whole signal f n Signal 2 D Wavelet 4 Scale stretch the wavelet and repeat steps 1 through 3 A
60. rmed on a computer using real world data must be performed on a discrete signal that is on a signal that has been measured at discrete time So what exactly is continuous about it What s continuous about the CWT and what distinguishes it from the discrete wavelet transform to be discussed in the following section is the set of scales and positions at which it operates Unlike the discrete wavelet transform the CWT can operate at every scale from that of the original signal up to some maximum scale that you determine by trading off your need for detailed analysis with available computational horsepower 1 22 1 23 T Wavelets A New Tool for Signal Analysis 1 24 Discrete Wavelet Transform Calculating wavelet coefficients at every possible scale is a fair amount of work and it generates an awful lot of data What if we choose only a subset of scales and positions at which to make our calculations It turns out rather remarkably that if we choose scales and positions based on powers of two so called dyadic scales and positions then our analysis will be much more efficient and just as accurate We obtain such an analysis from the discrete wavelet transform DWT For more information on DWT see Algorithms on page 6 23 An efficient way to implement this scheme using filters was developed in 1988 by Mallat see Mal89 in References on page 6 155 The Mallat algorithm is in fact a classic
61. s Set this option by clicking the Display icon in your desktop s Control Panel accessible through the Settings Control Panel submenu Select the Configuration option and then use the Font Size menu to change to Small Fonts You ll have to restart Windows for this change to take effect 1 5 T Wavelets A New Tool for Signal Analysis Wavelet Applications Wavelet Applications Fonts for Non Windows Platforms We recommend you set your operating system to use standard default fonts However for all platforms if you prefer to use large fonts some of the labels in the GUI figures may be illegible when using the default display mode of the toolbox To change the default mode to accept large fonts use the wtbxmngr function For more information see either the wtbxmngr help or its reference page Mouse Compatibility Wavelet Toolbox software was designed for three distinct types of mouse control Left Mouse Button Middle Mouse Button Right Mouse Button Make selections Display cross hairs to Translate plots up and Activate controls show position dependent down and left and information right o Shift Option Note The functionality of the middle mouse button and the right mouse button can be inverted depending on the platform For more information see Using the Mouse on page A 4 Wavelets have scale aspects and time aspects consequently every application has scale and time aspects To
62. s from disk into the Continuous Wavelet 1 D tool e Save wavelet coefficients from the Continuous Wavelet 1 D tool to disk Loading Signals into the Continuous Wavelet 1 D Tool To load a signal you ve constructed in your MATLAB workspace into the Continuous Wavelet 1 D tool save the signal in a MAT file with extension mat or other For instance suppose you ve designed a signal called warma and want to analyze it in the Continuous Wavelet 1 D tool save warma warma 2 17 2 Using Wavelets The workspace variable warma must be a vector sizwarma size warma Sizwarma 1 1000 To load this signal into the Continuous Wavelet 1 D tool use the menu option File gt Load Signal A dialog box appears that lets you select the appropriate MAT file to be loaded Note The first one dimensional variable encountered in the file is considered the signal Variables are inspected in alphabetical order Saving Wavelet Coefficients The Continuous Wavelet 1 D tool lets you save wavelet coefficients to disk The toolbox creates a MAT file in the current directory with the extension wc1 and a name you give it To save the continuous wavelet coefficients from the present analysis use the menu option File gt Save gt Coefficients A dialog box appears that lets you specify a directory and filename for storing the coefficients Consider the example analysis File gt Example Analysis gt with haar at scales 1 1 64 gt
63. t 1 D or 2 D de noising wdcbm Thresholds for wavelet 1 D using BirgDO Massart strategy wdencmp Wavelet de noising and compression wden Automatic wavelet de noising wthrmngr Threshold settings manager In this section you ll learn how to e Load a signal e Perform a single level wavelet decomposition of a signal e Construct approximations and details from the coefficients e Display the approximation and detail e Regenerate a signal by inverse wavelet transform e Perform a multilevel wavelet decomposition of a signal e Extract approximation and detail coefficients e Reconstruct the level 3 approximation e Reconstruct the level 1 2 and 3 details e Display the results of a multilevel decomposition e Reconstruct the original signal from the level 3 decomposition e Remove noise from a signal e Refine an analysis e Compress a signal e Show a signal s statistics and histograms Since you can perform analyses either from the command line or using the graphical interface tools this section has subsections covering each method One Dimensional Discrete Wavelet Analysis S s The final subsection discusses how to exchange signal and coefficient information between the disk and the graphical tools One Dimensional Analysis Using the Command Line This example involves a real world signal electrical consumption measured over the course of 3 days This signal is particularly interesting because of noise
64. tion Compression is a booming field and coding and de noising are promising For each of these areas the Wavelet Toolbox software provides a way to introduce learn and apply the methods regardless of the user s experience It includes a command line mode and a graphical user interface mode each very capable and complementing to the other The user interfaces help the novice to get started and the expert to implement trials The command line provides an open environment for experimentation and addition to the graphical interface In the journey to the heart of a signal s meaning the toolbox gives the traveler both guidance and freedom going from one point to the other wandering from a tree structure to a superimposed mode jumping from low to high scale and skipping a breakdown point to spot a quadratic chirp The time scale graphs of continuous analysis are often breathtaking and more often than not enlightening as to the structure of the signal Here are the tools waiting to be used Yves Meyer Professor Ecole Normale Sup rieure de Cachan and Institut de France Notes by Ingrid Daubechies Wavelet transforms in their different guises have come to be accepted as a set of tools useful for various applications Wavelet transforms are good to have at one s fingertips along with many other mostly more traditional tools Wavelet Toolbox software is a great way to work with wavelets The toolbox together with the power of M
65. tivity in ECG noise removal In EEGs a quick transitory signal is drowned in the usual one The wavelets are able to determine if a quick signal exists and if so can localize it There are attempts to enhance mammograms to discriminate tumors from calcifications Another prototypical application is a classification of Magnetic Resonance Spectra The study concerns the influence of the fat we eat on our body fat The type of feeding is the basic information and the study is intended to avoid taking a sample of the body fat Each Fourier spectrum is encoded by some of its wavelet coefficients A few of them are enough to code the most interesting features of the spectrum The classification is performed on the coded vectors Signal analysts already 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 F Amplitude Amplitude Fourier Transform Frequency Time For many signals Fourier analysis is extremely useful because the signal s frequency content is of great importance So why do we need other techniques like wavelet analysis Fourier analysis has a serious drawback In transforming to the frequency domain time information is lost Whe
66. ts of 1000 samples of data Then the resulting signals will each have 1000 samples for a total of 2000 These signals A and D are interesting but we get 2000 values instead of the 1000 we had There exists a more subtle way to perform the decomposition using wavelets By looking carefully at the computation we may keep only one point out of two in each of the two 2000 length samples to get the complete information This is the notion of downsampling We produce two sequences called cA and cD 1000 samples 500 coefs 1000 samples 500 coefs The process on the right which includes downsampling produces DWT coefficients To gain a better appreciation of this process let s perform a one stage discrete wavelet transform of a signal Our signal will be a pure sinusoid with high frequency noise added to it 1 25 T Wavelets A New Tool for Signal Analysis 1 26 Here is our schematic diagram with real signals inserted into it cD High Frequency LE HC mtr S 500 DWT coefficients ANVAN 1000 data point ERDAM 500 DWT coefficients cA Low Frequency The MATLAB code needed to generate s cD and cAis s sin 20 linspace 0 pi 1000 0 5 rand 1 1000 cA cD dwt s db2 where db2 is the name of the wavelet we want to use for the analysis Notice that the detail coefficients cD are small and consist mainly of a high frequency noise while the approximation coefficients cA contain much less
67. uct the details at levels 1 2 and 3 from C type D1 wrcoef d C L db1 1 D2 wrcoef d C L db1 2 D3 wrcoef d C L db1 3 One Dimensional Discrete Wavelet Analysis 9 Display the results of a multilevel decomposition To display the results of the level 3 decomposition type subplot 2 2 1 plot A3 title Approximation A3 subplot 2 2 2 plot D1 title Detail D1 subplot 2 2 3 plot D2 title Detail D2 subplot 2 2 4 plot D3 title Detail D3 Approximation A3 Detail D1 600 40 500 400 300 200 100 40 0 1000 2000 3000 4000 0 1000 2000 3000 4000 Detail D2 Detail D3 40 40 20 20 0 0 20 20 40 40 0 1000 2000 3000 4000 0 1000 2000 3000 4000 10 Reconstruct the original signal from the Level 3 decomposition To reconstruct the original signal from the wavelet decomposition structure type AO waverec C L db1 err max abs s A0O err D 5475e 013 2 35 9 One Dimensional Discrete Wavelet Analysis Using Wavelets 11 Crude de noising of a signal Original Using wavelets to remove noise from a signal requires identifying which component or components contain the noise and then reconstructing the signal without those components In this example we note that successive approximations become less and less noisy as more and more high frequency information is filtered out of the signal Level 3 Approximation The level 3 approxim
68. ur plots related to the complex continuous wavelet transform coefficients Real and imaginary parts Modulus and angle The cwt command can accept more arguments to define the different characteristics of the produced plots For more information see the cwt reference page Type c cwt Ccuspamax 1 2 64 cgau4 plot One Dimensional Complex Continuous Wavelet Analysis A plot appears Real part of Ca b fora 13579 Imaginary part of Ca b fora 13579 scales a scales a 200 400 600 800 1000 200 400 600 800 1000 time or space b time or space b Modulus of Ca b fora 13579 Angle of Ca b fora 13579 l scales a scales a 200 400 600 800 1000 200 400 600 800 1000 time or space b time or space b Of course coefficient plots generated from the command line can be manipulated using ordinary MATLAB graphics commands Complex Continuous Analysis Using the Graphical Interface We now use the Complex Continuous Wavelet 1 D tool to analyze the same cusp signal we examined using the command line interface in the previous section 1 Start the Complex Continuous Wavelet 1 D Tool From the MATLAB prompt type wavemenu The Wavelet Toolbox Main Menu appears 2 23 One Dimensional Complex Continuous Wavelet Analysis SUSE 2 Using Wavelets The continuous wavelet analysis tool for one dimensional signal data appears Wavelet Toolbox Main Menu Wavelet 1 D SVT Denoisin
69. uter software or commercial computer software documentation as such terms are used or defined in FAR 12 212 DFARS Part 227 72 and DFARS 252 227 7014 Accordingly the terms and conditions of this Agreement and only those rights specified in this Agreement shall pertain to and govern the use modification reproduction release performance display and disclosure of the Program and Documentation by the federal government or other entity acquiring for or through the federal government and shall supersede any conflicting contractual terms or conditions If this License fails to meet the government s needs or is inconsistent in any respect with federal procurement law the government agrees to return the Program and Documentation unused to The MathWorks Inc Trademarks MATLAB and Simulink are registered trademarks of The MathWorks Inc See www mathworks com trademarks for a list of additional trademarks Other product or brand names may be trademarks or registered trademarks of their respective holders Patents The MathWorks products are protected by one or more U S patents Please see www mathworks com patents for more information Revision History March 1997 September 2000 June 2001 July 2002 June 2004 July 2004 October 2004 March 2005 June 2005 September 2005 March 2006 September 2006 March 2007 September 2007 October 2007 March 2008 October 2008 March 2009 September 2009 First printing Second printing Online

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