ˇTable of Contents ˇMethods and Instrumentation | Identification and Analysis of Acoustic Emission Signals by Cohen's Class of Time Frequency DistributionLubo Pazdera, Jaroslav SmutnýDepartment of Physics and Department of Railway Construction and Structures, Faculty of Civil Engineering, Technical University of Brno, Czech Republic Contact |
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:
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.
(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 C_{x} (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 A_{x}(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 A_{x}(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 |
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.
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.
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