·Table of Contents ·Methods and Instrumentation | Denoising Method of AE Signal by Using Wavelet TransformJingRong Zhao Yukuan Ma and Hong GaoJILIN University of Technology, E-mail: myk@jut.edu.cn JingQiu Yang Changchun Huiyuan Test Technology Co.Ltd. P.R.China Contact |
Key words: AE (acoustic emission) signal, wavelet analysis, multi-scale propagation behavior,Demising
The AE technique plays more and more important role in NDT field, especially in material researching, pressure vessel evaluating and intensity watching and measuring for the component of plane. However the research for signal processing technique in AE detection is still a key problem in theory research and engineering application of AE detection technique. In recent years, all digital AE instruments have come out, these provided guarantee on hardware. So it is necessary for us starting from software to find new method for AE signal processing, especially to seek a new digital signal processing way for noise removing. Wavelet analysis is a significant branch of applied mathematics, which has developed in recent years. It is called mathematical microscope, for it can be applied to remove noise from signal, coding and encoding, edge detection, data compress, mode identification and so on. Wavelet analysis has been used in signal processing more and more widely, and it had been used for removing noise from signal in seismic data. Wavelet theory demonstrates that Lipschitz regularity of signal can be calculated its multi-scale behaviors of wavelet transform on modulus maxima figures. When we process signal using wavelet transform, because of the different Lipschitz regularity of signal and random noise, the change rules of the extremum point are different along with the change of scales. So according to this we can remove the extremum point corresponding to noise and then reconstruct signal. Thus purpose of denoising is achieved. In this paper we just take these into account, using wavelet analysis in AE detection, put forward a new algorithm of AE signal denoising.
A.Theory of wavelet transform
Notation:
L^{2}(R) denotes the Hibert space of measurable, square-integrable functions. The function is said to be a wavelet if and only when the following condition is satisfied.
(1) |
There is the dilation of Y (t) by the scale factor s.
In order to be used expediently in practice,s is scattered as discrete binary system ,i.e. Let ,then the wavelet is ,its wavelet transform is
(2) |
Hence its contrary transform is
(3) |
There x(t) satisfies .
Being dispersed in time domain farther, a discrete wavelet transform can be obtained. It exist an effective and fast algorithm
(4) |
(5) |
B. The relation between signal singularity and signal local maxima of wavelet transform modules
Singularities of signal and random noise are different. Their dyadic wavelet transform maximum modules have variable rules by variable scales. A remarkable property of wavelet transform is its ability to characterize the local regularity of functions.
Usually the local regularity of function is often measured with Lipschitz regularity.
Definition 1:
Let n be a positive integer and n<a£n+1. A function f(x) is said to be Lipschitz a, at x_{0}, if and only if there exists two constants A and h_{0}>0 ,and a polynomial of order n, P_{n}(x),such that for h>h_{0}
(6) |
Literature [1] indicated that a function f(x) that is continuously differentiable at a point is Lipschitz 1 at this point. If the derivative of f(x) is bounded but discontinuous at x_{0},f(x) is still Lipschitz 1 and x_{0}, and following Definition 1, we consider that if f(x) is Lipschitz not singular at x_{0}.
Theorem1:
Let f(x)ÎL^{2}(R)and[a,b] an interval of R. Let 0<a<1. For any e>0, f(x) is uniformly Lipschitz a over [a+e,b-e], if and only if for any e>0 there exists a constant A_{e} such that for xÎ[a+e,b-e] and s>0,
(7) |
In common use of dyadic scale of wavelet transform,equations above becomes
(8) |
(9) |
(10) |
(11) |
Due to the wavelet function has a compact support, is a constant,s^{2} is a constant too, the average amplitude of is in the inverse ratio of 2^{j}.
At a given scale 2^{j},wavelet transform is a random process of variable x,if suppose n(x) be gauss noise white,then is a gauss process either. Literature [1] made a deduction of its average density of maximum modules equal to
(12) |
here y^{(1)}(x) and y^{(1)}(x) are first and second derivative of y(x). This shows average density of gauss white noise is in the inverse ratio of scale 2^{j},that is to say the larger scale is ,the sparser its average density is. Hence equations (11),(12) become one of important characteristics to distinguish the maximum modules spread behavior of wavelet transform of signal and noise at different scale.
(12) |
In the equation above, N is noise power set before, J is the largest scale selected, Z is a constant, let it be 2. Using the threshold above one can get rid of the amplitude of maximum modules points under T_{0} on 2^{J}, for considering these maximum modules points dominated by noise.
Fig.2 shows the module maxima of wavelet transform of noisy signal. It shows that at the higher scales, many module maxima disappeared at corresponding lower scales. So we can conclude that these module maxima points are dominated by noise, which should be removed. While the other module maxima points correspond to the transformation of signal, which should be kept. The results of denoising is shown as Fig.3. The original signal and the processed signal are both given in Fig.3. It shows that the influence of noise was decreased greatly and the effect of waveform resuming of AE signal is fine. And the error between the reconstruct signal and the original is in acceptable range.
Fig 2: module maxima of wavelet transform of noisy signal |
Fig 1: noisy signal |
Fig 3: original signal and reconstruct signal |
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