EWGAE 2010 Vienna, 8th to 10th September Latest improvements on Freeware AGU-Vallen-Wavelet Jochen VALLEN 1, Hartmut VALLEN 2 1 Vallen Systeme GmbH, Schäftlarner Weg 26a, 82057 Icking, Germany jochen@vallen.de, tel.+49 (8178) 9674 400, fax +49 (8178) 9674 444 2 Vallen Systeme GmbH, Schäftlarner Weg 26a, 82057 Icking, Germany sales@vallen.de, tel.+49 (8178) 9674 400, fax +49 (8178) 9674 444 Keywords: Frequency analysis, Wavelet, Choi-Williams transform Abstract AGU-Vallen Wavelet is a free of cost tool available to everybody to analyze AE waveform data using wavelet analysis. The presentation starts with a general overview, compares different time-frequency transforms and finally discusses how to use Choi-Williams Transform to improve results in time and frequency domain. Introduction AGU-Vallen Wavelet has been started as project on IAES-15 in Tokyo, Japan in September 2000. Wavelet analysis for AE was a highly discussed topic at all international conferences. Especially in Japan many scientific institutes were working on implementing and evaluating this new technique in data analysis. Different institutes realized individual implementations what made it difficult to compare results among each other. A general tool was requested to process, display, export, and print the complex data results pleasingly and quickly in order to move the focus from implementation of common algorithms to evaluation of the method itself. Prof. Kanji Ono made the contact between Prof. Mikio Takemoto from Aoyama Gakuin University, Tokyo, Japan and the author and initiated a collaboration. Prof. Yoshihiro Mizutani, at that time a student of Prof. Takemoto improved a Wavelet algorithm based on [2] for implementation. The author implemented the algorithm in a complete software package and added a graphical interface for data display and export. In addition Prof. Mizutani also developed specific numerical methods for solving the plate wave equations used in the Dispersion software which also is part of the project. Dispersion curves can be used for graphical overlay with Wavelet data [2]. On AEWG-45 in Evanston, IL, USA this first version of AGU-Vallen Wavelet has been published in the internet [1] and presented to the Acoustic Emission Working Group (AEWG). The AEWG highly valued the project results and dignified Prof. Takemoto, Prof. Yoshihiro and the author with the AEWG Publication Award which was presented by Prof. Carpenter on IAES- 16 in Tokushima, Japan in November 2002. The success of the project is not least due to Prof. Marvin Hamstad, Denver University, CO, USA. As the first scientific user he accompanied the development of the program and contributed data samples created by finite element modelling (FEM) for verification of the results. Prof. Hamstad published the first paper on results using AGU-Vallen Wavelet [3] and continuously presented his progress on different conferences [4,5,6].
Content of AGU-Vallen Wavelet Software The software is distributed over the internet [1] as a self-extracting and self-installing package for Windows 2000/XP/Vista/7. It unpacks itself into folder c:\vallen\wavelet but doesn t make changes to the system. It is enough to delete the folder to uninstall the application. The package is a collection of free software tools to calculate different transforms in time and frequency domain on individual waveform data sets and convenient graphical interface for data analysis as well as export as bitmap or into MS-Excel and print features. For data input a waveform data file in a native format is required, but a data converter is also supplied. In detail the components are: 1. AGU-Vallen Wavelet: The main application for data transform, display and extensive analysis, print, and export. 2. Waveform Importer: A data conversion tool to import and convert waveform data from different ASCII formats into the compatible native format. 3. Dispersion Curve Calculator: A tool to calculate the plate wave dispersion curves for various materials. The curves can be saved and imported into AGU-Vallen Wavelet for data overlay [2]. AGU-Vallen Wavelet offers the following transforms, which are discussed later on: Fast Fourier Transform (FFT): Frequency analysis using different window functions (Rectangular, Hamming, Hanning, Trapezium or Welch). Wavelet Transform (WT): Time-frequency analysis using the Gabor Wavelet [2] Choi-Williams Transform (ChWT): Time-frequency analysis using a member of Cohen s class distribution [7]. Annotation: The abbreviation ChWT is used instead of CWT as CWT is widely used in literature for Continuous Wavelet Transform. Fast Fourier Transform (FFT) The FFT is an algorithm to show the frequency components of a certain signal. It s simple and fast and widely used for frequency analysis. The figure below shows on the left hand side one signal in time domain three times. The rectangle moving from left to right shows the portion of the signal the FFT is calculated of. On the right hand side the FFT result gives the frequency components of the portion of the signal identified by the rectangle. When moving the portion of interest over the signal the changes in frequency over time can be somehow estimated. Fig. 1: Moving an FFT window over a time signal in order to show time-frequency information
As soon as the time-frequency correlation is of higher interest the FFT shows its limitations: Decreasing the width of the signal portion in order to increase the time resolution is a restricted approach as the frequency resolution decreases, too. The frequency resolution is always half of the time resolution in samples, which by itself needs to be power of two. Zero padding for different window sizes smoothes the FFT result but won t increase the frequency resolution. Also the FFT expects the input signal to be continuous which means that in theory the signal continues to the right beyond the window as it starts from the left hand side of the window. A vertical step between the most left and the most right sample of the portion of the signal leads to high frequency components in frequency domain resulting from the induced step function. This effect is typically solved by using window functions in time domain like Hamming or Hanning to deflate the beginning and end of the signal to or close to zero. The FFT is an adequate method for frequency analysis as long as the accuracy for timefrequency correlation is sufficient. Wavelet Transform (WT) A Dirac pulse is used to show the performance of WT over FFT. A Dirac pulse is a very short time signal. For the example shown the pulse consists of one non-zero sample among zero samples. In time domain the Dirac pulse shows up as a sharp peak. In frequency domain the result of the FFT contains all frequencies with the same magnitude as a flat line, means all frequency components contain the same energy. Fig. 2.: Comparison of FFT and WT of a Dirac Pulse Fig. 3.: 3D representation of WT Fig. 2 shows the single peak top left, the FFT as flat line top right and the WT in the large colour contour diagram below with WT coefficients as magnitudes over frequency and time. Fig. 3 shows the 3D representation of the WT. The time resolution is limited by the wave length for lower frequencies. For higher frequencies the resolution and the magnitude increase to a sharp peak shown in red. Still the energy per frequency component is constant, means the integral over time of the magnitudes is equal for all frequencies. As a result the WT is capable to show time, frequency, and the magnitude of the frequency components in one single diagram. In 2D a color coding is used to express the magnitudes. For comparison the same color table is used in the 3D representation. Fig. 4 shows an artificially FEM calculated signal in a thin aluminium plate. The WT shows the arrival of the symmetric S 0 and anti-symmetric A 0 modes within the signal and illustrates the
power of the WT over FFT. For a better analysis the display can be shown either in linear or logarithmic scaling as shown in the 3D representations on the right. More information on wave mode analysis using WT can be found in [3,4,5,6]. The mathematical background of WT and a basic algorithm can be found in [2]. Fig.4: Left: WT display as 2D color contour plot, right: 3D representation with linear and logarithmic magnitude New graphical features The latest version of AGU-Vallen Wavelet now supports the Windows Multi Document Interface (MDI) which allows one to open several waveform data files at the same time for easier comparison of different input signals. Besides, each data file or waveform can be opened in multiple pages for convenient comparison of transforms with different settings. On user request the new 2D Time Projection and 2D-Frequency Projection diagrams have been added. Fig. 5: Time Projection: The upper diagram shows a cut along the line in the lower diagram
The Time Projection shown in Fig. 5 works like a narrow band frequency. A frequency marker shown as horizontal bar can be vertically moved with the mouse to pick a certain frequency. The Time Projection diagram is updated in real time and shows the narrow band filtered portion of the original input signal. Fig. 6: Frequency Projection The Frequency Projection shown in Fig. 6 works similar. A time marker, shown as a vertical bar in the right hand diagram, is moved horizontally with the mouse to pick a certain time. The Frequency Projection displays the WT coefficients for all frequencies at that time in the left hand diagram. The Frequency Projection diagram is rotated counter clockwise in order to match the frequency scaling in both diagrams. In addition the Frequency Projection shows a line for the WT coefficient maxima at all times (blue line). New: Choi-Williams Transform (ChWT) A Dirac pulse is used to show the difference between WT and ChWT. The advantage of the ChWT over WT shows up in Fig. 7 in the tremendously better time resolution for lower frequencies. Even more, the time resolution is independent from frequency. This is of special importance if the result is to be used as input for arrival time determination and discrimination of different wave modes within the same signal. Fig.7: Comparison of FFT and ChWT of a Dirac Pulse
Fig. 8 shows an artificial sine function increasing its frequency in discrete steps. Real world signals never have a discrete change in frequency nor are pure sine signals, however this example works to show how the transforms interpret the point of frequency change in time differently. The WT is shown on the left, ChWT on the right. Obviously the resolution is improved not only in time domain but in frequency domain, too, which is pretty much appreciated. Fig. 9 shows the 3D representation of WT and ChWT. Fig.8: Comparison of WT (left) and ChWT (right) of a sine function at 50, 200, 400, 800 khz Fig. 9: 3D representation of WT (left) and ChWT (right) of the same signal as above About Choi-Williams Transform (ChWT) ChWT is a member of the Cohen s class distribution function. This class of distributions utilizes a bilinear transformation depending on two variables: time and frequency. The general mathematical definition is as follows: C ( t, f ) = x A ( η, τ ) φ( η, τ ) e x j2π ( ηt τf ) dη dτ
This function consists of two parts: The Ambiguity Function τ * τ j2πtη Ax ( η, τ ) = x( t + ) x ( t ) e dt, 2 2 which is the general auto-correlation function extended for a non-stationary process, and the Kernel Function φ ( η, τ ). Fig. 10: Generation of cross terms (Image from www.wikipedia.org) Fig. 11: Masking out the cross terms (Image from www.wikipedia.org) Applying the Ambiguity Function in time and frequency returns the desired independent terms in time and frequency domain. These terms are called autoterms as each dimension (either time or frequency) affects itself. But there are additional terms resulting from the combination of both dimensions, which are called cross-terms. These cross-terms are mathematic results from the Ambiguity Function but show up as unwanted artefacts in the final display and for signal interpretation these results are not predictable (Fig. 10). The Kernel Function is used in order to mask out or filter the unwanted cross-terms. As the cross-terms are two dimensional, the Kernel Function needs to be two dimensional, too, in order to diminish the effects of interfering crossterms and to induce the desired properties to sustain the validity of the final time-frequency distribution (Fig. 11). In literature different Window functions have been proposed, each with its own merits and draw backs. Some of the most typical are: Wigner-Ville Distribution: φ ( η, τ ) = 1 2 α ( ητ ) Exponential Distribution: φ( η, τ ) = e sin( πητ ) 2 2πατ Cone-Shape Distribution: φ( η, τ ) = e πητ For AGU-Vallen Wavelet the proposed Exponential Distribution according to the publication of Choi and Williams [7] is implemented. So the term Choi-Williams Transform (ChWT) is used equivalently to the Exponential Distribution. It needs to be noted that the exponential distribution doesn t perform perfectly in removing the cross-term and improving the auto-terms. In addition the exponential function requires a damping parameter α which needs to be heuristically adjusted. Fig. 12 and 13 show the influence of the exponential damping parameter on the transform result of the same signal. A higher value has a smoothing effect on the auto-terms but intensifies the cross-terms.
Cross-terms Fig. 12: ChWT: Damping parameter = 2 Fig. 13: ChWT: Damping parameter = 50 Conclusions AGU-Vallen Wavelet is a ready-to-use piece of software free of cost to be used to analyze waveform data in time and frequency domain. The recently implemented Choi-Willliams Transform (ChWT) provides a significant better resolution in time and frequency domain, which can be of great benefit for plate wave mode discrimination and wave mode specific arrival time determination. ChWT has trade-offs in cross-term artefacts and more processing time compared to Wavelet Transform (WT). References 1. Public Internet Link to home page of AGU-Vallen Wavelet: http://www.vallen.de/wavelet, Vallen Systeme GmbH, Germany 2. Suzuki H., Kinjo T., Hayashi Y., Takemoto M., Ono K., Appendix by Hayashi Y: Wavelet Transform of Acoustic Emission Signals, Journal of Acoustic Emission, Vol. 14, No.2 (1996, April-June), pp. 69-84 3. Hamstad M.A.: An illustrated Overview of the Use and Value of a Wavelet Transformation To Acoustic Emission Technology, Denver 2002, http://www.vallen.de/wavelet 4. Hamstad M.A., Downs K.S., O Gallagher A.: Practical Aspects of Acoustic Emission Source Location by a Wavelet Transform, Journal of Acoustic Emission, 21, 2003, pp. 70-94 5. Hamstad M.A.: Comparison of Wavelet Transform and Choi-Williams Distribution to determine Group Velocities for different Acoustic Emission Sensors, Journal of Acoustic Emission, 26, 2008, pp. 40-59 6. Hamstad M.A.: On Determination of accurate Lamb-Mode Group-Velocity Arrival Times with different types of Acoustic Emission Sensors, 28 th Conference on Acoustic Emission Testing, Cracow, 2008, Poland, pp. 178-183 7. Choi H., and Williams W.J., Senior Member IEEE: Improved Time-frequency Representation of Multicomponent Signals using Exponential Kernals, Publication in IEEE Transactions on Acoustics, Speech and Signal Processing, 1989, pp. 862-871