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BV0049-11.qxp 14-05-2007 09:02 Page 1 Technical Review No. 2 – 1996 Non-stationary Signal Analysis using Wavelet T ransform, Short-time Fourier T ransform and Wigner-V ille Distribution HEADQUARTERS: DK-2850 Nærum · Denmark · Telephone: +45 4580 0500 Fax: +45 4580 1405 · www.bksv.com · [email protected] Australia (+61) 2 9889-8888 · Austria (+43) 1 865 74 00 · Brazil (+55) 11 5188-8161 · Canada (+1) 514 695-8225 China (+86)10680 29906 · Czech Republic (+420) 2 6702 1100 · Finland (+358) 9 521300 · France (+33) 1 69 90 71 00 Germany (+49) 421 17 87 0 · Hong Kong (+852) 2548 7486 · Hungary (+36) 1 215 83 05 · Ireland (+353) 1 807 4083 Italy (+39) 0257 68061 · Japan (+81) 3 5715 1612 · Netherlands (+31) 318 55 9290 · Norway (+47) 66 771155 Poland (+48) 22 816 75 56 · Portugal (+351) 21 4169 040 · Republic of Korea (+82) 2 3473 0605 · Singapore (+65) 6377 4512 Slovak Republic (+421) 25 443 0701 · Spain (+34) 91 659 0820 · Sweden (+46) 33 22 56 22 · Switzerland (+41) 44 880 7035 Taiwan (+886) 2 2502 7255 · United Kingdom (+44) 14 38 739 000 · USA (+1) 800 332 2040 Local representatives and service organisations worldwide ISSN 007–2621 BV 0049–11

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Previously issued numbers of Brüel & Kjær Technical Review 1 – 1996 Calibration Uncertainties & Distortion of Microphones. Wide Band Intensity Probe. Accelerometer Mounted Resonance Test 2 – 1995 Order Tracking Analysis 1 – 1995 Use of Spatial Transformation of Sound Fields (STSF) Techniques in the Automative Industry 2 – 1994 The use of Impulse Response Function for Modal Parameter Estimation Complex Modulus and Damping Measurements using Resonant and Non-resonant Methods (Damping Part II) 1 – 1994 Digital Filter Techniques vs. FFT Techniques for Damping Measurements (Damping Part I) 2 – 1990 Optical Filters and their Use with the Type 1302 & Type 1306 Photoacoustic Gas Monitors 1 – 1990 The Brüel & Kjær Photoacoustic Transducer System and its Physical Properties 2 – 1989 STSF — Practical instrumentation and application Digital Filter Analysis: Real-time and Non Real-time Performance 1 – 1989 STSF — A Unique Technique for scan based Near-Field Acoustic Holography without restrictions on coherence 2 – 1988 Quantifying Draught Risk 1 – 1988 Using Experimental Modal Analysis to Simulate Structural Dynamic Modifications Use of Operational Deflection Shapes for Noise Control of Discrete Tones 4 – 1987 Windows to FFT Analysis (Part II) Acoustic Calibrator for Intensity Measurement Systems 3 – 1987 Windows to FFT Analysis (Part I) 2 – 1987 Recent Developments in Accelerometer Design Trends in Accelerometer Calibration 1 – 1987 Vibration Monitoring of Machines 4 – 1986 Field Measurements of Sound Insulation with a Battery-Operated Intensity Analyzer Pressure Microphones for Intensity Measurements with Significantly Improved Phase Properties Measurement of Acoustical Distance between Intensity Probe Microphones Wind and Turbulence Noise of Turbulence Screen, Nose Cone and Sound Intensity Probe with Wind Screen 3 – 1986 A Method of Determining the Modal Frequencies of Structures with Coupled Modes Improvement to Monoreference Modal Data by Adding an Oblique Degree of Freedom for the Reference (Continued on cover page 3)

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Technical Review No. 2 – 1996

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Contents Non-stationary Signal Analysis using Wavelet Transform, Short-time Fourier Transform and Wigner-Ville Distribution .................................... 1 Svend Gade and Klaus Gram-Hansen Copyright © 1996, Brüel & Kjær A/S All rights reserved. No part of this publication may be reproduced or distributed in any form, or by any means, without prior written permission of the publishers. For details, contact: Brüel & Kjær Sound & Vibration Measurement A/S, DK-2850 Nærum, Denmark. Editor: Harry K. Zaveri Photographer: Peder Dalmo Layout: Judith Sarup Printed by Highlight Tryk A/S

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Non-stationary Signal Analysis using Wavelet Transform, Short-time Fourier Transform and Wigner-Ville Distribution by Svend Gade, * Klaus Gram-Hansen Abstract While traditional spectral analysis techniques based on Fourier Transform or Digital Filtering provide a good description of stationary and pseudo-station- ary signals, they face some limitations when analysing highly non-stationary signals. These limitations are overcome using Time-frequency analysis tech- niques such as Wavelet Transform, Short-time Fourier Transform, and Wigner-Ville distribution. These techniques, which yield an optimum resolu- tion in the time and frequency domain simultaneously, are described in this article and their advantages and benefits are illustrated through examples. Résumé Si les techniques d’analyse spectrale traditionnelles basée sur la Transformée de Fourier ou le filtrage numérique fournissent une bonne description des signaux stationnaires et pseudo- stationnaires, elles présentent cependant cer- taines limites dans le cas de signaux non stationnaires. Ces problèmes peuvent être contournés à l’aide de méthodes d’analyse telles que la Transformée d’Ondelette, la Transformée de Fourier courte durée ou la Distribution Wigner- Ville. Ces techniques, qui procurent une résolution optimale dans les domaines temporel et fréquentiel simultanément, sont décrites dans cet article, et leurs avantages illustrés par des exemples. * Gram & Juhl Studsgade 10, baghuset, 8000 Århus C, Denmark 1

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Zusammenfassung Traditionelle Methoden der Spektralanalyse auf der Basis von Fourier-Trans- formation oder Digitalfiltern liefern zwar eine gute Beschreibung stationärer und pseudostationärer Signale, sind jedoch zur Analyse stark nichtstationä- rer Signale nur bedingt geeignet. Mit Hilfe von Zeit-Frequenz-Analysemetho- den wie Wavelet-Transformation, Kurzzeit-Fourier-Transformation und Wigner-Ville-Verteilung lassen sich diese Begrenzungen überwinden. Diese Techniken gewähren eine optimale Auflösung gleichzeitig im Zeit- und im Fre- quenzbereich. Dieser Artikel beschreibt die Methoden und illustriert Vorzüge und Nutzen anhand von Beispielen. General Introduction A number of traditional analysis techniques can be used for the analysis of non-stationary signals and they can roughly be categorised as follows: 1) Divide the signal into quasi-stationary segments by proper selection of analysis window a) Record the signal in a time buffer (or on disk) and analyse after- wards: Scan Analysis b) Analyse on-line and store the spectra for later presentation and postprocessing: Multifunction measurements 2) Analyse individual events in a cycle of a signal and average over several cycles: Gated measurements 3) Sample the signal according to its frequency variations: Order Tracking measurements The introduction of Wavelet Transform (WT), Short Time Fourier Transform (STFT) and Wigner-Ville distribution (WVD) offers unique tools for non-sta- tionary signal analysis. The procedure used is for the time being as described in 1a) above, although in the future faster analysis systems will certainly offer real-time WT and STFT processing. These techniques yield an optimum resolution in both time and frequency domain simultaneously. The general features, advantages and benefits are presented and discussed in this article. The Wavelet Transform is especially promising for acoustic work, since it offers constant percentage bandwidth (e.g., one third octaves) resolution. Traditional spectral analysis techniques, based on Fourier Transform or Digital Filtering, provide a good description of stationary and pseudo-station- ary signals. Unfortunately, these techniques face some limitations when the 2

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signals to be analysed are highly non-stationary (i.e., signals with time-vary- ing spectral properties). In such cases, the solution would be to deliver an instantaneous spectrum for each time index of the signal. The tools which attempt to do so are called Time-frequency analysis techniques. Introduction to the Short-time Fourier Transform and Wavelet Transform The idea of the Short-time Fourier Transform, STFT, is to split a non-station- ary signal into fractions within which stationary assumptions apply and to carry out a Fourier transform (FFT/DFT) on each of these fractions. The sig- nal, s (t) is split by means of a window, g ( t – b), where the index, b represents the time location of this window (and therefore the time location of the corre- sponding spectrum). The series of spectra, each of them related to a time index, form a Time-frequency representation of the signal. See Fig. 1. g(t - b) s(t) time b 941095e Fig. 1. The Short-time Fourier Transform (STFT). The Window, g(t-b) extracts spectral information from the signal, s(t), around time b by means of the Fourier Transform Note that the length (and the shape) of the window, and also its translation steps, are fixed: these choices have to be made before starting the analysis. The recently introduced Wavelet Transform (WT) is an alternative tool that deals with non-stationary signals. The analysis is carried out by means of a special analysing function ψ, called the basic wavelet. During the analysis this 3

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wavelet is translated in time (for selecting the part of the signal to be ana- lysed), then dilated/expanded or contracted/compressed using a scale parame- ter, a (in order to focus on a given range or number of oscillations). When the wavelet is expanded, it focuses on the signal components which oscillate slowly (i.e., low frequencies); when the wavelet is compressed, it observes the fast oscillations (i.e., high frequencies), like those contained in a discontinuity of a signal. See Fig. 2. ψb,a(t) s(t) time b 941096e Fig. 2. The Wavelet Transform (WT). The Wavelet, ψ , extracts time-scaled information from the signal, s(t), around the time b by means of inner products between the signal and scaled (parameter a) versions of the wavelet Due to this scaling process (compression-expansion of the wavelet), the WT leads to a time-scale decomposition. As seen both STFT and WT are local transforms using an analysing (weight- ing) function. Short-time Fourier Transform The Fast Fourier Transform (FFT) was (re)introduced by Cooley and Tukey in 1962, and has become the most important and widely used frequency analysis tool, Ref.[1]. Over the years there has been a tendency to develop FFT-analys- ers with increasing number of spectral lines, i.e., 400 lines, 800 lines and now- adays 1600 – 6400 lines FFTs are on the market. The Brüel & Kjær Multichannel Analysis System Type 3550 analyzer even offers up to 25 600 line Fourier Spectra. 4

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Unfortunately, a large transform is not very suitable when dealing with con- tinuous non-stationary signals, and as a consequence many modern FFT-ana- lysers also offer small transform sizes, e.g. 50 lines and 100 lines. STFT provides constant absolute bandwidth analysis, which is often pre- ferred for vibration signals in order to identify harmonic components. STFT offers constant resolution in time as well as in the frequency domain, irrespec- tive of the actual frequency. The STFT is defined as the Fourier Transform (using FFT) of a windowed time signal for various positions, b, of the window. See Eq. ( 1) and Fig. 1. + ∞ –jπf(t – b) Sb = ∫ s(t)g (t – b)e dt –∞ = 〈s, gb, f〉 (1) jπf(t–b) with gb, f(t) = g(t– b)e This can also be stated in terms of inner products (< >) between the signal and the window, where s is the signal, g is the window, b is the time location parameter, f is frequency and t is time. The inner product between two time- functions, f ( t) and h ( t) is defined as the time integrated (from minus infinity to plus infinity) product between the two time signals, where the second signal has been complex conjungated. Time functions that are real can be converted into complex functions by using the Hilbert Transform. The result is a scalar : + ∞ < f(t), h(t) > = f(t) • h * (t)dt (2) ∫ –∞ Actually, the use of STFT for Time-frequency analysis goes back to Gabor from his work about communications dated 1946, Ref. [2]. In the fifties, the method became known as the “spectrogram” and found applications in speech analysis. The STFT is a true Time-frequency analysis tool. Fig.3 shows the STFT (Transform size, N = 1024) of the response signal of a gong excited by a hammer (the gong is damped by the user’s hand at 120 ms). The modal frequencies are clearly seen and damping properties can be extracted using the decay method. See Ref. [15]. 5

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Fig. 3. A Short-time Fourier Transform of a free vibration decay measurement of a gong. 2Δ f = 50 Hz, 2Δ t = 6.3 ms. A slice cursor can be used to extract the decay curve of the reso- nances for damping calculations Wavelet Transform It was not until 1982 that the Wavelet Transform, WT was introduced in sig- nal analysis by the geophysicist J. Morlet, Ref. [3]. Since then it has received great deal of attention, especially in mathematics. In the nineties we have also seen an increasing interest in the field of sound and vibration measure- ments. The WT is defined from a basic wavelet, ψ, which is an analysing function located in both time and frequency. From the basic wavelet, a set of analysing functions is found by means of scalings (parameter a) and translations (param- eter b). 6

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