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Advances in Mechanical Engineering
Volume 2013 (2013), Article ID 540172, 9 pages
Adaptive Approach for Boundary Effects Reduction in Rotating Machine Signals Analysis
Key Laboratory of Fiber Optic Sensing Technology and Information Processing, Wuhan University of Technology, Hubei 430070, China
Received 14 March 2013; Accepted 4 July 2013
Academic Editor: Zude Zhou
Copyright © 2013 Hang Su et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Boundary effects are caused by incomplete data in the boundary regions when the analysis window gets closer to the edge of a signal. Various extension schemes have been developed to handle the boundaries of finite length signals to reduce the boundary effects. Zero padding, periodic extension, and symmetric extension are some basic extension methods. However, these solutions have drawbacks. In this paper, we consider the problem of handling the boundary effects due to improper extension methods in the wavelet transform for the application of fault diagnosis of rotating machine. An extension algorithm based on curve fitting with properties that make it more suitable for boundary effects reduction is presented. This extension algorithm could preserve the time-varying characteristics of the signals and be effective to reduce distortions appearing at the boundary. Then, an interpolation approach is used in the boundary effects region to further alleviate the distortions. Procedures for realization of these two algorithms and relative issues are presented. Several experimental tests show that the proposed algorithms are efficient to alleviate the boundary effects in comparison to the existing extension methods.
Wavelet transform analysis has been presented as a time-frequency analysis and processing method for over the past two decades [1, 2]. But it has still received increased attention in recent years [3–5]. Wavelet transform analysis has been widely used for the purpose of denoising, data compression, feature recognition, system nonlinearities detection, and so on [4–7].
The wavelet transform is calculated as shifting the wavelet function in time along the input signal and calculating the convolution of them. In most practical applications, the signals of interest have finite support. As the wavelet gets closer to the edge of the signal, computing the convolution requires the nonexistent values beyond the boundary [8–10]. This creates boundary effects caused by incomplete data in the boundary regions. Since the analysis wavelet extends into a region with no available data at both boundaries of the signal, thus, the results of wavelet transform in these boundary effects regions have questionable accuracy. Actually, the particular impacts of boundary effects become increasingly significant for some systems that may possess longer period sequence and thus require higher frequency resolutions.
To deal with boundary effects, the boundaries should be treated differently from the other parts of the signal. If not properly made, distortion would appear at the boundaries . Two alternatives to deal with boundary effects can be found. The first one is to accept the loss of data and truncate those unfavorable results at boundaries after convolution between signal and wavelet. But simply neglecting these regions in analysis yields to a considerable loss of data which is not allowed in many situations where the edges of the signal contain critical information. The other one is the artificial extension at boundaries before processing signals. In fact, there is another approach that employs the usual wavelet filters for the interior of the signal and constructs different boundary wavelets at the ends of the signal. This method has been shows to be merged into the class of signal extension .
Various extension schemes have been developed to deal with the boundaries of finite length signals [11–14]. Zero padding, periodic extension, and symmetric extension are basic extension methods. Each method has its disadvantages [3, 10]. Computing the wavelet transform of an extension signal is equivalent to using the corresponding boundary wavelets. The boundary wavelets corresponding to zero padding and periodic extension have no vanishing moments at the boundaries. Therefore, the transform values behave as if signals were discontinuous at the boundaries. They introduce a singularity in the signal. And boundary wavelets of symmetric extension have one vanishing moment and avoid the discontinuous at the boundaries. So the symmetric extension introduces a singularity in the first derivation. The wavelets must be symmetric to ensure no distortion in the transform values if the reflection is symmetric. But the Haar wavelet is the only symmetric wavelet with a compact support that has been found so far. One goal of this paper is to seek an extension scheme that preserves the property of vanishing moments.
In addition, these basic extension schemes are usually exploited to the application of data coding which focuses on the procedures of analysis and synthesis using filter banks [15–20]. However, when it comes to particular applications of the fault diagnosis of rotating machine, the above-mentioned methods do not have the ability to recover the significant features. They only make simple assumptions about the signal’s characteristics outside the boundaries. Zero padding assumes that the signals outside boundary are zero; periodic extension assumes that the signals are periodic; and symmetric extension assumes that the signals are symmetric about the boundaries. Many rotating machine signals of interest could not be easily included in the above three categories. So we need a new extension mode appropriate to the requirements of the application of rotating machine signals analysis. In this paper, a new extension mode based on curve fitting technique will be introduced for the rotating machine signals analysis. This extension mode extends signal according to the time-varying characteristics of the rotating machine signals inside of the boundaries so that distortions due to improper extensions could be reduced.
Features appearing near the boundaries of transform values will contain information from outside the support of the rotating machine signal which is synthetic. In other word, the wavelet transform resulting at the boundaries will be affected by the adding data no matter whichever extension mode is employed. Therefore, we will consider the problem from a perspective way that is different from extension method to alleviate these effects. In the paper, we will employ an interpolation processing in the region of the boundary effects to reduce the distortions. It will show that improvement can be obtained by such processing.
The paper is organized as follows. Next section gives a brief review of the boundary effects in the basic extension methods. A general matrix formulation that is common to all signal extension methods is also included. In Section 3, we give depth analysis of the significant importance of smooth extension and present the design method for adaptive smooth extension with properties that make it more suitable than other extensions for nonstationary rotating machine signals analysis. In Section 4, we develop a new algorithm based on interpolation technique for further boundary effects reduction along with some discussion on the implement of this technique. In Section 5, we present the method of testing and the results concerning the performance of the proposed methods applied on both linear and nonlinear rotating machine signals. The performance of proposed adaptive extension method based on curve fitting and interpolation is shown to be superior to all of the other methods. Section 6 summarizes the results obtained throughout the paper.
2. Boundary Effects in the Rotating Machine Signals Analysis Using Wavelet Scalogram
The need for a rotating machine signal time-frequency analysis comes from the incomplete of either time domain or frequency domain analysis to fully describe the behavior of nonstationary rotating machine signals. The time-frequency representation of a rotating machine signal for time-frequency analysis provides information about how the frequency content varies with time, thus providing an ideal approach to examine, analyse, and study nonstationary rotating machine signals. Time-frequency representation is an image of a two-dimensional time-frequency representation mapped from one signal. A number of methods have been developed to obtain the energy distribution function with respect to both the time and frequency. Wavelet transform is one of most notably tools. Wavelets have the great advantage of being able to isolate the fine details in a rotating machine signal. Very small wavelets can be used to identify very fine details in a signal, while very large wavelets can identify coarse details. Wavelet theory is capable of revealing aspects of data that other signal analysis techniques fail to be present like trends, breakdown points, and discontinuities in higher derivatives and self-similarity.
But as mentioned in the previous section, wavelet transform suffers from boundary effects like other signal analysis techniques which involve convolution operation. The boundary effects would lead to serious distortion at both boundaries of signal which makes it hard to distract the right information particularly on the start and the end of signals. Therefore, this section will first explore the effect of basic extension methods, which include zero padding, periodic extension and symmetric extension, on the wavelet transform in order to design a suited extension method that is able to minimize boundary effects.
We start with a general formula of various extension modes. We denote vectors by bold lower case letters. Subscripts , , and represent left, central, and right, respectively. Matrices are denoted by bold upper case letters. We use a subscript, such as , to denote the size of a matrix.
A finite signal with length is , . Then we can express this signal in a form as where and are vectors consisting of the first and last components of the signal. is the central part. Denote the extension vector of as where similarly and are the left and right extension vectors of length . Generalized expression for signal extension methods is given by where is the extension matrix. The basic extension methods are all linear extension. Hence (3) can be written in the form where is an identity matrix; and are respective left and right extension matrices.
For the zero padding extension, the extension matrix is where is an zero matrix. Since for the periodic extension and the extension matrices of the periodic extension are
Similar result is available for the extension matrix of the symmetric extension where (or ) is an exchange matrix where the 1 elements reside on the counterdiagonal and all other elements are zero.
In order to illustrate the boundary effects of various basic methods, we consider a linear frequency modulation signal with constant amplitude and frequency varied with time from 0.1 to 0.4 (normalized frequency). The sampling frequency used is Hz s with 300 data points. We perform different extension methods on the test data and extract the instantaneous frequency from wavelet transform of extension data. The estimation error of the instantaneous frequency obtained from three basic extension methods is shown in Figure 1. The symmetric extension performs better than zero and periodic methods. This is because symmetric extension have one vanishing moment and zero padding and periodic extension have no vanishing moments. This paper is to seek an extension scheme that preserves the property of vanishing moments.
Define the moments of wavelet function as As a consequence of the Fourier transform properties, we can obtain where is the Fourier transform of . If has order multiple zeros at , that is, then we can find
If a wavelet function satisfies (11), then we say that this wavelet function has vanishing moments.
Assume that signal is a polynomial of degree , which is given by where are constant coefficients. Additionally, we assume that has vanishing moments. Equation (11) indicates that
In other words, the wavelet transform of is identical to zero. If can be expanded into a high-order polynomial of degree with , then the terms of the polynomial with degree lower than contribute nothing to the wavelet transform which only reflects the terms with degree higher than (high-frequency component). Such a wavelet has the advantages to capture the high-frequency component and breakpoints of signals. Therefore, is required to have as high as possible vanishing moments so that is smooth at to possess a satisfied band-pass property.
3. Boundary Effects Reduction via Adaptive Smooth Extension
It has been shown that every basic extension method has its own drawbacks. We should seek a method representing the feature of rotating machine signal. Moreover, smooth extension is also critical to the reduction of boundary effects.
3.1. Adaptive Extension Method
In the following, we will investigate a new extension mode which could characterize rotating machine signal better. On the one hand, the signal used in Section 2 is comprised by many harmonic oscillations, and on the other hand, it is very common to use the Fourier series to represent such harmonic oscillations. Thus, the Fourier series can be consider as a new mode to extend signal to preserve the harmonic oscillations.
The Fourier series model is given by where is a constant term, , , and are parameters that needed to be estimated by the curve fitting, and is the number of harmonics in the data.
In summary, the following are the steps of the proposed adaptive algorithm for rotating machine signal extension.(1) Initialize the number of harmonics; for example, set .(2) Main iteration: increment by 1, and apply these steps. Perform data transformations to obtain a linear or simple model.(i)Find the above model parameters to minimize the summed square of residual defined as the difference between the real data value and the fitted response value , producing results , and .(ii)Update the fitted response value using , , and .(iii)If is smaller than some predetermined threshold, stop. Otherwise, apply the iteration.(3) Extend the producing Fourier series to define the data beyond the borders.
3.2. Smooth Extension
From the perspective of convolution operation, the wavelet transform of a rotating machine signal could be interpreted as the output of a system whose unit impulse response is the scale wavelet function where is scale. Let us consider a low-pass (smooth) function . Set
We use and as the mother wavelets. Then computing the first derivative of a rotating machine signal after smoothing is equivalent to processing this signal using the first derivative of the smooth function . Similarly, computing the second derivative of a signal after smoothing is equivalent to processing this signal using the second derivative of the smooth function . This result can be generalized to the higher order. Mathematically, a point of a function with zero first derivative corresponds to extreme value while zero second derivative corresponds to inflection point. Hence, the wavelet transform is able to reflect the extreme and inflection points of a rotating machine signal if the wavelet is original from a smooth function.
An improper extension maybe results in extra transient component referred to singular points which are defined as points with derivative on the right and the left with different signs, that is, the points at which their derivative is discontinuous or not defined and finding the amplitudes of the jumps. In other words, singular points represents the extreme and inflection points presenting in the rotating machine signal. It is easy to obtain that the singular points of rotating machine signals are indicated by the amplitudes of their wavelet transform, that is, zero-crossing points or maximum points of the transform. In the case of rotating machine signals extension, an extreme point due to extension would lead to zero point or very small value in the wavelet transform at the corresponding location. More ordinary case is that extension introduces a step at the boundary leading to very large wavelet transform amplitude. For example, the result that wavelet transform of the rotating machine signal using function is very large indicates the inflection point of .
An unsmooth extension at or leads to wavelet transform modulus maximum at the same points which is the reason of distortion. Hence, we should select an extension mode that is as smooth as possible at the boundaries to avoid distortion.
4. Boundary Effects Reduction via Interpolation
Whichever extension method is employed to reduce the boundaries distortion phenomenon, the extension parts would definitely affect the analysis results which are determined by both original and extension rotating machine signals. If the extension parts do not properly reflect the trend of the original rotating machine signal, it will fail to produce satisfactory or perfect results. Nevertheless, it is well known that the rotating machine signals in the application of time-frequency analysis are usually random and it is difficult to estimate the past and future of the rotating machine signals based on the present data. Hence, this problem should be seen from a perspective that is broader than devising a convenient extension for the rotating machine signal. Apart from the Fourier series extension method, an additional goal of this paper is to propose an approach to shorten the width of the boundary effect region. This approach is based on interpolation in the boundary effect region to reduce the boundary effects.
Figure 2 explains the principle of reduction of boundary effects using interpolation method. Without interpolation, the convolution is computed between wavelet and data with length , from to .
After interpolation, the convolution is still computed between wavelet and data. However, the end point of these data has become to if is odd. As shown in Figure 2(b), the length of wavelet is the half of that before interpolation and becomes shorter compared to the original rotating machine signal. It is easy to show that the boundary effects region which is decided by the length of wavelet also becomes shorter in consequence of the interpolation procedure. Therefore the boundary effects are alleviated by exploiting interpolation.
In practical applications, we only require employing interpolation in the boundary effects region to obtain good results without heavy computational burden. Interpolation can be considered as an expansion of the extension method towards the interior of rotating machine signal. Comparing with extension methods, it is easier and more accurate to estimate the rotating machine signal value between two points than predict the data from the view of probability.
4.1. The Range of Interpolation
Performing interpolation processing at the boundaries could further reduce boundary effects. In the implement, we should consider the range of interpolation processing, that is, how long of the rotating machine signal should be involved in the interpolation processing. A range which is too long or too short would yield an expensive computation or an inaccurate result. We first discuss the range of interpolation and then explore how to determine it in practice.
Let us assume that rotating machine signal has a singular point at . It is obvious that the singular point at will not impact the whole time-scale plane but only the neighborhood of . We refer to it as cone of influence of .
The range of interpolation depends on the cone of influence. For the sake of simplicity, suppose that the wavelet that we use has a support . Then the scaled wavelet has support . We define the cone of influence as the set of points containing in the support from the whole time-scale plane. Thus, the cone of influence of is
In the cone of influence, the performance of wavelet transform is impacted by the singular point introduced by extension. We refer the cone of influence as the region of boundary effect where the interpolation processing should be performed. It is noticed that the range of interpolation is proportional to the scales factor . Figure 3 illustrates the length of interpolation required at different scale on the time-scale plane.
4.2. The Implement of Interpolation
Based on the previous discussion, algorithm for the interpolation processing for boundary effect reduction can be summarized as follows.(1)Obtain the wavelet transform of the rotating machine signal.(2)We find the scale corresponding to the maximum amplitude of the wavelet transform at .(3)The range of interpolation processing is determined by the scale from the above step. Figure 3 shows a different range of interpolation processing at different scales.
5. Numerical Examples
In order to validate the results given in Sections 3 and 4, we present the numerical examples of the proposed algorithms. The performance of the proposed methods has been assessed by means of tests on generic synthetic rotating machine signals. The purpose of the test is to establish the measurement accuracy of the proposed methods as well as their advantages in boundary effects reduction over the basic methods. The test consists of two parts which involve the proposed extension method and interpolation preprocessing. Two rotating machine signals exhibiting linear and nonlinear instantaneous frequency laws are used for evaluating the performance of the algorithms.
5.1. Performance Assessment of Fourier Series Extension
First consider that the linear rotating machine signal was presented in Section 2. Some results that illustrate the performance of the Fourier series extension in the instantaneous frequency estimation are shown in Figure 4. For comparison purposes, the extension algorithm is compared with symmetric extension which is superior over the other two basic methods. It is apparent that the results provided by the Fourier series extension method are in better agreement with the theoretical values in Figure 4(a). As we have done in the Section 2, the error between theoretical and estimated wavelet ridges are shown in Figure 4(b) to illustrate the effect of Fourier series extension. It can be observed that Fourier series extension has less singularity appearing at the boundary than symmetric extension.
In the nonlinear case, we consider a logarithmic frequency modulated rotating machine signal with same samples as the linear case, and its instantaneous frequency is given by
We set , , and . The total rotating machine signal length and sample period used are and . The proposed algorithm is successfully applied on this nonlinear rotating machine signal. Figure 5 illustrates the results provided by the Fourier series extension and symmetric extension applied on the nonlinear rotating machine signal. It can be seen that the results are similar to the results of linear rotating machine signal. Figure 5(b) shows that the Fourier series extension is indeed efficient to reduce boundary effects for complicated rotating machine signals with time-varying instantaneous frequency laws.
5.2. Performance Assessment of the Interpolation Method
Several tests have been conducted in order to assess the capability of the proposed interpolation preprocessing to further reduce the boundary effects. All the basic extension methods and the Fourier series extension will be considered in this section. The performance of the interpolation preprocessing is examined using the same two classes of rotating machine signals.
We plot the comparison results of symmetric extension and the Fourier extension with and without interpolation processing on the linear rotating machine signal and logarithmic rotating machine signal in Figures 6 and 7, respectively. It can be clearly observed that the interpolation processing is indeed able to further reduce the distortion resulted from the boundary effects for both rotating machine signals.
To show the effect of the interpolation preprocessing, compression test results of the linear rotating machine signal for different extension methods with and without interpolation processing are provided in Table 1. MSE is the mean square error. All of the values are calculated from true normalized frequency and estimating normalized frequency. In order to display the performance of boundary effects reduction, we only calculate the data from the boundary part. The number of data participating to the calculation is 50 points.
It can be concluded that the Fourier series extension with the interpolation processing provides the best performance among the four methods mentioned in this study. Furthermore, the interpolation processing is able to produce a better accuracy of the time-frequency characteristics estimate no matter which extension method is applied.
In this paper, we have investigated the problem of dealing with the boundary effects that would arise in the analysis of rotating machine signals. Basic methods including zero padding, periodic extension, and symmetric extension were shown to provide unsatisfied performance to reduce the boundary effects. We derived a generalized expression for various extension methods. The relationship between smooth and the boundaries effect has been stressed. A smooth extension scheme using the Fourier series to avoid distortion appearing at the boundaries was proposed. This extension technique possesses the property of preserving the harmonic oscillations of the time-varying signal that makes it more suitable than the other methods for the time-frequency analysis application. A new algorithm based on interpolation technique was proposed from new perspectives to further reduce the boundary effects. It has been shown that the range of interpolation is determined by the scale factor which maximized the amplitude of the wavelet transform at the boundaries. By comparing the results of the analysis, it has been shown that the adaptive smooth extension with the interpolation processing provided the best performance in the study. Although we have restricted the analysis to the wavelet transform, the proposed methods can be applied on any time-frequency distributions that involve convolution operation.
This work is supported by the National High Technology Research and Development Program (863 Program) of China under Grant no. 2012AA041203.
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