Home

Wavelet Analysis Tools User Manual

1. Figure 4 15 Denoising Comparison In Figure 4 15 you can see that the UWT outperforms the DWT in signal denoising because the denoised signal in the Denoising with UWT graph is smoother In the Browse tab of the NI Example Finder you can view this example by selecting Toolkits and Modules Wavelet Analysis Getting Started Denoise 1D Real Signal VI You also can view the follow examples that use the UWT based method to denoise complex signals and 2D signals e Toolkits and Modules Wavelet Analysis Getting Started Denoise 1D Complex Signal VI e Toolkits and Modules Wavelet Analysis Getting Started Denoise Image VI Refer to the Finding Example VIs section of Chapter 1 Introduction to Wavelet Signal Processing for information about launching the NI Example Finder Wavelet Analysis Tools User Manual 4 16 ni com Chapter 4 Signal Processing with Discrete Wavelets Better Peak Detection Capability Peaks often imply important information about a signal You can use the UWT to identify the peaks in a noise contaminated signal The UWT based peak detection method is more robust and less sensitive to noise than the DWT based method because the UWT based method involves finding zero crossings in the multiscale UWT coefficients The UW T based method first finds zero crossings among the coefficients with coarse resolution and then finds zero crossings among the coefficients with finer resolution Finding zero c
2. 400 300 200 100 o1 o2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 Time s Figure 3 8 AWT Based Scalogram of the HypChirps Signal National Instruments Corporation 3 9 Wavelet Analysis Tools User Manual Chapter 3 Signal Processing with Continuous Wavelets Figure 3 9 shows the tiling of the AWT based time frequency representation that provides fine frequency resolution at low frequencies and fine time resolution at high frequencies Frequency Time Figure 3 9 Tiling of Wavelet Based Time Frequency Representation Similar to the CWT the AWT also adds excess information redundancy and is computationally intensive Moreover you cannot reconstruct the original signal from the AWT coefficients Wavelet Normalization Energy versus Amplitude In wavelet analysis wavelets at different scales often have the same energy Because both the center frequency and the bandwidth of a wavelet are inversely proportional to the scale factor the wavelet at a larger scale has a higher magnitude response than a wavelet at a smaller scale Figure 3 10 shows the Fourier magnitude spectra of different wavelets with energy normalization Energy Normalization Time Figure 3 10 Magnitude Spectra of Wavelets with Energy Normalization Wavelet Analysis Tools User Manual 3 10 ni com Chapter 3 Signal Processing with Continuous Wavelets However some re
3. G0 Blue Cross HO Red Circle EAE x 0 0000 y 0 0000 to N No N n Imaginary k cn oo uu PE o N Zeroes at x Wavelet Analysis Tools User Manual Figure 5 9 Zeroes of GO and HO In Figure 5 9 notice that besides the four zeroes assigned to Ho z at T the Zeroes of GO and H0 graph also contains six more zeroes that belong to Ho z Note that two zeroes are on the negative half plane and do not appear on this graph To make the number of zeroes of Go z close to that of Ho z click either of the two zeroes 0 of Ho z near the bottom of the Zeroes of GO and HO graph and switch the two zeroes to those of Go z Go z now has six zeroes and Ho z has eight zeroes Figure 5 10 shows the design result Notice that the analysis and synthesis scaling functions are similar and the analysis and synthesis wavelets also are similar which means the designed wavelet is near orthogonal The symmetry of the filter banks also preserves the linear phase property 5 10 ni com Chapter 5 Interactively Designing Discrete Wavelets i Configure Wavelet Design Wavelet Type Wavelet and Filter Banks O Orthogonal S Biorthogonal 12 Analysis scaling 6 Analysis wavelet Product of lowpass PO GO HO PO type Maxflat Positive Equiripple O General Equiripple ae 1 4 4 4 amp We i 2 3 4 5 6 1 Zero pairs at amp PD of taps Passband a Analysis lowpass GO 28 Analysis highpass G1 19 0
4. Transient features generally are not smooth and are of short duration Because wavelets are flexible in shape and have short time durations the wavelet signal processing methods can capture transient features precisely Figure 2 5 shows an ECG signal and the peaks detected with the wavelet transform based method This method locates the peaks of the ECG signal precisely Peaks ECG Signal Amplitude 1 1 1 1 1 1 1 1 1 1 1 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Time Figure 2 5 Peaks in the ECG Signal Wavelet Analysis Tools User Manual 2 4 ni com Chapter 2 Understanding Wavelet Signal Processing In the Browse tab of the NI Example Finder you can view this example by selecting Toolkits and Modules Wavelet Analysis A pplications ECG QRS Complex Detection VI Refer to the Finding Example VIs section of Chapter 1 Introduction to Wavelet Signal Processing for information about launching the NI Example Finder The LabVIEW Wavelet Analysis Tools provide many types of wavelets such as the Daubechies Haar and Coiflet wavelets Refer to Chapter 4 Signal Processing with Discrete Wavelets for information about these wavelets and applying them to the wavelet transforms Multiple Resolutions Signals usually contain both low frequency components and high frequency components Low frequency components vary slowly with time and require fine frequency resolution but coarse time resolution High frequency
5. 30 05 05 im ae me emm et cn Feri Wt es oh Gy u 2 X f f Arbitrary Minimum Phase Linear Phase O B Spline Synthesis scaling Synthesis wavelet 13 17 Zeroes at x GO 4 E Factorization Type of GO 0 1 0 0 2 1 1 it iot fue fen uei ed ent i L L L I D I LE OME2 S364 5IbEZSH twp vA eh Gh i sr y Be 2 03 Synthesis lowpass HO 08 Synthesis highpass H1 x 0 32888 y 0 0000 i 0 1 E nin imum 001213141516178 Zeroes of GO and HO G0 Blue Cross HO Red Circle o N 05 a Imaginary ND NM in Frequency response 0 o tn lt 50 S 907 1 1 D 0 0 2 0 4 0 6 Normalized frequency xs Figure 5 10 The Designed FBI Wavelet Because of the near orthogonality and linear phase properties of the FBI wavelet you can apply this wavelet to many kinds of signal and image processing for example image compression in JPEG2000 The FBI wavelet is called bior4_4 because both the analysis and synthesis lowpass filters Go z and H z have four zeroes at n This wavelet also is known as CDF 9 7 because the lengths of the analysis and synthesis highpass filters are nine and seven respectively National Instruments Corporation 5 11 Wavelet Analysis Tools User Manual Integer Wavelet Transform Many signal samples you encounter in real world applications are encoded as integers such as the signal amplitudes encoded by analog to digital A D convert
6. Representation Wavelets are localized in both the time and frequency domains because wavelets have limited time duration and frequency bandwidth The wavelet transform can represent a signal with a few coefficients because of the localization property of wavelets Figure 2 3 shows the waveform of the Doppler signal Doppler Signal Amplitude Figure 2 3 Waveform of the Doppler Signal National Instruments Corporation 2 3 Wavelet Analysis Tools User Manual Chapter 2 Understanding Wavelet Signal Processing Figure 2 4 shows the discrete wavelet transform DWT coefficients of the Doppler signal Refer to Chapter 4 Signal Processing with Discrete Wavelets for more information about the DWT DWT Coefficients a D Amplitude 1 1 1 1 1 1 1 1 1 1 1 1 50 100 150 200 250 300 350 400 450 S00 550 600 Time Figure 2 4 DWT Coefficients of the Doppler Signal In Figure 2 4 most of the DWT coefficients are zero which indicates that the wavelet transform is a useful method to represent signals sparsely and compactly Therefore you usually use the DWT in some signal compression applications Transient Feature Detection Transient features are sudden changes or discontinuities in a signal A transient feature can be generated by the impulsive action of a system and frequently implies a causal relationship to an event For example heartbeats generate peaks in an electrocardiogram ECG signal
7. Similarly D denotes the output of the filtering operations 000 1 in which the total number of 0 is L The impulse response of 000 1 converges asymptotically to the mother wavelet and the impulse response of 000 0 converges to the scaling function in the wavelet transform The DWT is invertible meaning that you can reconstruct the signal from the DWT coefficients with the inverse DWT The inverse DWT also is implemented with filter banks by cascading the synthesis filter banks Figure 4 3 shows the inverse DWT using filter banks D 72H Hz efta H Holz D 12 Hi Zz Reconstructed D Signal T72L H z _ xc s A A Li e e e 2TH Hoz Figure 4 3 Inverse Discrete Wavelet Transform Use the WA Discrete Wavelet Transform VI to compute the DWT of 1D and 2D signals Use the WA Inverse Discrete Wavelet Transform VI to compute the inverse DWT of 1D and 2D signals Refer to the LabVIEW Help available by selecting Help Search the LabVIEW Help for information about these VIs Wavelet Analysis Tools User Manual 4 4 ni com Chapter 4 Signal Processing with Discrete Wavelets Discrete Wavelet Transform for Multiresolution Analysis The DWT is well suited for multiresolution analysis The DWT decomposes high frequency components of a signal with fine time resolution but coarse frequency resolution and decomposes low freq
8. can find the examples using the NI Example Finder Select Help Find Example to launch the NI Example Finder You also can select the Examples or Find Examples options on the Getting Started window which appears when you launch LabVIEW to launch the NI Example Finder Related Signal Processing Tools In signal processing you usually categorize signals into two types stationary and nonstationary For stationary signals you assume that the spectral content of stationary signals does not change as a function of time space or some other independent variable For nonstationary signals you assume that the spectral content changes over time space or some other independent variable For example you might work under the assumption that an engine vibration signal is stationary when an engine is running at a constant speed and nonstationary when an engine is running up or down Nonstationary signals are categorized into two types according to how the spectral content changes over time evolutionary and transient The spectral contents of evolutionary signals change over time slowly Evolutionary signals usually contain time varying harmonics The time varying harmonics relate to the underlying periodic time varying characteristic of the system that generates signals Evolutionary signals also can contain time varying broadband spectral contents Transient signals are the short time events in a nonstationary signal such as peaks edges breakdo
9. coefficients of either the DWT or the UWT to detect the discontinuities in the HeaviSine signal by locating the peaks in the coefficients However if the HeaviSine signal is shifted by 21 samples all of the first level DWT detail coefficients become very small Therefore you cannot use the first level DWT detail coefficients to detect the discontinuities in the shifted HeaviSine signal Because of the translation invariant property of the UWT you can use the first level UWT detail coefficients to detect the discontinuities of the shifted HeaviSine Signal The first level UWT detail coefficients of the shifted HeaviSine Signal are simply the shifted version of the first level UWT detail coefficients of the original HeaviSine signal Better Denoising Capability Denoising with the UWT also is shift invariant The denoising result of the UWT has a better balance between smoothness and accuracy than the DWT The DWT based method is more computationally efficient than the UWT based method However you cannot achieve both smoothness and accuracy with the DWT based denoising method Use the Wavelet Denoise Express VI or the WA Denoise VI to reduce noise in 1D signals with both the UWT based and DWT based methods The UWT based method supports both real and complex signals The DWT based method supports only real signals You also can use the WA Denoise VI to reduce noise in 2D signals with the UWT based method Refer to the LabVIEW Help available b
10. enlarge the energy of noise suppressed in the wavelet domain However the filters associated with orthogonal wavelets are not linear phase filters Linear phase filters maintain a constant time delay for different frequencies and are necessary in many signal and image feature extraction applications such as peak detection and image edge detection Biorthogonal wavelets can be linear phase and are suitable for applications that require linear phase filters You also can use the Wavelet Design Express VI to design a customized wavelet Refer to Chapter 5 Interactively Designing Discrete Wavelets for information about wavelet design Discrete Wavelet Transform Unlike the discrete Fourier transform which is a discrete version of the Fourier transform the DWT is not really a discrete version of the continuous wavelet transform Instead the DWT is functionally different from the continuous wavelet transform CWT To implement the DWT you use discrete filter banks to compute discrete wavelet coefficients Two channel perfect reconstruction PR filter banks are a common and efficient way to implement the DWT Figure 4 1 shows a typical two channel PR filter bank system T G z r 2r o 1t2rF HG Reconstructed 5 Signal Signal p Er gt Goz Hl2t 4 t 412H Holz Figure 4 1 Two Channel Perfect Reconstruction Filter Banks The signal X z fi
11. information about STFT spectrograms Wavelet Analysis Tools User Manual 3 4 ni com Chapter 3 Signal Processing with Continuous Wavelets The CWT has the following general disadvantages e The CWT adds excess redundancy and is computationally intensive so you usually use this transform in offline analysis applications e The CWT does not provide the phase information of the analyzed signal For applications in which the phase information is useful use the AWT Refer to the Analytic Wavelet Transform section of this chapter for information about the AWT e You cannot reconstruct the original signal from the CWT coefficients For applications that require signal reconstruction use the discrete wavelet tools Refer to Chapter 4 Signal Processing with Discrete Wavelets for information about the discrete wavelet tools Application Example Breakdown Point Detection One useful CWT application is the detection of abrupt discontinuities or breakdown points in a signal Figure 3 4 shows an example that detects the breakdown points in a noise contaminated signal using the WA Continuous Wavelet Transform VI noise free w Signal noisy 1 200 Scalogram 20 10 8 10 0 1 D NI 200 400 600 800 1024 Time CWT Coefficients Cumulation breakdown locations DNI cum Figure 3 4 Breakdown Points in the Noise Contaminated HeaviSine Signal National Instruments Corporation 3 5 Wavelet A
12. is to ensure smoothness higher order and linear phase first and then pursue near orthogonality Using the Wavelet Design Express VI you can design a wavelet with specific properties For example you can complete the following steps to design the FBI wavelet which is linear phase and near orthogonal 1 National Instruments Corporation Place the Wavelet Design Express VI on the block diagram The Configure Wavelet Design dialog box as shown in Figure 5 1 automatically launches Select Biorthogonal as the Wavelet Type because only biorthogonal wavelets have the linear phase property In the Product of lowpass G0 H0 section select Maxflat as the PO type and set the value of Zero pairs at n PO to 4 When you set parameters on the left hand side of the configuration dialog box plots of the designed wavelet and the associated filter banks interactively appear on the right hand side 5 9 Wavelet Analysis Tools User Manual Chapter 5 Interactively Designing Discrete Wavelets In the Factorization Type of G0 section select Linear Phase as the Filter type and set the value of Zeroes at x GO to 4 because the wavelet must be near orthogonal meaning that Go z and Ho z have the same or almost the same amount of zeroes By setting the value of Zeroes at T GO to 4 you can ensure that both Go z and Ho z have the same amount of zeroes at 7 Figure 5 9 shows the zeroes of Go z and Ho z Zeroes of GO and HO
13. large coefficients only around discontinuities So the wavelet transform is a useful tool to convert signals to sparse representations In the NI Example Finder refer to the ECG Compression VI for more information about performing wavelet transform based compression on electrocardiogram ECG signals Extracting relevant features is a key step when you analyze and interpret signals and images Signals and images are characterized by local features such as peaks edges and breakdown points The wavelet transform based methods are typically useful when the target features consist of rapid changes such as the sound caused by engine knocking Wavelet signal processing is suitable for extracting the local features of signals because wavelets are localized in both the time and frequency domains Figure 1 3 shows an image and the associated edge maps detected at different levels of resolutions using the wavelet transform based method Conventional methods process an image at a single resolution and return a binary edge map The wavelet transform based method processes an image at multiple levels of resolution and returns a series of grey level edge maps at different resolutions Edge Level 1 Edge Level 2 Figure 1 3 Image Edge Detection Wavelet Analysis Tools User Manual 1 4 ni com Chapter 1 Introduction to Wavelet Signal Processing A large level value corresponds to an edge map wi
14. to an integer representation before entropy based encoding As a result compression with the DWT is lossy meaning that some information is lost when you compress a signal using the DWT and that you typically cannot reconstruct the original signal perfectly from the coefficients of the DWT National Instruments Corporation 6 1 Wavelet Analysis Tools User Manual Chapter 6 Integer Wavelet Transform The IWT however provides lossless compression You can use the IWT to convert integer signal samples into integer wavelet coefficients and you can compress these integer coefficients by entropy based encoding without further quantization As a result you can reconstruct the original signal perfectly from a compressed set of IWT coefficients Figures 6 1 shows an example of lossless compression with the IWT Original Image Reconstructed Image Maximum Difference 0 IWT Coefficients Histogram Original Image 1 IWT Coefficients Ok 5k Number um 1 1 T 66 0 200 400 600 687 Grey Level Se Figure 6 1 Lossless Image Compression In the Histogram graph most of the elements in the IWT Coefficients plot are zero meaning that you can obtain a high compression ratio using the IWT of this image You can reconstruct the image perfectly with the inverse IWT as shown in the Reconstructed Image graph The Maximum Difference value of 0 indicates that the reconstructed image retains all the inform
15. wavelet tools to perform wavelet transforms on signals that are defined in continuous time Unlike discrete wavelet tools which operate on sampled data signals continuous wavelet tools operate on signals that are defined for all time over a time region of interest though the computations are done numerically in discrete time The LabVIEW Wavelet Analysis Tools provide two continuous wavelet tools the continuous wavelet transform CWT and the analytic wavelet transform AWT The AWT retains both the magnitude and phase information of signals in the time scale or time frequency domain whereas the CWT retains only the magnitude information The CWT is simpler because the results of the CWT are real values if both the wavelet and the signal are real The results of the AWT normally are complex values From a mathematical point of view both the CWT and AWT add informational redundancy because the number of the resulting wavelet coefficients in the time scale or time frequency domain is larger than the number of time samples in the original signal Excess redundancy generally is not desirable because more computations and more memory are required to process signals with excess redundancy However excess redundancy can be helpful for some applications such as singularity and cusp extraction time frequency analysis of nonstationary signals and self similarity analysis of fractal signals This chapter explains both the CWT and the AWT in detail
16. with Discrete Wavelets Figure 4 5 shows the multiresolution results for a signal using the DWT Signal Approximation Level 1 Detail Level 1 Mire AVA Approximation Level 2 Detail Level 2 Amplitude 45 onm mw I i 1 1 Figure 4 5 DWT Based Multiresolution Analysis You can see that the approximation at level 1 is the summation of the approximation and detail at level 2 The approximation at level 2 is the summation of the approximation and detail at level 3 As the level increases you obtain lower frequency components or large scale approximation and detail of the signal In the Browse tab of the NI Example Finder you can view a multiresolution analysis example by selecting Toolkits and Modules Wavelet Analysis Getting Started Multiresolution Analysis VI Refer to the Finding Example VIs section of Chapter 1 Introduction to Wavelet Signal Processing for information about launching the NI Example Finder Wavelet Analysis Tools User Manual 4 6 ni com Chapter 4 Signal Processing with Discrete Wavelets Use the Multiresolution Analysis Express VI to decompose and reconstruct a signal at different levels and with different wavelet types Refer to the LabVIEW Help available by selecting Help Search the LabVIEW Help for information about this Express VI 2D Signal Processing The preceding sections introduce the DWT in 1D signal processing Using the Wavelet Analysis Tools
17. you can extend the DWT to 2D signal processing Figure 4 6 shows the PR filter bank implementation of the 2D DWT which applies the filter banks to both rows and columns of an image Rows Columns pA G z HL 2 H gt high high gt ae Het Lp Gz 112 high ow gt A Gu H4 2 low high Lp Gyz Hi2 Go z 2 gt low low Figure 4 6 2D Discrete Wavelet Transform As Figure 4 6 shows when decomposing 2D signals with two channel PR filter banks you process rows first and then columns Consequently one 2D array splits into the following four 2D arrays e low low e low high e high low e high high Each array is one fourth of the size of the original 2D array National Instruments Corporation 4 7 Wavelet Analysis Tools User Manual Chapter 4 Signal Processing with Discrete Wavelets Figure 4 7 shows an example of decomposing and reconstructing an image file with the 2D DWT and the inverse 2D DWT Image Reconstructed Image Figure 4 7 Example of 2D Discrete Wavelet Transform The source image is decomposed into the following four sub images e ow low Shows an approximation of the source signal with coarse resolution e ow high Shows the details at the discontinuities along the column direction high low Shows the details at the discontinuities along the
18. 00 Time Time Scale 2 Shift 50 2 Amplitude 1 1 1 1 50 100 150 200 250 300 350 400 450 500 550 Time 1 1 1 1 1 1 1 50 100 150 200 250 300 350 400 450 500 550 Time Figure 2 2 Dilations and Translations of the db02 Wavelet Wavelet Analysis Tools User Manual 2 2 ni com Chapter 2 Understanding Wavelet Signal Processing The wavelet transform computes the inner products of a signal with a family of wavelets The wavelet transform tools are categorized into continuous wavelet tools and discrete wavelet tools Usually you use the continuous wavelet tools for signal analysis such as self similarity analysis and time frequency analysis You use the discrete wavelet tools for both signal analysis and signal processing such as noise reduction data compression peak detection and so on Refer to Chapter 3 Signal Processing with Continuous Wavelets for information about the continuous wavelet tools Refer to Chapter 4 Signal Processing with Discrete Wavelets for information about the discrete wavelet tools Benefits of Wavelet Signal Processing Wavelet signal processing is different from other signal processing methods because of the unique properties of wavelets For example wavelets are irregular in shape and finite in length Wavelet signal processing can represent signals sparsely capture the transient features of signals and enable signal analysis at multiple resolutions Sparse
19. 008 National Instruments Corporation All rights reserved Important Information Warranty The media on which you receive National Instruments software are warranted not to fail to execute programming instructions due to defects in materials and workmanship for a period of 90 days from date of shipment as evidenced by receipts or other documentation National Instruments will at its option repair or replace software media that do not execute programming instructions if National Instruments receives notice of such defects during the warranty period National Instruments does not warrant that the operation of the software shall be uninterrupted or error free A Return Material Authorization RMA number must be obtained from the factory and clearly marked on the outside of the package before any equipment will be accepted for warranty work National Instruments will pay the shipping costs of returning to the owner parts which are covered by warranty National Instruments believes that the information in this document is accurate The document has been carefully reviewed for technical accuracy In the event that technical or typographical errors exist National Instruments reserves the right to make changes to subsequent editions of this document without prior notice to holders of this edition The reader should consult National Instruments if errors are suspected In no event shall National Instruments be liable for any damages arising out of or re
20. Analysis Tools provide the following discrete wavelet tools e Discrete wavelet transform DWT e Wavelet packet decomposition and arbitrary path decomposition e Undecimated wavelet transform UWT You can use the discrete wavelet tools to perform signal analysis and signal processing including multiresolution analysis denoising compression edge detection peak detection and others This chapter introduces the commonly used discrete wavelets and describes discrete filter banks that you use to implement the wavelet transforms This chapter also explains each discrete wavelet tool in detail and provides application examples Selecting an Appropriate Discrete Wavelet The Wavelet Analysis Tools provide the following commonly used discrete wavelets e Orthogonal wavelets Haar Daubechies dbxx Coiflets coifx and Symmlets symx Biorthogonal wavelets FBI and Biorthogonal biorx x x indicates the order of the wavelet The higher the order the smoother the wavelet National Instruments Corporation 4 1 Wavelet Analysis Tools User Manual Chapter 4 Signal Processing with Discrete Wavelets Orthogonal wavelets are suitable for applications such as signal and image compression and denoising because the wavelet transform with orthogonal wavelets possesses the same amount of energy as that contained in the original data samples The energy conservative property ensures that the inverse wavelet transform does not
21. LabVIEW Advanced Signal Processing Toolkit Wavelet Analysis Tools User Manual June 2008 lt 7 NATIONAL 371533B 01 ANiNsrRUMENTS Worldwide Technical Support and Product Information ni com National Instruments Corporate Headquarters 11500 North Mopac Expressway Austin Texas 78759 3504 USA Tel 512 683 0100 Worldwide Offices Australia 1800 300 800 Austria 43 662 457990 0 Belgium 32 0 2 757 0020 Brazil 55 11 3262 3599 Canada 800 433 3488 China 86 21 5050 9800 Czech Republic 420 224 235 774 Denmark 45 45 76 26 00 Finland 358 0 9 725 72511 France 01 57 66 24 24 Germany 49 89 7413130 India 91 80 41190000 Israel 972 3 6393737 Italy 39 02 41309277 Japan 0120 527196 Korea 82 02 3451 3400 Lebanon 961 0 1 33 28 28 Malaysia 1800 887710 Mexico 01 800 010 0793 Netherlands 31 0 348 433 466 New Zealand 0800 553 322 Norway 47 0 66 90 76 60 Poland 48 22 3390150 Portugal 351 210 311 210 Russia 7 495 783 6851 Singapore 1800 226 5886 Slovenia 386 3 425 42 00 South Africa 27 0 11 805 8197 Spain 34 91 640 0085 Sweden 46 0 8 587 895 00 Switzerland 41 56 2005151 Taiwan 886 02 2377 2222 Thailand 662 278 6777 Turkey 90 212 279 3031 United Kingdom 44 0 1635 523545 For further support information refer to the Technical Support and Professional Services appendix To comment on National Instruments documentation refer to the National Instruments Web site at ni com info and enter the info code feedback 2005 2
22. RE NOT DESIGNED WITH COMPONENTS AND TESTING FOR A LEVEL OF RELIABILITY SUITABLE FOR USE IN OR IN CONNECTION WITH SURGICAL IMPLANTS OR AS CRITICAL COMPONENTS IN ANY LIFE SUPPORT SYSTEMS WHOSE FAILURE TO PERFORM CAN REASONABLY BE EXPECTED TO CAUSE SIGNIFICANT INJURY TO A HUMAN 2 IN ANY APPLICATION INCLUDING THE ABOVE RELIABILITY OF OPERATION OF THE SOFTWARE PRODUCTS CAN BE IMPAIRED BY ADVERSE FACTORS INCLUDING BUT NOT LIMITED TO FLUCTUATIONS IN ELECTRICAL POWER SUPPLY COMPUTER HARDWARE MALFUNCTIONS COMPUTER OPERATING SYSTEM SOFTWARE FITNESS FITNESS OF COMPILERS AND DEVELOPMENT SOFTWARE USED TO DEVELOP AN APPLICATION INSTALLATION ERRORS SOFTWARE AND HARDWARE COMPATIBILITY PROBLEMS MALFUNCTIONS OR FAILURES OF ELECTRONIC MONITORING OR CONTROL DEVICES TRANSIENT FAILURES OF ELECTRONIC SYSTEMS HARDWARE AND OR SOFTWARE UNANTICIPATED USES OR MISUSES OR ERRORS ON THE PART OF THE USER OR APPLICATIONS DESIGNER ADVERSE FACTORS SUCH AS THESE ARE HEREAFTER COLLECTIVELY TERMED SYSTEM FAILURES ANY APPLICATION WHERE A SYSTEM FAILURE WOULD CREATE A RISK OF HARM TO PROPERTY OR PERSONS INCLUDING THE RISK OF BODILY INJURY AND DEATH SHOULD NOT BE RELIANT SOLELY UPON ONE FORM OF ELECTRONIC SYSTEM DUE TO THE RISK OF SYSTEM FAILURE TO AVOID DAMAGE INJURY OR DEATH THE USER OR APPLICATION DESIGNER MUST TAKE REASONABLY PRUDENT STEPS TO PROTECT AGAINST SYSTEM FAILURES INCLUDING BUT NOT LIMITED TO BACK UP OR SHUT DOWN MECHANISMS BECAUSE EACH END USER SYSTEM I
23. S CUSTOMIZED AND DIFFERS FROM NATIONAL INSTRUMENTS TESTING PLATFORMS AND BECAUSE A USER OR APPLICATION DESIGNER MAY USE NATIONAL INSTRUMENTS PRODUCTS IN COMBINATION WITH OTHER PRODUCTS IN A MANNER NOT EVALUATED OR CONTEMPLATED BY NATIONAL INSTRUMENTS THE USER OR APPLICATION DESIGNER IS ULTIMATELY RESPONSIBLE FOR VERIFYING AND VALIDATING THE SUITABILITY OF NATIONAL INSTRUMENTS PRODUCTS WHENEVER NATIONAL INSTRUMENTS PRODUCTS ARE INCORPORATED IN A SYSTEM OR APPLICATION INCLUDING WITHOUT LIMITATION THE APPROPRIATE DESIGN PROCESS AND SAFETY LEVEL OF SUCH SYSTEM OR APPLICATION Contents About This Manual Gonventions te e E OU UE EE EU E Ri MM UE UN EE vii Related Documentation iate cere ors Hon eee eie tip e Nee viii Chapter 1 Introduction to Wavelet Signal Processing Wavelet Signal Processing Application Areas sese 1 1 Multiscale Analysis estamos iet R S 1 2 Noise Reduction eoa e ER tte er RH E te ripe 1 3 Compression dace ca cg E UP e PP eeu TP tego 1 3 Feature EXtractioni o e eet edi e e HERD eL P eng 1 4 Overview of LabVIEW Wavelet Analysis Tools esee 1 5 Finding Example Vets eee Ra t E EEEE RR ERE Re 1 6 Related Signal Processing Tools eese 1 6 Chapter 2 Understanding Wavelet Signal Processing Wavelet and Wavelet Transform esses eene nne 2 1 Benefits of Wavelet Signal Processing eene rennes 2 3 Sparse Representation 4 i
24. Wavelet Design Express VI automatically generates the zeroes for Ho z and Go z You cannot switch the zeroes between Go z and Ho z The minimum phase filter possesses minimum phase lag When Po z is maximally flat and Go z is minimum phase the resulting wavelets are the Daubechies wavelets Zeroes of GO and HO G0 Blue Cross HO Red Circle HoA p Daubechies x 3 0407 y 0 0000 225 g m2 o iS 1 05 fe x 01 n x 1 P 1 1 1 D 1 2 332 Figure 5 6 Minimum Phase Filter National Instruments Corporation 5 7 Wavelet Analysis Tools User Manual Chapter 5 Interactively Designing Discrete Wavelets Wavelet Analysis Tools User Manual Linear Phase Any zero and its reciprocal must belong to the same filter as shown in Figure 5 7 When you switch a zero of Go z to that of H z the reciprocal of the zero also switches to Ho z When you switch a zero of Ho z to that of Go z the reciprocal of the zero also switches to Go z This option is available only if the filter is biorthogonal Zeroes of GO and HO G0 Blue Cross HO Red Circle wi 2 x 3 0407 y 0 0000 w N e Imaginary N N in hota te oc B x Figure 5 7 Linear Phase Filter In the time domain a linear phase implies that the coefficients of the filter are symmetric or antisymmetric Linear phase filters have a constant group delay for all frequencies This property is required in many signal and image f
25. al world applications require that you use a uniform amplitude response to measure the exact amplitude of the signal components as shown in Figure 3 11 Uu K Uu Uu Uu 1 D 0 00 0 01 002 0 03 0 04 0 05 0 06 0 07 0 08 0 09 0 10 Time Figure 3 11 Magnitude Spectra of Wavelets with Amplitude Normalization With the WA Analytic Wavelet Transform VI you can analyze a signal based on amplitude normalization by selecting amplitude in the normalization list If you set scale sampling method to even freq and set normalization to amplitude the WA Analytic Wavelet Transform VI generates the scalogram of the HypChirps signal as shown in Figure 3 12 Scalogram Figure 3 12 Scalogram with Amplitude Normalization Notice that the magnitude at high frequencies small scales also has been enlarged With amplitude normalization you can obtain the precise magnitude evolution over time for each hyperbolic chirp National Instruments Corporation 3 11 Wavelet Analysis Tools User Manual Signal Processing with Discrete Wavelets Although you can use numerical algorithms to compute continuous wavelet coefficients as introduced in Chapter 3 Signal Processing with Continuous Wavelets to analyze a signal the resulting wavelet coefficients are not invertible You cannot use those wavelet coefficients to recover the original data samples For applications that require signal reconstruction the LabVIEW Wavelet
26. analysis methods including the linear discrete Gabor transform and expansion the linear adaptive transform and expansion the quadratic Gabor spectrogram and the quadratic adaptive spectrogram The Time Frequency Analysis Tools also include VIs to extract features from a signal such as the mean instantaneous frequency the mean instantaneous bandwidth the group delay and the marginal integration For both evolutionary signals and transient signals use the Wavelet Analysis Tools Refer to the Overview of LabVIEW Wavelet Analysis Tools section of this chapter for information about the Wavelet Analysis Tools 1 7 Wavelet Analysis Tools User Manual Understanding Wavelet Signal Processing This chapter introduces wavelets and the wavelet transform and describes the benefits of wavelet signal processing in detail Wavelet and Wavelet Transform Just as the Fourier transform decomposes a signal into a family of complex sinusoids the wavelet transform decomposes a signal into a family of wavelets Unlike sinusoids which are symmetric smooth and regular wavelets can be either symmetric or asymmetric sharp or smooth regular or irregular Figure 2 1 shows a sine wave the db02 wavelet and the FBI wavelet Sine Wave Amplitude 1 1 1 300 400 500 Time 1 Uu u Uu 1 100 125 150 175 200 Time FBI Wavelet Amplitude 1 1 200 300 400 500 Time Figure 2 1 Sine Wave versus Wavelets N
27. and provides an application example that uses the CWT National Instruments Corporation 3 1 Wavelet Analysis Tools User Manual Chapter 3 Signal Processing with Continuous Wavelets Continuous Wavelet Transform Mathematically the CWT computes the inner products of a continuous signal with a set of continuous wavelets according to the following equation oo WT a SM 9 sw coat oo where Vua T v Ez9 WT a is the resulting wavelet coefficients Y a denotes a continuous wavelet where u is the shift factor and a is the scale factor of the wavelet V a is the complex conjugate of y a For the continuous time signal s t the scale factor must be a positive real number whereas the shift factor can be any real number If the continuous wavelet y a meets the admissibility condition you can use the computed wavelet coefficients to reconstruct the original signal s t However you seldom use the above integration to compute the CWT because of the following reasons e The majority of real world signals that you encounter are available as discrete time samples The analytical form of the signal s t usually is not accessible e The closed form solution of the integration does not exist except for very special cases For these reasons you usually select a set of discrete values for the scales and shifts of the continuous wavelets and then compute the CWT numerically Use the WA Continuous Wavelet Trans
28. and the path 11 to the decomposition of the signal that the Engine Knocking Sound graph contains The Enhanced Sound graph shows the signal reconstructed from the path 11 The high amplitude components around 0 6 0 8 and 1 0 in the Enhanced Sound graph indicate where the ignition malfunction of the engine occurs In the Browse tab of the NI Example Finder you can view this example by selecting Toolkits and Modules Wavelet Analysis A pplications Engine Knocking Detection VI Refer to the Finding Example VIs section of Chapter 1 Introduction to Wavelet Signal Processing for information about launching the NI Example Finder Undecimated Wavelet Transform Unlike the DWT which downsamples the approximation coefficients and detail coefficients at each decomposition level the UWT does not incorporate the downsampling operations Thus the approximation coefficients and detail coefficients at each level are the same length as the original signal The UWT upsamples the coefficients of the lowpass and highpass filters at each level The upsampling operation is equivalent to dilating wavelets The resolution of the UWT coefficients decreases with increasing levels of decomposition National Instruments Corporation 4 13 Wavelet Analysis Tools User Manual Chapter 4 Signal Processing with Discrete Wavelets Use the WA Undecimated Wavelet Transform VI and the WA Inverse Undecimated Wavelet Transform VI to decompose and reconstruct 1D o
29. any form electronic or mechanical including photocopying recording storing in an information retrieval system or translating in whole or in part without the prior written consent of National Instruments Corporation National Instruments respects the intellectual property of others and we ask our users to do the same NI software is protected by copyright and other intellectual property laws Where NI software may be used to reproduce software or other materials belonging to others you may use NI software only to reproduce materials that you may reproduce in accordance with the terms of any applicable license or other legal restriction Trademarks National Instruments NI ni com and LabVIEW are trademarks of National Instruments Corporation Refer to the Terms of Use section on ni com 1legal for more information about National Instruments trademarks Other product and company names mentioned herein are trademarks or trade names of their respective companies Members of the National Instruments Alliance Partner Program are business entities independent from National Instruments and have no agency partnership or joint venture relationship with National Instruments Patents For patents covering National Instruments products refer to the appropriate location Help Patents in your software the patents txt file on your media or ni com patents WARNING REGARDING USE OF NATIONAL INSTRUMENTS PRODUCTS 1 NATIONAL INSTRUMENTS PRODUCTS A
30. ation Type for Po Z essent 5 6 Example Designing the FBI Wavelet eeeseeeeeseeeeeeeeennee nennen 5 9 Chapter 6 Integer Wavelet Transform Application Example Lossless Compression eene 6 1 Appendix A Technical Support and Professional Services Wavelet Analysis Tools User Manual vi ni com About This Manual Conventions This manual provides information about the wavelet analysis tools in the LabVIEW Advanced Signal Processing Toolkit including different types of methods that you can use to perform wavelet signal processing theoretical basis for each type of method and application examples based on the wavelet transform based methods bold italic monospace The following conventions appear in this manual The symbol leads you through nested menu items and dialog box options to a final action The sequence File Page Setup Options directs you to pull down the File menu select the Page Setup item and select Options from the last dialog box This icon denotes a note which alerts you to important information Bold text denotes items that you must select or click in the software such as menu items and dialog box options Bold text also denotes parameter names Italic text denotes variables emphasis a cross reference or an introduction to a key concept Italic text also denotes text that is a placeholder for a word or value that you must supply Te
31. ation of the original image In the Browse tab of the NI Example Finder you can view this example by selecting Toolkits and Modules Wavelet Analysis Applications Lossless Medical Image Compression VI Refer to the Finding Example VIs section of Chapter 1 Introduction to Wavelet Signal Processing for information about launching the NI Example Finder Wavelet Analysis Tools User Manual 6 2 ni com Technical Support and Professional Services Visit the following sections of the award winning National Instruments Web site at ni com for technical support and professional services National Instruments Corporation Support Technical support resources at ni com support include the following Self Help Technical Resources For answers and solutions visit ni com support for software drivers and updates a searchable KnowledgeBase product manuals step by step troubleshooting wizards thousands of example programs tutorials application notes instrument drivers and so on Registered users also receive access to the NI Discussion Forums at ni com forums NI Applications Engineers make sure every question submitted online receives an answer Standard Service Program Membership This program entitles members to direct access to NI Applications Engineers via phone and email for one to one technical support as well as exclusive access to on demand training modules via the Services Resource Center NI offers complementary
32. ational Instruments Corporation 2 1 Wavelet Analysis Tools User Manual Chapter 2 Understanding Wavelet Signal Processing In Figure 2 1 you can see that the Sine Wave is symmetric smooth and regular The db02 Wavelet is asymmetric sharp and irregular The FBI Wavelet is symmetric smooth and regular You also can see that a sine wave has an infinite length whereas a wavelet has a finite length For different types of signals you can select different types of wavelets that best match the features of the signal you want to analyze Therefore you can perform wavelet signal processing and generate reliable results about the underlying information of a signal The family of wavelets contains the dilated and translated versions of a prototype function Traditionally the prototype function is called a mother wavelet The scale and shift of wavelets determine how the mother wavelet dilates and translates along the time or space axis A scale factor greater than one corresponds to a dilation of the mother wavelet along the horizontal axis and a positive shift corresponds to a translation to the right of the scaled wavelet along the horizontal axis Figure 2 2 shows the db02 mother wavelet and the associated dilated and translated wavelets with different scale factors and shift values Mother Wavelet Scale 2 Shift 0 mc Amplitude Amplitude 1 D D 1 D 1 D D D D 1 50 100 150 200 250 300 350 400 450 500 550 1 100 150 2
33. components vary quickly with time and require fine time resolution but coarse frequency resolution You need to use a multiresolution analysis MRA method to analyze a signal that contains both low and high frequency components Wavelet signal processing is naturally an MRA method because of the dilation process Figure 2 6 shows the wavelets with different dilations and their corresponding power spectra Wavelets Power Spectra of Wavelets le n 1 1 1 1 1 1 1 1 1 1 1 0 50 100 150 200 250 300 350 400 450 500 a 64 u 300 i edd f a 32 u 200 Ms r PIS H J a 16 u 100 Time Frequency National Instruments Corporation 2 5 Figure 2 6 Wavelets and the Corresponding Power Spectra The Wavelets graph contains three wavelets with different scales and translations The Power Spectra of Wavelets graph shows the power spectra of the three wavelets where a and u represent the scale and shift of the wavelets respectively Figure 2 6 shows that a wavelet with a small scale has a short time duration a wide frequency bandwidth and a high Wavelet Analysis Tools User Manual Chapter 2 Understanding Wavelet Signal Processing central frequency This figure also shows that a wavelet with a large scale has a long time duration a narrow frequency bandwidth and a low central frequency The time duration and frequency bandwidth determine the time and frequency resolutions of a wavelet respect
34. eature extraction applications such as peak detection and image edge detection B Spline This option is available only if Wavelet Type is Biorthogonal and P0 type is Maxflat In this case the analysis lowpass filter Go z and the synthesis lowpass filter H z are defined by the following equations respectively ak 12p k Gg Z 1 2 Hg Z 1 2 Q z where k is specified with the Zeroes at n GO control and p is determined by the Zero pairs at x P0 control The Wavelet Design Express VI automatically generates the zeroes of Go z and Ho z based on the settings for k and p You cannot switch the zeroes between Go z and Ho z Figure 5 8 shows an example of B Spline factorization 5 8 ni com Chapter 5 Interactively Designing Discrete Wavelets Zeroes of GO and HO G0 Blue Cross HO Red Circle eR x 3 0407 y 0 0000 28 25 Imaginary A ENIMS en o eo en o e oo Figure 5 8 B Spline Filter Refer to LabVIEW Help available by selecting Help Search the LabVIEW Help for information about the Wavelet Design Express VI Example Designing the FBI Wavelet Different signal processing applications require different properties of wavelets For image compression you need a wavelet that is smooth linear phase and orthogonal However as discussed in the Selecting the Wavelet Type section of this chapter you cannot achieve all those properties simultaneously One thing you can do
35. er Go z and a synthesis lowpass filter Ho z the Wavelet Design Express VI automatically generates the corresponding analysis highpass filter G z and synthesis highpass filter H 1 z The following sections describe each of the steps in the wavelet design process and the controls you use to complete the steps You also can select Help Show Context Help or press the lt Ctrl H gt keys for more information about controls and indicators on the Configure Wavelet Design dialog box Selecting the Wavelet Type Use the Wavelet Type control on the Configure Wavelet Design dialog box to select the wavelet type You can choose from the following two wavelet types Orthogonal default and Biorthogonal The wavelet transform with orthogonal wavelets is energy conserving meaning that the total energy contained in the resulting coefficients and the energy in the original time samples are the same This property is helpful for signal and image compression and denoising But the filters associated with orthogonal wavelets are not linear phase Linear phase is a helpful property for feature extraction applications The filters associated with biorthogonal wavelets can be linear phase Designing the Product P z The auxiliary function Po z denotes the product of Go z and Ho z as shown in the following equation Po Z Gg Z Hg Z You usually use one of the following three types of filters for Po z e Maximally flat e General equi
36. ers and color intensities of pixels encoded in digital images For integer encoded signals an integer wavelet transform IWT can be particularly efficient The IWT is an invertible integer to integer wavelet analysis algorithm You can use the IWT in the applications that you want to produce integer coefficients for integer encoded signals Compared with the continuous wavelet transform CW T and the discrete wavelet transform DWT the IWT is not only computationally faster and more memory efficient but also more suitable in lossless data compression applications The IWT enables you to reconstruct an integer signal perfectly from the computed integer coefficients Use the WA Integer Wavelet Transform VI which implements the IWT with the lifting scheme to decompose an integer signal or image Use the WA Inverse Integer Wavelet Transform VI which implements the inverse IWT with the inverse lifting scheme to reconstruct an integer signal or image from the IWT coefficients Refer to the LabVIEW Help available by selecting Help Search the LabVIEW Help for information about these VIs This chapter describes an application example that uses the IWT to compress an image file Application Example Lossless Compression When you apply the DWT to integer signal samples you convert the original integer signal samples to floating point wavelet coefficients In signal compression applications you typically further quantize these coefficients
37. ex Detection VI e Toolkits and Modules Wavelet Analysis Getting Started Peak Detection Wavelet vs Normal VI Refer to the Finding Example VIs section of Chapter 1 Introduction to Wavelet Signal Processing for information about launching the NI Example Finder Wavelet Analysis Tools User Manual 4 18 ni com Interactively Designing Discrete Wavelets Both the discrete wavelet transform and the inverse discrete wavelet transform are implemented using a set of cascaded two channel perfect reconstruction PR filter banks Refer to Chapter 4 Signal Processing with Discrete Wavelets for more information about discrete wavelets and filter banks The WA Wavelet Filter VI already contains a collection of predefined wavelets including orthogonal wavelets Haar Daubechies Coiflets Symmlets and biorthogonal wavelets FBI Biorthogonal You can apply the predefined wavelets directly to signal processing applications If you cannot find a wavelet that best matches the signal you can use the Wavelet Design Express VI to design a customized discrete wavelet The design of discrete wavelets is essentially the design of two channel PR filter banks This chapter describes the steps that you can follow when using the Wavelet Design Express VI to design discrete wavelets and provides an example of designing the FBI wavelet National Instruments Corporation 5 1 Wavelet Analysis Tools User Manual Chapter 5 Interactively Designing D
38. form VI to compute the CWT by specifying a set of integer values or arbitrary real positive values for the scales and a set of equal increment values for the shifts Refer to the LabVIEW Help available by selecting Help Search the LabVIEW Help for information about this VI Shie Qian Introduction to Time Frequency and Wavelet Transforms Upper Saddle River New Jersey Prentice Hall PTR 2001 Wavelet Analysis Tools User Manual 3 2 ni com Chapter 3 Signal Processing with Continuous Wavelets Figure 3 1 shows the procedure that the WA Continuous Wavelet Transform VI follows Kei vH AR He Signal Amplitude 1 1 1 Time Shifting Wavelets Scale 1 Dilating Figure 3 1 Procedure of the Continuous Wavelet Transform The procedure involves the following steps 1 Shifts a specified wavelet continuously along the time axis 2 Computes the inner product of each shifted wavelet and the analyzed signal Dilates the wavelet based on the scale you specify 4 Repeats steps 1 through 3 till the process reaches the maximum scale you specify The output of the CWT is the CWT coefficients which reflect the similarity between the analyzed signal and the wavelets You also can compute the squares of the CWT coefficients and form a scalogram which is analogous to the spectrogram in time frequency analysis In signal processing scalograms are useful in pattern matching applications and discon
39. he P type control to specify the Po z type When Wavelet Type is set to Orthogonal you can set Po z either to Maxflat default for a maximally flat filter or to Positive Equiripple When Wavelet Type is set to Biorthogonal you can set Po z to Maxflat default Positive Equiripple or General Equiripple Because all filters including Po z Go z and Ho z are real valued finite impulse response FIR filters the zeroes of these filters are mirror symmetric about the x axis in the z plane Therefore for any zero Zi a corresponding complex conjugate z always exists If z is complex meaning that if z is located off of the x axis you always can find a corresponding zero on the other side of the x axis as shown in Figure 5 4 National Instruments Corporation 5 5 Wavelet Analysis Tools User Manual Chapter 5 Interactively Designing Discrete Wavelets As a result you only need to see the top half of the z plane to see all of the zeroes that are present After you select z the Wavelet Design Express VI automatically includes the complex conjugate z zl Figure 5 4 Zero Distribution of Real Valued FIR Filters Two parameters are associated with equiripple filters of taps and Passband Use the ft of taps control to define the number of coefficients of Po z Because Po z is a type I FIR filter the length of Po z must be odd Use the Passband control to define the normalized passband freque
40. ing is denoising or reducing noise in a signal The wavelet transform based method can produce much higher denoising quality than conventional methods Furthermore the wavelet transform based method retains the details of a signal after denoising Figure 1 2 shows a signal with noise and the denoised signal using the wavelet transform based method Noisy Signal Amplitude aa Uu J ACN k ENT I 4 U WAAN AH V VV in net i Lea teu Figure 1 2 Noise Reduction With the wavelet transform you can reduce the noise in the signal in the Noisy Signal graph The resulting signal in the Denoised Signal graph contains less noise and retains the details of the original signal In the NI Example Finder refer to the Noise Reduction VI for more information about performing wavelet transform based denoising on signals In many applications storage and transmission resources limit performance Thus data compression has become an important topic in information theory Usually you can achieve compression by converting a source signal into a sparse representation which includes a small number National Instruments Corporation 1 3 Wavelet Analysis Tools User Manual Chapter 1 Feature Extraction Introduction to Wavelet Signal Processing of nonzero values and then encoding the sparse representation with a low bit rate The wavelet transform as a time scale representation method generates
41. iscrete Wavelets Figure 5 1 shows the Configure Wavelet Design dialog box of the Wavelet Design Express VI I Configure Wavelet Design Wavelet Type Orthogonal Biorthogonal Product of lowpass PO GO HO PO type Maxflat Zero pairs atn PO 4 of taps Pas 2 gl 19 General E quiripple Positive Equiripple Factorization Type of GO Filter type OArbitrary Minimum Phase B Spline Zeroes at GO Zeroes of GO and HO G0 Blue Cross HO Red Circle A 2 x 0 0000 y 0 0000 3 9 Imaginary w nN Wavelet and Filter Banks 14 0 4 0 8 Analysis scaling Analysis wavelet uv B amp Analysis lowpass GO 0 1 2 1 E Analysis highpass G1 4 Synthesis scaling 0 1 E 3 Synthesis lowpass H0 Frequency response T L g 907 1 90 7 1 1 0 0 2 0 4 1 0 6 Normalized frequency x i 1 Figure 5 1 Configure Wavelet Design Dialog Box On the left hand side of the configuration dialog box you can specify attributes of the wavelet that you want to design On the right hand side of the configuration dialog box you can see the real time plots of the designed wavelet Refer to the LabVIEW Help available by selecting Help Search the LabVIEW Help for more information about the Wavelet Design Express VI Wavelet Analysis Tools User Manual 5 2 ni com Chapter 5 Figure 5 2 shows the wavelet design process Biorthog
42. ively A long time duration means coarse time resolution A wide frequency bandwidth means coarse frequency resolution Figure 2 7 shows the time and frequency resolutions of the three wavelets with three boxes in the time frequency domain The heights and widths of the boxes represent the frequency and time resolutions of the wavelets respectively This figure shows that a wavelet with a small scale has fine time resolution but coarse frequency resolution and that a wavelet with a large scale has fine frequency resolution but coarse time resolution gt o c o 2 o LL az 264 L 4 a 232L41 73 cJ d Vi l MED EETA asde een E a E kea pou Ay lt r east settee Time a 16 a 32 a 64 u 100 u 200 u 300 Figure 2 7 Time and Frequency Resolutions of Wavelets The fine frequency resolution of large scale wavelets enables you to measure the frequency of the slow variation components in a signal The fine time resolution of small scale wavelets enables you to detect the fast variation components in a signal Therefore wavelet signal processing is a useful multiresolution analysis tool Refer to the Discrete Wavelet Transform for Multiresolution Analysis section of Chapter 4 Signal Processing with Discrete Wavelets for information about performing multiresolution analysis Wavelet Analysis Tools User Manual 2 6 ni com Signal Processing with Continuous Wavelets You can use continuous
43. lated to this document or the information contained in it EXCEPT AS SPECIFIED HEREIN NATIONAL INSTRUMENTS MAKES NO WARRANTIES EXPRESS OR IMPLIED AND SPECIFICALLY DISCLAIMS ANY WARRANTY OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE CUSTOMER S RIGHT TO RECOVER DAMAGES CAUSED BY FAULT OR NEGLIGENCE ON THE PART OF NATIONAL INSTRUMENTS SHALL BE LIMITED TO THE AMOUNT THERETOFORE PAID BY THE CUSTOMER NATIONAL INSTRUMENTS WILL NOT BE LIABLE FOR DAMAGES RESULTING FROM LOSS OF DATA PROFITS USE OF PRODUCTS OR INCIDENTAL OR CONSEQUENTIAL DAMAGES EVEN IF ADVISED OF THE POSSIBILITY THEREOF This limitation of the liability of National Instruments will apply regardless of the form of action whether in contract or tort including negligence Any action against National Instruments must be brought within one year after the cause of action accrues National Instruments shall not be liable for any delay in performance due to causes beyond its reasonable control The warranty provided herein does not cover damages defects malfunctions or service failures caused by owner s failure to follow the National Instruments installation operation or maintenance instructions owner s modification of the product owner s abuse misuse or negligent acts and power failure or surges fire flood accident actions of third parties or other events outside reasonable control Copyright Under the copyright laws this publication may not be reproduced or transmitted in
44. le by selecting Toolkits and Modules Wavelet Analysis A pplications Breakdown Point Detection VI Refer to the Finding Example VIs section of Chapter 1 Introduction to Wavelet Signal Processing for information about launching the NI Example Finder Analytic Wavelet Transform The AWT is a wavelet transform that provides both the magnitude and phase information of signals in the time scale or time frequency domain The magnitude information returned by the AWT describes the envelopes of signals The phase information encodes the time related characteristics of signals for example the location of a cusp You usually use the magnitude information for time frequency analysis and phase information for applications such as instantaneous frequency estimation Wavelet Analysis Tools User Manual 3 6 ni com Chapter 3 Signal Processing with Continuous Wavelets The AWT computes the inner products of the analyzed signal and a set of complex Morlet wavelets This transform is called the analytic wavelet transform because the complex Morlet wavelets are analytic that is the power spectra of the Morlet wavelets are zero at negative frequencies The resulting AWT coefficients are complex numbers These coefficients measure the similarity between the analyzed signal and the complex Morlet wavelets The AWT is just one type of complex continuous wavelet transform Use the WA Analytic Wavelet Transform VI to compute the AWT Refer to the LabVIEW He
45. llection of commonly used discrete wavelets such as the Daubechies Haar Coiflet and biorthogonal wavelets Refer to the Selecting an Appropriate Discrete Wavelet section of Chapter 4 Signal Processing with Discrete Wavelets for information about the collection of discrete wavelets You also can create a discrete wavelet that best matches the signal you analyze using the Wavelet Design Express VI Refer to Chapter 5 Interactively Designing Discrete Wavelets for information about designing a wavelet The Wavelet Analysis Tools contain Express VIs that provide interfaces for signal processing and analysis These Express VIs enable you to specify parameters and settings for an analysis and see the results immediately For example the Wavelet Denoise Express VI graphs both the original and denoised signals You can see the denoised signal immediately as you select a wavelet specify a threshold and set other parameters The Wavelet Analysis Tools also provide Express VIs for multiresolution analysis wavelet design and wavelet packet decomposition National Instruments Corporation 1 5 Wavelet Analysis Tools User Manual Chapter 1 Introduction to Wavelet Signal Processing Finding Example VIs The Wavelet Analysis Tools also provide example VIs that you can use and incorporate into the VIs that you create You can modify an example VI to fit an application or you can copy and paste from one or more examples into a VI that you create You
46. lp available by selecting Help Search the LabVIEW Help for information about this VI Scale and Frequency Wavelets are functions of time and scale so you can consider a wavelet transform as a tool that produces a time scale representation of signals You also can consider the time scale representation of signals as a time frequency representation because wavelets with different scales measure the corresponding frequency components in the signal The frequency of a wavelet is inversely proportional to the scale factor Refer to the Multiple Resolutions section of Chapter 2 Understanding Wavelet Signal Processing for information about the relationship between the scale factor and the frequency of a wavelet Using the WA Analytic Wavelet Transform VI you can specify different settings for the scale factor to compute the AWT When you set scale sampling method to even scale this VI computes the wavelet coefficients at evenly distributed integer scales You usually use the even scale option to obtain the time scale representation of a signal When you set scale sampling method to even freq this VI computes the wavelet coefficients at scales with evenly distributed frequencies Notice that the scales are not evenly distributed You usually use the even freq option to obtain the time frequency representation of a signal Because the time and frequency resolutions of wavelets are adaptive the AWT provides adaptive time and frequency resolution
47. ly applies the lowpass and highpass filters to either the approximation or the detail coefficients at each level You can consider arbitrary path decomposition as a band pass filter which you can implement by cascading filter banks Figure 4 12 shows an example arbitrary path decomposition Figure 4 12 Arbitrary Path Decomposition In this example the decomposition path is 011 because the signal first enters a lowpass filter 0 then a highpass filter 1 and finally a highpass filter 1 again The results on the paths 1 00 and 010 also can be saved for reconstruction purpose The paths 1 00 and 010 define residual paths Use the WA Arbitrary Path Decomposition VI and the WA Arbitrary Path Reconstruction VI to decompose and reconstruct a signal according to different paths and wavelet types Refer to the LabVIEW Help available by selecting Help Search the LabVIEW Help for information about these two VIs Wavelet Analysis Tools User Manual 4 12 ni com Chapter 4 Signal Processing with Discrete Wavelets Figure 4 13 shows an application of the arbitrary path decomposition in detecting engine knocking due to an ignition system malfunction Engine Knocking Sound 2 9E 4 2 0E 4 0 0E 0 Amplitude 2 0E 4 3 3E 4 0 0 Time Enhanced Sound 5 8E 3 2 5E 3 a O 0E 0 2 5E 3 58E 37 0 0 Figure 4 13 Engine Knocking Detection This example applies the bior3 7 wavelet
48. membership for a full year after purchase after which you may renew to continue your benefits For information about other technical support options in your area visit ni com services or contact your local office at ni com contact Training and Certification Visit ni com training for self paced training eLearning virtual classrooms interactive CDs and Certification program information You also can register for instructor led hands on courses at locations around the world System Integration If you have time constraints limited in house technical resources or other project challenges National Instruments Alliance Partner members can help To learn more call your local NI office or visit ni com alliance A 1 Wavelet Analysis Tools User Manual Appendix A Technical Support and Professional Services If you searched ni com and could not find the answers you need contact your local office or NI corporate headquarters Phone numbers for our worldwide offices are listed at the front of this manual You also can visit the Worldwide Offices section of ni com niglobal to access the branch office Web sites which provide up to date contact information support phone numbers email addresses and current events Wavelet Analysis Tools User Manual A 2 ni com
49. nalysis Tools User Manual Chapter 3 Signal Processing with Continuous Wavelets The Signal graph in Figure 3 4 shows the HeaviSine signal which is a common wavelet test signal contaminated with white noise The HeaviSine signal is a sinusoid with two breakdown points one at 51 and the other at 481 The CWT precisely shows the positions of the two breakdown points by doing the following steps 1 Computes the CWT using the Haar wavelet 2 Calculates the squares of the CWT coefficients of the signal and forms a scalogram as shown in the Scalogram graph in Figure 3 4 3 Cumulates the CWT coefficients along the scale axis and forms a cumulation plot as shown in the CWT Coefficients Cumulation graph in Figure 3 4 4 Detects the peak locations in the CWT Coefficients Cumulation graph The peak locations are where the breakdown points exist Breakdown points and noise can generate large values in the resulting coefficients Breakdown points generate large positive or negative coefficients at all scales Noise generates positive coefficients at some scales and negative coefficients at other scales If you accumulate the coefficients at all scales the coefficients of breakdown points are enlarged while the coefficients of noise at different scales counteract one another Therefore the peaks in the CWT Coefficients Cumulation graph correspond only to the breakdown points In the Browse tab of the NI Example Finder you can view this examp
50. ncy 0 of Po z The value of must be less than 0 5 Longer filters improve the sharpness of the transition band and the magnitude of the attenuation in the stopband at the expense of extra computation time for implementation Selecting the Factorization Type for P z After you determine Po z the next step is to specify how Po z is factorized into the analysis lowpass filter Go z and the synthesis lowpass filter Ho z respectively Use the Factorization Type of G0 control to specify the factorization type The factorizing process is not unique For a given Po z you have the following four options for creating Go z and H z e Arbitrary No specific constraints are associated with this filter Figure 5 5 shows an example of arbitrary factorization The blue crosses represent the zeroes of Go z and the red circles represent the Wavelet Analysis Tools User Manual 5 6 ni com Chapter 5 Interactively Designing Discrete Wavelets zeroes of Ho z Click on the zero you want to select to switch the zero from that of Go z to that of Ho z and vice versa Zeroes of GO and HO G0 Blue Cross HO Red Circle EB 79 x 2 7367 y 0 0000 E 25 Imaginary a UNS en o eo en o 5 Em co Figure 5 5 Arbitrary Filter Minimum Phase AIl of the zeroes of Go z are contained inside the unit circle as shown in Figure 5 6 All the zeroes of Ho z are the reciprocal of the zeroes of Go z The
51. ndow 64 you obtain coarse frequency resolution and fine time resolution Therefore you can distinguish the frequency components of the HypChirps signal at higher frequencies with a short window However you cannot distinguish the Wavelet Analysis Tools User Manual 3 8 ni com Chapter 3 Signal Processing with Continuous Wavelets two frequency components at both low and high frequencies in either of the STFT spectrograms Figure 3 7 shows the tiling of the STFT based time frequency representation Frequency Frequency Time Time Figure 3 7 Tiling of STFT Based Time Frequency Representation In Figure 3 7 you can see that the STFT spectrogram has uniform time frequency resolution across the whole time frequency domain You can balance the time frequency resolution by adjusting the window length The left tiling diagram provides better frequency resolution in the STFT Spectrogram Window Length 256 graph of Figure 3 6 The right tiling diagram shows better time resolution in the STFT Spectrogram Window Length 64 graph of Figure 3 6 However you cannot achieve high time resolution and frequency resolution simultaneously Figure 3 8 shows the AWT based time frequency representation of the HypChirps signal In the Scalogram graph you can distinguish the two frequency components at both low and high frequencies Scalogram 500
52. oe ate ties tere e e een 2 3 Transient Feature Detection eese teen eene tnnt nnne neenon 2 4 Multiple Resolutions eee e ttes 2 5 Chapter 3 Signal Processing with Continuous Wavelets Continuous Wavelet Transform sese eere retener 3 2 Application Example Breakdown Point Detection esses 3 5 Analytic Wavelet Transform esee en eene rennen E 3 6 Scale and Brequency Re ERR er E 3 7 Wavelet Normalization Energy versus Amplitude sss 3 10 National Instruments Corporation V Wavelet Analysis Tools User Manual Contents Chapter 4 Signal Processing with Discrete Wavelets Selecting an Appropriate Discrete Wavelet sese 4 1 Discrete Wavelet Transform ees Rege p eraot the hove eee 4 2 Discrete Wavelet Transform for Multiresolution Analysis 4 5 2D S1gnal Processing ere n ettet sd 4 7 Wavelet Packet Decomposition essseseeeeeeeeeeeeene ener reete rennen 4 0 Arbitrary Path Decomposition eese nennen 4 12 Undecimated Wavelet Transform eese eren nnne 4 13 Benefits of Undecimated Wavelet Transform sese 4 14 Chapter 5 Interactively Designing Discrete Wavelets Selecting the Wavelet Type eter tre tease E Feb eR ERN Pa eH ds 5 4 Designing the Prod ct Do 2 ce Hm ree totis aolet et reserve etek ons 5 4 Selecting the Factoriz
53. ollowing resources offer useful background information on the general concepts discussed in this documentation These resources are provided for general informational purposes only and are not affiliated sponsored or endorsed by National Instruments The content of these resources is not a representation of may not correspond to and does not imply current or future functionality in the Wavelet Analysis Tools or any other National Instruments product Wavelet Analysis Tools User Manual Mallat Stephane A Wavelet Tour of Signal Processing 2nd ed San Diego California Academic Press 1999 Qian Shie Introduction to Time Frequency and Wavelet Transforms Upper Saddle River New Jersey Prentice Hall PTR 2001 viii ni com Introduction to Wavelet Signal Processing Wavelets are functions that you can use to decompose signals similar to how you use complex sinusoids in the Fourier transform to decompose signals The wavelet transform computes the inner products of the analyzed signal and a family of wavelets In contrast with sinusoids wavelets are localized in both the time and frequency domains so wavelet signal processing is suitable for nonstationary signals whose spectral content changes over time The adaptive time frequency resolution of wavelet signal processing enables you to perform multiresolution analysis on nonstationary signals The properties of wavelets and the flexibility to select wavelets make wavelet signal p
54. onal Orthogonal Po z Go z Ho z Maximum Flat zlyPQ General Equiripple Positive Equiripple P e gt 0 Maximum Flat 1 zb ep Go z 1 z gt Ho z Positive Equiripple P e gt 0 Step 1 Wavelet Type Step 2 Step 3 Product of Lowpass Factorization B spline Go z 1 zk Hoz 1 21 OR Linear Phase G o z has to contain both zero z and its reciprocal 1 z Arbitrary Linear Phase Go z has to contain both zero z and its reciprocal 1 z Arbitrary Linear Phase Go z has to contain both zero z and its reciprocal 1 z Arbitrary Minimum Phase Daubechies Go z contains all zeros z 1 Arbitrary Minimum Phase G z contains all zeros z 1 Arbitrary Interactively Designing Discrete Wavelets Figure 5 2 Design Procedure for Wavelets and Filter Banks Using the Wavelet Design Express VI you need to complete the following steps to design wavelets 1 2 National Instruments Corporation Select the wavelet type Design the product of lowpass filters Po z where the auxiliary function Po z is the product of Go z and Ho z Select the factorization type to factorize Po z into Go z and Ho z 5 8 Wavelet Analysis Tools User Manual Chapter 5 Interactively Designing Discrete Wavelets After you create an analysis lowpass filt
55. ows the decomposition of the Piece Polynomial signal The resulting histogram of the wavelet packet coefficients is similar to the histogram of the discrete wavelet coefficients meaning that the DWT and the wavelet packet decomposition have similar compression performance for the Piece Polynomial signal Wavelet Analysis Tools User Manual 4 10 ni com Chapter 4 Signal Processing with Discrete Wavelets Time DWT Coefficients y Histogram Wavelet Packet Coefficients Amplitude Figure 4 10 Decomposition of the Piece Polynomial Signal Figure 4 11 shows the decomposition of the Chirps signal The resulting histogram of the wavelet packet coefficients is more compact than the histogram of the DWT coefficients Therefore the wavelet packet decomposition can achieve a higher compression ratio for signals like the Chirps signal Signal o a D z x L NI D 0 8 1 023 Time DWT Coefficients iN Histogram Wavelet Packet Coefficients Amplitude Figure 4 11 Decomposition of the Chirps Signal National Instruments Corporation 4 11 Wavelet Analysis Tools User Manual Chapter 4 Signal Processing with Discrete Wavelets Arhitrary Path Decomposition Traditional wavelet packet decomposition iteratively applies the lowpass and highpass filters to both the approximation and the detail coefficients The arbitrary path decomposition as a special case of the wavelet packet decomposition iterative
56. r 2D signals Refer to the LabVIEW Help available by selecting Help Search the LabVIEW Help for information about these two VIs Benefits of Undecimated Wavelet Transform This section describes the unique features of the UWT by comparing the UWT with the DWT Translation Invariant Property Unlike the DWT the UWT has the translation invariant or shift invariant property If two signals are shifted versions of each other the UWT results for the two signals also are shifted versions of each other The translation invariant property is important in feature extraction applications Figure 4 14 shows an example that detects discontinuities in the HeaviSine signal with both the DWT and the UWT HeaviSine Signal Signals Shifted HeaviSine Signal Nu 1 1 1 1 1 1 1 1 1 Du i00 200 300 400 500 600 700 800 900 1023 Time HeaviSine Signal First Level DWT Detail Coefficients Shifted HeaviSine Signal N 1 Amplitude hon Me ar 1 1 1 D 1 1 1 1 Li 50 100 150 200 250 300 350 400 450 511 o Time HeaviSine Signal First Level UWT Detail Coefficients Shifted HeaviSine Signal 2 Amplitude T m o i 1 1 1 1 1 1 NI i00 200 300 400 500 600 700 800 900 1023 Time Figure 4 14 Discrete Wavelet Transform versus Undecimated Wavelet Transform Wavelet Analysis Tools User Manual 4 14 ni com Chapter 4 Signal Processing with Discrete Wavelets You can use the first level detail
57. ripple halfband Positive equiripple halfband The maximally flat filter is defined by the following equation 21 2 Py d zb Qu Wavelet Analysis Tools User Manual 5 4 ni com Chapter 5 Interactively Designing Discrete Wavelets The Zero pairs at n PO control on the Configure Wavelet Design dialog box specifies the value of the parameter p which determines the number of zeroes placed at x on the unit circle The more the zeroes at 7 the smother the corresponding wavelet The value of p also affects the transition band of the frequency response A large value of p results in a narrow transition band In the time domain a narrower transition band implies more oscillations in the corresponding wavelet When you specify a value for the parameter p you can review the frequency response shown on the right hand side of the Configure Wavelet Design dialog box In a general equiripple halfband filter halfband refers to a filter in which T where denotes the stopband frequency and denotes the passband frequency as shown in Figure 5 3 A PO 0 Q T 2 Ws n Figure 5 3 Halfband Filter The positive equiripple halfband filter is a special case of general equiripple halfband filters The Fourier transform of this type of filter is always nonnegative Positive equiripple halfband filter is appropriate for orthogonal wavelets because the auxiliary function Po z must be nonnegative Use t
58. rocessing a beneficial tool for feature extraction applications Refer to the Benefits of Wavelet Signal Processing section of Chapter 2 Understanding Wavelet Signal Processing for information about the benefits of wavelet signal processing This chapter describes the application areas of wavelet signal processing and provides an overview of the LabVIEW Wavelet Analysis Tools Wavelet Signal Processing Application Areas You can use wavelets in a variety of signal processing applications such as analyzing signals at different scales reducing noise compressing data and extracting features of signals This section discusses these application areas by analyzing signals and images with the Wavelet Analysis Tools The Wavelet Analysis Tools provide example VIs for each application area In the Browse tab of the NI Example Finder you can view these example VIs by selecting Toolkits and Modules Wavelet Analysis Applications Refer to the Finding Example VIs section of this chapter for information about launching the NI Example Finder National Instruments Corporation 1 1 Wavelet Analysis Tools User Manual Chapter 1 Introduction to Wavelet Signal Processing Multiscale Analysis Multiscale analysis involves looking at a signal at different time and frequency scales Wavelet transform based multiscale analysis helps you understand both the long term trends and the short term variations of a signal simultaneously Figure 1 1 sho
59. rossings among the coefficients with coarse resolution enables you to remove noise from a signal efficiently Finding Zero crossings among the coefficients with finer resolution improves the precision with which you can find peak locations The WA Multiscale Peak Detection VI uses the UWT based method This VI detects peaks in offline and online signals You can use this VI in the following ways e Once for an offline signal e Continuously for a block of signals e Continuously for signals from streaming data sources Refer to the LabVIEW Help available by selecting Help Search the LabVIEW Help for information about this VI Figure 4 16 shows an example that uses the WA Multiscale Peak Detection VI to detect peak in an electrocardiogram ECG signal The UWT based method locates the peaks of the ECG signal accurately regardless of whether the peaks are sharp or rounded Peaks ECG Signal Amplitude 1 1 J t 1 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Time Figure 4 16 Peak Detection in a Noisy ECG Signal National Instruments Corporation 4 17 Wavelet Analysis Tools User Manual Chapter 4 Signal Processing with Discrete Wavelets In the Browse tab of the NI Example Finder you can view the following examples by selecting e Toolkits and Modules Wavelet Analysis Applications ECG Heart Rate Monitor Online VI e Toolkits and Modules Wavelet Analysis Applications ECG QRS Compl
60. row direction high high Shows the details at the discontinuities along the diagonal direction You can apply the decomposition iteratively to the low low image to create a multi level 2D DWT which produces an approximation of the source signal with coarse resolution You can determine the appropriate number of decomposition levels for a signal processing application by evaluating the quality of the decomposition at different levels Wavelet Analysis Tools User Manual 4 8 ni com Chapter 4 Signal Processing with Discrete Wavelets Use the Multiresolution Analysis 2D Express VI to decompose and reconstruct a 2D signal Refer to the LabVIEW Help available by selecting Help Search the LabVIEW Help for information about this Express VI Figure 4 8 shows an example of image compression using the 2D DWT with the FBI wavelet Original Image Reconstructed Image Original Image DWT Coefficients Histogram 15k 12 5k 10k 7 5k Number 1 1 1 1 i 1 50 75 100 1 150 1 200 Amplitude Figure 4 8 Example of Image Compression The histogram of the DWT Coefficients plot shows that the majority of the DWT coefficients are small meaning that you can use a small number of large DWT coefficients to approximate the image and achieve data compression Wavelet Packet Decomposition As discussed in the Discrete Wavelet Transform section of this chapter you can approximate the DWT using fil
61. rst is filtered by a filter bank consisting of Go z and G z The outputs of Go z and G z then are downsampled by a factor of 2 After some processing the modified signals are upsampled by a factor of 2 and filtered by another filter bank consisting of Ho z and H z Wavelet Analysis Tools User Manual 4 2 ni com Chapter 4 Signal Processing with Discrete Wavelets If no processing takes place between the two filter banks the sum of outputs of Ho z and H z is identical to the original signal X z except for the time delay This system is a two channel PR filter bank where Go z and G z form an analysis filter bank and Ho z and H z form a synthesis filter bank Traditionally Go z and Ho z are lowpass filters and G z and H z are highpass filters The subscripts 0 and 1 represent lowpass and highpass filters respectively The operation 12 denotes a decimation of the signal by a factor of two Applying decimation factors to the signal ensures that the number of output samples of the two lowpass filters equal the number of original input samples X z Therefore no redundant information is added during the decomposition Refer to the LabVIEW Digital Filter Design Toolkit documentation for more information about filters You can use the two channel PR filter bank system and consecutively decompose the outputs of lowpass filters as shown in Figure 4 2 Signal gt Di 12 D gt Giz L2 g
62. s Conventional time frequency analysis methods such as the short time Fourier transform STFT only provide uniform time and frequency resolutions in the whole time frequency domain Refer to the Time Frequency Analysis Tools User Manual for information about the STFT and other conventional time frequency analysis methods National Instruments Corporation 3 7 Wavelet Analysis Tools User Manual Chapter 3 Signal Processing with Continuous Wavelets Figure 3 5 shows a common wavelet test signal the HypChirps signal This signal contains two frequency components which are hyperbolic functions over time The frequency components change slowly at the beginning and rapidly at the end HypChirps Signal 1 1 D D D D 1 0 01 02 03 04 05 0 6 0 7 0 8 09 1 Time s Figure 3 5 The HypChirps Signal Figure 3 6 shows two representations of this HypChirps signal in the time frequency domain based on the STFT method STFT Spectrogram Window Length 256 500 D L D D 1 J 01 02 03 04 05 06 07 08 09 Time s D 1 Lu U 1 1 1 i 0 1 02 0 3 0 4 05 06 07 08 0 9 Time s Figure 3 6 STFT Spectrograms of the HypChirps Signal In Figure 3 6 if you use a relatively long window 256 you obtain fine frequency resolution and coarse time resolution Therefore you can distinguish the frequency components of the HypChirps signal at lower frequencies with a long window If you use a short wi
63. t 2 o Gaz Hie 2 Figure 4 2 Discrete Wavelet Transform Lowpass filters remove high frequency fluctuations from the signal and preserve slow trends The outputs of lowpass filters provide an approximation of the signal Highpass filters remove the slow trends from the signal and preserve high frequency fluctuations The outputs of highpass filters provide detail information about the signal The outputs of lowpass filters and highpass filters define the approximation coefficients and detail coefficients respectively Symbols A and D in Figure 4 2 represent the approximation and detail information respectively You also can call the detail coefficients wavelet coefficients because detail coefficients approximate the inner products of the signal and wavelets This manual alternately uses the terms wavelet coefficients and detail coefficients depending on the context National Instruments Corporation 4 3 Wavelet Analysis Tools User Manual Chapter 4 Signal Processing with Discrete Wavelets The Wavelet Analysis Tools use the subscripts 0 and 1 to describe the decomposition path where 0 indicates lowpass filtering and 1 indicates highpass filtering For example D in Figure 4 2 denotes the output of two cascaded filtering operations lowpass filtering followed by highpass filtering Therefore you can describe this decomposition path with the sequence 01
64. ter banks When the decomposition is applied to both the approximation coefficients and the detail coefficients the operation is called wavelet packet decomposition National Instruments Corporation 4 9 Wavelet Analysis Tools User Manual Chapter 4 Signal Processing with Discrete Wavelets Figure 4 9 shows the wavelet packet decomposition tree Figure 4 9 Wavelet Packet Decomposition Tree at Level Three The numbers indicate the path of each node The path is a combination of the characters 0 and 1 where 0 represents lowpass filtering followed by a decimation with a factor of two and 1 represents highpass filtering followed by a decimation with a factor of two Based on Figure 4 9 you can represent a signal with different sets of sequences or different decomposition schemes such as 1 01 001 000 1 00 010 O11 0r 000 001 010 011 100 101 110 111 As the decomposition level increases the number of different decomposition schemes also increases The DWT is useful in compressing signals in some applications The wavelet packet decomposition also can compress signals and provide more compression for a given level of distortion than the DWT does for some signals such as signals composed of chirps For example the wavelet packet decomposition and the DWT with the sym8 wavelet decomposition level 4 and periodic extension are applied to the Piece Polynomial signal and the Chirps signal Figure 4 10 sh
65. th low resolution You can obtain the global profile of the image in a low resolution edge map and the detailed texture of the image in a high resolution edge map You also can form a multiresolution edge detection method by examining the edge maps from the low resolution to the high resolution With the multiresolution edge detection method you can locate an object of interest in the image reliably and accurately even under noisy conditions In the NI Example Finder refer to the Image Edge Detection VI for more information about performing wavelet transform based edge detection on image files Overview of LabVIEW Wavelet Analysis Tools The Wavelet Analysis Tools provide a collection of Wavelet Analysis VIs that assist you in processing signals in the LabVIEW environment You can use the Continuous Wavelet VIs the Discrete Wavelet VIs and the Wavelet Packet VIs to perform the continuous wavelet transform the discrete wavelet transform the undecimated wavelet transform the integer wavelet transform and the wavelet packet decomposition You can use the Feature Extraction VIs to detrend and denoise a signal You also can use these VIs to detect the peaks and edges of a signal Refer to the LabVIEW Help available by selecting Help Search the LabVIEW Help for information about the Wavelet Analysis VIs The Wavelet Analysis Tools provide a collection of commonly used continuous wavelets such as Mexican Hat Meyer and Morlet and a co
66. tinuity detections If a signal contains different scale characteristics over time the scalogram can present a time scale view National Instruments Corporation 3 3 Wavelet Analysis Tools User Manual Chapter 3 Signal Processing with Continuous Wavelets of the signal which is more useful than the time frequency view of that signal Figure 3 2 shows a test signal the Devil s Staircase fractal signal An important characteristic of a fractal signal is self similarity Devil s Staircase 1 1 1 1 F J 1 50 100 150 200 250 300 350 400 450 500 550 600 640 Time s Figure 3 2 Devil s Staircase Signal Figure 3 3 shows the scalogram and the STFT spectrogram of the fractal signal respectively Scalogram 128 100 80 60 Scale 40 20 Oey 1 1 1 1 1 1 1 1 D 1 D 1 1 0 50 100 150 200 250 300 350 400 450 500 550 600 640 Time s STFT Spectrogram 0 5 0 47 em 2 0 37 i S 02 c D pei 1 1 1 1 1 1 1 i 1 1 100 150 200 250 300 350 400 450 500 550 600 640 Time s Figure 3 3 Scalogram versus STFT Spectrogram of the Devil s Staircase Signal In Figure 3 3 you can see the self similarity characteristic of the signal clearly in the Scalogram graph but not in the STFT Spectrogram graph The STFT Spectrogram graph displays the conventional time frequency analysis result of the signal Refer to the Time Frequency Analysis Tools User Manual for more
67. uency components with fine frequency resolution but coarse time resolution Figure 4 4 shows the frequency bands of the DWT for the db08 wavelet Al PM D1 Frequency Band of 1 Level DWT j D LU I E 0 15 02 025 09 05 Normalized Frequency Frequency Band of 2 Level DWT 8E 6 6E 6 B 2 4E 6 t 2E 6 1 1 1 0 15 02 025 0 3 0 5 Normalized Frequency Frequency Band of 3 Level DWT 8E 6 6E 6 g 2 4E 6 i 2E 6 1 1 1 1 1 D 1 O15 02 025 03 0 35 0 5 Normalized Frequency Figure 4 4 Frequency Bands of the Discrete Wavelet Transform You can see that the central frequency and frequency bandwidth of the detail coefficients decrease by half when the decomposition level increases by one For example the central frequency and frequency bandwidth of D are half that of D You also can see that the approximation at a certain resolution contains all of the information about the signal at any coarser resolutions For example the frequency band of A covers the frequency bands of A and D3 DWT based multiresolution analysis helps you better understand a signal and is useful in feature extraction applications such as peak detection and edge detection Multiresolution analysis also can help you remove unwanted components in the signal such as noise and trend National Instruments Corporation 4 5 Wavelet Analysis Tools User Manual Chapter 4 Signal Processing
68. wn points and start and end of bursts Transient signals usually vary over time and you typically cannot predict the occurrence exactly The LabVIEW Advanced Signal Processing Toolkit contains the following tools and toolkit that you can use to perform signal analysis and processing e Wavelet Analysis Tools e LabVIEW Time Series Analysis Tools e LabVIEW Time Frequency Analysis Tools e LabVIEW Digital Filter Design Toolkit Wavelet Analysis Tools User Manual 1 6 ni com Chapter 1 Introduction to Wavelet Signal Processing To extract the underlying information of a signal effectively you need to choose an appropriate analysis tool based on the following suggestions National Instruments Corporation For stationary signals use the Time Series Analysis Tools or the Digital Filter Design Toolkit LabVIEW also includes an extensive set of tools for signal processing and analysis The Time Series Analysis Tools provide VIs for preprocessing signals estimating the statistical parameters of signals building models of signals and estimating the power spectrum the high order power spectrum and the cepstrum of signals The Digital Filter Toolkit provides tools for designing analyzing and simulating floating point and fixed point digital filters and tools for generating code for DSP or FPGA targets For evolutionary signals use the Time Frequency Analysis Tools which include VIs and Express VIs for linear and quadratic time frequency
69. ws a multiscale analysis of a Standard amp Poor s S amp P 500 stock index during the years 1947 through 1993 The S amp P 500 Index graph displays the monthly S amp P 500 indexes The other three graphs are the results of wavelet analysis The Long Term Trend graph is the result with a large time scale which describes the long term trend of the stock movement The Short Term Variation and Medium Term Variation graphs describe the magnitudes of the short term variation and medium term variation respectively S amp P 500 Index 150 Amplitude a S e e t I 1 1 450 500 556 Month Medium Term Variation 10 a 1 lh A v A VY Amplitude o 1 D a 1 l 1 Li Li 1 I D 1 I 1 D n 50 100 150 200 250 300 350 450 500 556 Month D e o 1 o Long Term Trend 150 Amplitude a S e e t t 1 1 1 1 1 1 LLI 200 250 300 350 400 450 500 556 Month 1 I RET au ba m I D D 1 I 1 I I I I 150 200 250 300 450 500 556 Month Figure 1 1 Multiscale Analysis of the S amp P 500 Stock Index Wavelet Analysis Tools User Manual 1 2 ni com Noise Reduction Compression Chapter 1 Introduction to Wavelet Signal Processing In the NI Example Finder refer to the Multiscale Analysis VI for more information about performing wavelet transform based multiresolution analysis on stock indexes One of the most effective applications of wavelets in signal process
70. xt in this font denotes text or characters that you should enter from the keyboard sections of code programming examples and syntax examples This font is also used for the proper names of disk drives paths directories programs subprograms subroutines device names functions operations variables filenames and extensions National Instruments Corporation vij Wavelet Analysis Tools User Manual About This Manual Related Documentation The following documents contain information that you might find helpful as you read this manual LabVIEW Help available by selecting Help Search the LabVIEW Help Getting Started with LabVIEW available by selecting Start All Programs National Instruments Lab VIEW x x LabVIEW Manuals where x x is the version of LabVIEW you installed and opening LV Getting Started pdf This manual also is available by navigating to the labview manuals directory and opening LV Getting Started pdf The LabVIEW Help includes all the content in this manual LabVIEW Fundamentals available by selecting Start All Programs National Instruments LabVIEW x x LabVIEW Manuals where x x is the version of LabVIEW you installed and opening LV Fundamentals pdf This manual also is available by navigating to the Llabview manuals directory and opening LV Fundamentals pdf The LabVIEW Help includes all the content in this manual The LabVIEW Digital Filter Design Toolkit documentation B Note The f
71. y selecting Help Search the LabVIEW Help for information about these VIs The denoising procedure in the Wavelet Denoise Express VI and the WA Denoise VI involves the following steps 1 Applies the DWT or the UWT to noise contaminated signals to obtain the DWT coefficients or the UWT coefficients The noise in signals usually corresponds to the coefficients with small values 2 Selects an appropriate threshold for the DWT coefficients or the UWT coefficients to set the coefficients with small values to zero The Wavelet Denoise Express VI and the WA Denoise VI provide methods that automatically select the thresholds The bound of noise reduction with these methods is 3 dB To achieve better denoising performance for a signal you can select an appropriate threshold manually by specifying the user defined thresholds parameter of the WA Denoise VI 3 Reconstructs the signal with the inverse DWT or the inverse UWT National Instruments Corporation 4 15 Wavelet Analysis Tools User Manual Chapter 4 Signal Processing with Discrete Wavelets Figure 4 15 shows the denoising results of a noisy Doppler signal with both the DWT based method and the UWT based method Both methods use the level 5 wavelet transform and the soft threshold NA 0 Signal Amplitude Denoising with UWT 104 m wA I pw y Amplitude e 5 D rA o 1 107 gl IA o I VY UN LU Amplitude za on TY Time

Wavelet Analysis Tools User Manual

Contents

Download Pdf Manuals

Related Search

Related Contents