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## Identification and Analysis of Acoustic Emission Signals by Cohen's Class of Time Frequency Distribution

Lubo Pazdera, Jaroslav Smutný
Department of Physics and Department of Railway Construction and Structures, Faculty of Civil Engineering, Technical University of Brno, Czech Republic
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### Abstract

This article describes application of the acoustic emission method on real acoustic emission signals of a building structure. The part of this contribution demonstrates also some lesser-known methods for creating functions that represent the energy of the signal simultaneously in time and frequency.

### Introduction

The Acoustic Emission method is the technique from the non-destructive testing methods. It detects active defects into a test specimen. These failures inside material rise in consequence of local strain. However, an acoustic emission signal is strongly non-stationary.

There are many mathematical techniques for analysis of acoustic emission signals, but not all of them provide satisfactory results. Given a time series, one can readily see how the "energy" of the signal is distributed during time. By performing a Fourier transform to obtain the spectrum, one can readily see how the "energy" of the signal is distributed in frequency. For a stationary signal, there is usually no need to go beyond the time or frequency domains.

But most real acoustic emission signals have characteristics that change over time, and the individual domains of time and frequency do not provide the means for extracting this information [4].

Time-frequency representation combines time domain and frequency domain analyses to yield a potentially more revealing picture of the temporal localization of signal's spectral characteristics. Time-frequency representation may be divided into two groups by nature of their transform:

• Linear methods (including the Short-time Fourier transform and the Wavelet transform)
• Quadratic methods (of which the Wigner-Ville transform or distribution is fundamental)

The main disadvantage of linear time-frequency transform is that the time frequency resolution is limited to the Heisenberg bound.

Quadratic methods present the second fundamental class of time frequency distributions. Quadratic methods are based upon estimating of an instantaneous power spectrum (or energy) using a bilinear operation on the signal itself. Well, an alternative approach to time-frequency representation is by means a quadratic (or bilinear) transform. The class of all quadratic time-frequency distributions to time shifts and frequency-shift is called Cohen's class.

Cohen's class transform seems to be one of acceptable analytic techniques for classification and identification acoustic emission sources (place with plastic deformation) from acoustic emission hits.

### Cohen's class of time frequency distribution

Professor Cohen generalised the definition of the time frequency distributions in such a way so as to include a wide variety of different distribution [1]. These different distributions can be represented in several ways. Cohen's class definition like the Fourier transform, with respect to t , of the generalised local correlation function is most frequent. With a two-dimensional kernel, the bilinear time frequency distribution of the Cohen class is defined according to the equation:

 (1)

where x is the signal, t is the time, t is the time location parameter, w is angular frequency, q is shift frequency parameter, y (q , t ) is called the kernel of the time frequency distribution. The distribution Cx (t,w , y ) from Cohen's class can be interpreted as the two-dimensional Fourier transform of a weighted version of the ambiguity function of the signal

 (2)

where R(t, t ) is local correlation function and Ax(q , t ) is the ambiguity function of the signal x(t), given by equation:

 (3)

We note that all integrals run form -¥ to ¥ . The weighted function y (q , t ) is called the kernel. It determines the specific properties of the distribution. The product Ax(q , t )× y (q , t ) is known as the characteristic function [2, 3]. Since the ambiguity function is a bilinear function of the signal, it exhibits cross components, which, if allowed to pass into time frequency distribution, can reduce auto-component resolution, obscure the true signal feature, and make interpretation of the distribution difficult.

Therefore, the kernel is often selected to weigh the ambiguity function so as that the auto-components, which are centred at the origin of the (q , t ) ambiguity plane, are passed, while the cross-components, which are located away from origin, are suppressed. That is, in order to suppress cross-components y (q , t ), which should be the frequency response of a two-dimensional low-pass filter. When a low pass kernel is employed, there is a trade-off between cross-components suppression and auto-component concentration. Generally, as the pass-band region of the kernel is made smaller, the amount of cross-component suppression increases, but at the expense of auto-component concentration. There are definitions of the kernels for various time frequency distributions in the table 1.

 Distribution Kernel function y(q, t) Rihaczech Page Choi-Williams Parameter s   controls the cut off frequency of the filter Margenau-Hill Table 1: Definitions of the kernels for various time frequency distributions

### Using of the Cohen's class of time frequency distribution to the practical analysis

Described time-frequency analyses were used to test acoustic emission signals generated by thermal load of isolation building materials. The tested Dekalit had the density, 851 kg× m-3, and the form of blocks of dimensions: length, 16 cm, width, 4 cm, height, 1.2 cm. The sample was clamped at one side into a heat source of adjustable power input up to 500 watts. Temperature sensors were disposed on the sample symmetrically on their spacing being 120 mm. The acoustic emission sensor was placed in the middle of the sample.

The temperature was indicated by two Ni-5000 B-1k nickel resistance temperature sensors (manufactured by Tesla Lanskroun, Czech Republic). They were placed on the ends of the sample. The acoustic emission signal was measured by B&K 4344 accelerometers (Bruel&Kjear, Denmark). Except for the heating set-up and turn off the whole of the measurement was controlled by a computer. Data acquisition was implemented using a Datalab DL-912 Transient Recorder (England).

The measurement evaluation was primarily concentrated on the response, i.e., the impulse frequency in the acoustic emission signal and the length variations as induced by the specimen temperature gradient.

The evaluation of the measurement results shows that the acoustic emission method is a suitable tool for non-destructive studies of the effect thermal processes, which can have influence on building materials.

### Analysis of experiment

In the Fig. 1 there is temperature history of the both ends of the tested sample. The place marked "hit" means that the acoustic emission burst type signal was recorded at temperature 140 oC of the heated end and 23 oC of the free end of the tested sample. The sampled time was 31 minutes. From many recorded acoustic emission hits there is shown this hit.

 Fig 1: Time history of temperature on the free and heated end. In the place marked "hit" there was recorded an acoustic emission burst signal

There are three charts in Fig. 2 and Fig. 3. The upper chart shows time history of measured voltage on the acoustic emission sensor. Frequency depended on the power spectral density, which was computed from the whole time history of the signal, which is in the left chart. Values of spectral density are recomputed to maximal values and displayed as attenuation in deci-Bell (dB). Thus the maximal spectral density value is 0 dB as common in many professional acoustic emission instruments. The biggest chart shows the time-frequency distribution of the signal computed by Page distribution (see Fig. 2) and Margenau-Hill distribution (see Fig. 3). We note that values of power spectral density in time frequency (3D) chart are displayed by contours.

 Fig 2: Page distribution of the tested acoustic emission signal

 Fig 3: Margenau-Hill distribution of the tested acoustic emission signal

The Page distribution (see Fig. 2) determines significant frequency components on the frequency range from 10 kHz to 30 kHz, 40 kHz, 75 kHz and 95 kHz. The low frequencies component is contained into signal for longer time than the other frequency components. From significant frequencies the longest time is offered to frequency component 30 kHz.

The Margenau-Hill distribution (see Fig. 3) determines significant frequency components on the frequency range from 10 kHz to 30 kHz, 40 kHz, 75 kHz and 95 kHz. The low frequencies component is contained into signal for longer time than the other frequency components. From significant frequencies the longest time is offered to frequency component 30 kHz.

Both time-frequency distributions give similar results of the spectrum. There is more information than the frequency distribution and time history can give.

### Conclusions

It is possible on the base of executed analyses of measured signals and by comparison of used methods of time-frequency analyses to form the following conclusion:

• Methods of time-frequency analyses enlarge the information about the given technical occurrence by stating the time localisation of frequency components, i.e. they determine the size of power spectral density by appropriate frequencies at the given moment.
• Measurement and analysis of non-stationary signals with the use of time-frequency methods supply the new view to transfer and non-stationary characteristics by the acoustic emission measurement.
• From stated mathematical means of signal analysis it is possible to use for time localisation of occurrence of frequency components stationary and non-stationary signals of linear or quadratic distribution.
• The Cohen's class which gathers all the quadratic time-frequency distributions covariant by shift in time and in frequency, offers a wide set of powerfull tools to analyse non-stationary signals. The basic idea is to devise a joint function of time and frequency that describes the energy density or intenzity of a signal simultanneously in time and in frequency. The most important element of this class is propably the Wigner-Ville distribution, which satisfies many desirable properties. Rihaczek, Page and Choi-Williams distributions present next methods of the time frequency analysis applicable in valuation of acoustic emission very well.
• Unlike linear transform, such as the Short time Fourier transform and Wavelet transform, bilinear time frequency distributions can provide simultaneous high resolution in both the time and frequency domains.

### Acknowledgement

This research has been supported by the research project CEZ J22/98 No.~261100007 ("Theory, reliability and mechanism of failure statically and dynamically loaded building construction") and by Grant Agency of Czech Republic No~103/97/P140 "Study, Analysis and Evaluation of Acoustic Emission Signals Applied on Thin-Wall Systems".

### Literature

1. Cohen L.: Time-frequency distributions - a review, Proc. IEEE, vol. 77, no. 7, pp. 941-981, July 1989
2. O'Neill J.C.: Quadratic Functions for Time-Frequency Analysis with Applications to Signal Adaptive Kernel Design, SPIE - Advanced Signal Processing Algorithms, 1997
3. O'Neill J.C., Williams W.J.: Distributions in the Discrete Cohen Classes, Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP, vol. 3, pp. 1581-1584, 12-15 May 1998, Seattle, WA
4. Smutny J.: Analysis of Vibration since Rail Transport by using Wigner-Ville Transform, TRANSCOM 99 - 3-rd European Conference of Young Research and Science Workers in Transport and Telecommunications, Zilina, June 1999, Slovak Republic, pp. 101-104, ISBN 80-7100-616-5

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