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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

A least square method based on data fitting is proposed to construct a new lifting wavelet, together with the nonlinear idea and redundant algorithm, the adaptive redundant lifting transform based on fitting is firstly stated in this paper. By variable combination selections of basis function, sample number and dimension of basis function, a total of nine wavelets with different characteristics are constructed, which are respectively adopted to perform redundant lifting wavelet transforms on low-frequency approximate signals at each layer. Then the normalized ^{P}

Roller bearings are among the most commonly used components in modern production facilities. Breakdowns caused by running wear and inappropriate operation will not only lead to huge economic losses for enterprises, but potentially to serious casualties. Therefore, in order to avoid the occurrence of accidents, state monitoring and effective state feature extraction of roller bearings is of great importance.

With the fast development of modern signal processing technology, many theories and algorithms have been applied in signal analysis and feature extraction. Wavelet transform, due to its multi-resolution analysis, is being widely used in the processing of various complex non-stationary signals, while the lifting algorithm, which apparently is much superior, further propels the research and engineering applications of wavelet analysis.

In 1996, the new idea of the lifting algorithm was first presented by Sweldens [

Based on the existing theoretical and applied research on lifting algorithms, the adaptive redundant lifting wavelet transform based on data fitting is proposed here and the paper is organized as follows: in Section 2, a new way to construct lifting wavelets with variable characteristics based on the least square method of data fitting is specifically introduced, and the adaptive redundant lifting wavelet analysis is then presented as well. Section 3 discusses the power spectrum estimation, which is used to select the optimal node-signal for further processing. In Section 4, an experimental case and a practical one are respectively given to test the effectiveness of this proposed method. Finally, some conclusions are drawn.

In this section, a new way for wavelet construction based on data fitting is introduced first. Then different wavelets constructed by this method are used for adaptive redundant lifting wavelet analysis.

Since the proposition of the lifting scheme, how to improve the characteristics of wavelets through the design of lifting operator based on existing biorthogonal filter has been extensively researched. Though some progress has been made, however, the most commonly method at present is to construct symmetrical wavelets by an interpolation algorithm, so the question remains: are there any other ways to make the construction of asymmetrical wavelets or wavelets with special characteristics through the design of lifting operators more flexible and simpler to do, in order to satisfy the demands of practical applications?

When studying the new-sample prediction problem in the process of interpolating subdivisions in 2000, Sweldens

Choosing a function _{1}, _{2},…_{M}_{1}, _{2},…_{M}, M_{0},_{1},⋯, _{N}} denote the function class, then the fitting function is obtained as:
_{k}_{j}_{L}

In _{Ñ}_{L̃}_{L} ≥ _{L̃}

Both the process of keeping one sample out of two in the Mallat algorithm for classical wavelet and the split step in lifting wavelet transform are a down sampling process which will result in the decrease of both samples and the contained information with an increase of the decomposition layer; in addition the sampling frequency of the wavelet coefficients will no longer satisfy the Nyquist theorem, causing frequency aliasing and non-translational invariance. To solve all these problems generated during down-sampling and also control the computational complexity, Holschneider

Suppose the original signal is _{j}_{j}

Predict:

Update:

The reconstruction process is divided into three steps: undo update, undo predict and merge. The first two steps are quite simple: with the direction of signal flow and operators during the decomposition reversed. The merge step refers to taking the average of _{u}_{p}

One of the major applications of wavelet analysis is to utilize its excellent local characterization of signals both in the time domain and the frequency domain, to well capture the transient components in non-stationary signals. However, the wavelet transform is still a kind of “basis function” operation. For various complex components in signals, a wavelet is not always the optimal one to capture all the transient components, so the following question is raised: is it possible to select different wavelets according to specific features of various components in signals for analysis? For a classical wavelet, it's known from two-scale equation that the analyzing wavelet is obtained by dilation and translation of the mother wavelet, while the wavelet construction via the lifting algorithm is totally completed in the time domain and independent of Fourier transform, which just provides a solution to the above problem. In 1997, Claypoole

In light of the flexibility and convenience for wavelet construction by using algorithms based on data fitting and also according to the linear idea stated above, it is considered here to select different wavelets for approximate signals that need to be decomposed to complete the lifting wavelet transform, thus the adaptive redundant lifting wavelet analysis is proposed.

From _{k}

In this paper, a total of three basis functions,

Algebraic: _{k}^{k}, k

Transcendental expression: _{k}^{k}^{(0.1·(k+1)·x)},

Transcendental expression: _{k}^{k}^{(0.1·(k+1)·x)} · cos(0.1 · (

Combinations for the selected (_{Ñ}

From the above figures it can be shown that, with the increase of

Next, the above nine wavelets with different characteristics are chosen to perform the adaptive redundant lifting analysis of the experimental signals and engineering signals, respectively. The objective function is built on the decomposition results at each layer to determine the optimal wavelet that best matches the features of the approximate signals.

There are many ways to represent signals. How to realize more effective signal representation?

In this paper, ^{P}^{P}

In subsequent signal analysis, the value of ^{P}_{i,j}_{i,j}_{i,j}_{i,j}_{+1} and _{i,j}_{+1} are obtained. For comparison, normalized ^{P}_{i,j}_{+1} and _{i,j}_{+1} are calculated:

^{P}_{i,j}_{i,j}

When wavelet analysis is applied in signal denoising or compression, it usually takes the low-frequency approximate signals and high-frequency detailed signals that result after decomposition and some processing to make the complete reverse reconstruction. But instead of that, if some node-signal can be used for single branch reconstruction according to the need, then the filtering characteristic of the wavelet transform can be well reflected and adopted. Concrete steps for signal branch reconstruction of node-signal in adaptive redundant lifting wavelet transform are as follows:

Low-frequency approximate signals _{i, j+1}:

High-frequency detailed signals _{i, j+1}:

_{opt, j}

_{opt, j}

_{i,i}

_{i,j}

_{+1}and

_{i,j}

_{+1}are obtained, respectively.

When damage occurs to roller bearings, the collision between the damaged spot and the surface of roller bearing component will cause an impulsive pulse force which leads to unilateral vibration attenuation AM signals, together with the resonance phenomenon, resulting in the high frequency peaks with concentrated energy appearing in spectrum. Since in the wavelet transform, different node signals after decomposition correspond to different frequency ranges, it's easy to come up with the idea that the node-signal to which the frequency ranges with high frequency peaks falling are corresponding should be selected for single branch reconstruction and further demodulation, so that the characteristic components containing failure information can be more effectively extracted.

As the decomposition continues, the number of node-signals obtained also gradually increases. Then which node-signal should be selected for single branch reconstruction so it contains the information of the frequency range with the high frequency peaks? Since each node-signal is according to different frequency range, the power spectrum estimation is proposed here.

For a signal

Fourier transform on

Square of the amplitude of

From the energy concentration characteristics of high frequency peaks it can be seen that the node-signal with maximum power should be selected for single branch reconstruction. Step (2) indicates that power spectrum estimation is associated with signal length. If node signals _{i,j}_{i,j}_{i,j}_{i,j}_{i,2} or _{i,2}, which is very unreasonable and also violates the original intention of selecting the node-signals by power spectrum estimation.

Therefore, in order to solve the problem discussed above, the original signal is used directly for power spectrum estimation, which consists of two steps:

Based on the rules for frequency ranges division of node signals in the wavelet transform, the original signal is taken for subsection power spectrum estimation. When the decomposition is carried out

All the subsection powers are taken for comparison to determine the maximum and then its corresponding frequency range, whose corresponding node signal is finally selected for single branch reconstruction. Let the analysis frequency of _{s}_{s}^{j}_{i,j+1} will be selected, while if it falls within the interval [_{s}^{j}, f_{s}^{j−1}], then node signal _{i,j+1} will be selected. For easy analysis, subsection power spectrum estimation is done in the order of [_{s}^{j}, f_{s}^{j}^{−1}], [0, _{s}^{j}_{s}^{j}^{+1}, _{s}^{j}_{i,j+1} and 2_{i,j+1}.

In sum, the steps for failure feature extraction algorithm of roller bearings stated in this paper are as follows:

Nine wavelets constructed based on data fitting are adopted to complete the redundant wavelet lifting transform on the signal that's to be decomposed;

Normalized ^{P}_{i,j+1} and high-frequency detailed signal _{i,j+1} which are obtained through decomposition are calculated, in order to determine the optimal wavelet which best matches the features of decomposed node signal _{i,j}

The original signal is taken for subsection power spectrum estimation, then the node-signal corresponding to the frequency range of the maximum subsection power is selected for single branch reconstruction;

Hilbert demodulation analysis on the signal resulting from single branch reconstruction is conducted, in order to extract the characteristic information of early roller bearing failure.

Meanwhile, the result obtained by the method presented here, the spectrum of the original signal and the Hilbert demon spectra are compared and analyzed, just to verify the superiority of this new method.

The failure signals collected from a bearing test-bed and a field measurement signal from a steel plant are selected, respectively, for analysis to prove the feasibility and effectiveness of the new method.

The signal of a roller bearing with failure collected from a bearing test-bed is taken for processing and analysis using the method proposed in this paper. The diagram of the bearing test-bed is shown below (

In the diagram above, at the left end O is a motor, by which the shaft is driven through coupling C to bring about the rotation of three rotors R1, R2 and R3. Ends A and B are the bearing seats, while bearings with different failures can be easily replaced at end B. In the experiment, bearings of type 6307 with failures on the rolling element were mounted at end B and a sensor was vertically placed over end B to collect signals of vibration acceleration through a data collector. During the acquisition process, the motor speed was 1,496 rpm, the sampling number was set at 8,192 and the sampling frequency was 15,360 Hz. From the above parameters, the fault characteristic frequency of rolling element _{roller}

Firstly, time domain analysis and spectrum analysis were performed on the signal, with the results shown in

Next, a three layer adaptive redundant lifting wavelet transform was performed on the above vibration signal, with ^{P}_{i, j}

In ^{P}^{35}), respectively. Thus, the optimal wavelet which best matches node-signal _{i,j}_{i}_{,1}: wavelet constructed by 3, (6,5); (2) _{i}_{,2}: wavelet constructed by 3, (6,5); (3) _{i}_{,3}, wavelet constructed by 1, (8,7). From the results it can be seen that the wavelet constructed by 3, (6,5) is the optimal one which best matches both node signals _{i}_{,1} and _{i}_{,2}. Compared with the most widely used symmetrical lifting wavelet, the asymmetrical lifting wavelet is more suitable for feature matching of node signals.

The results of subsection power spectrum estimation of the original vibration signal are presented in

It can be seen from _{i,3} is selected for single branch reconstruction and Hilbert demodulation. Meanwhile, the local spectrum and Hilbert demon spectra of original signal are obtained for comparative analysis:

From

No characteristic information related to _{roller}

From the demon spectra, frequency component 298.1 Hz can be detected, which is very close to the triple frequency 297.667 Hz of _{roller}

_{roller}

From the above analysis it can be seen that the new method presented here is more effective in roller bearing fault feature extraction.

To verify the effectiveness of this new method in engineering applications, we take a vibration signal of a high speed wire finishing mill in a steel plant for analysis. The driving chain of the high speed wire finishing mill is shown in

In ^{2}. To detect the early equipment fault of the vibration acceleration signal at this measuring point the signal at 5 am on July 1, 2009 is selected for analysis. Related parameters are: the motor speed is 883 r/min; the sampling frequency is 10,000 Hz and the sampling number is 2048.

First, time domain analysis and spectrum analysis are performed on the signal, respectively, with the results shown in

Next, the adaptive redundant lifting wavelet transform is used to decompose the vibration acceleration signal into three layers. The ^{P}_{i,j}

^{P}^{29}), respectively Thus, the optimal wavelets which best matches node signal _{i,j}_{i,1}: wavelet constructed by 3, (4,3); (2) _{i,2}: wavelet constructed by 3, (6,5); (3) _{i,3}: wavelet constructed by 3, (6,65). It's easy to see that all the optimal wavelets which best match node signals _{i,1}, _{i,2} and _{i,3} are constructed by basis function (3). Compared with the most extensively applied symmetrical lifting wavelet, the asymmetrical lifting wavelet better matches the features of the node signals.

Then the subsection power spectrum estimation is performed on the vibration acceleration signals at the measuring points. The results are: _{i,4} is selected for single branch reconstruction and Hilbert demodulation. At the same time, the local spectrum and Hilbert demon spectra are acquired for comparison:

Through comparative analysis, the following results can be obtained:

In

In the local spectrum, the frequency component at 439.5 Hz is detected;

Only frequency component 146.5 Hz is detected in the demon spectra.

The 146.5 Hz frequency component extracted by the method discussed in this paper is very close to 145.695 Hz which is the fault characteristic frequency of the rolling element of the roller bearing that's at the north output end of the speed increasing box (marked in a red circle in

The results indicate that the method proposed here can also extract fault features of roller bearings in the industrial production field more effectively.

In this paper, a least square method based on data fitting is proposed to construct new lifting wavelets, and combined with the nonlinear idea and redundant algorithm, the adaptive redundant lifting wavelet transform based on fitting is presented for the first time. Through the steps of redundant lifting wavelet decomposition, determination of optimal wavelet for node-signal at each layer, subsection power spectrum analysis of original signals, single branch reconstruction of node signals and demodulation, the weak fault features of roller bearing are finally extracted. Then this method is applied in the analysis of experimental signals and engineering signals, respectively, and the spectrum analysis and demodulation analysis are done for comparison. Results indicate that this method can not only detect the roller bearing faults of a bearing test-bed, but also identify well the early faults of bearings in an industrial production field. Compared with spectrum analysis and demodulation analysis, this method is superior in roller bearing fault feature extraction. Furthermore, the selection results of the optimal wavelet for each node-signal determined by a normalized ^{P}

This work is supported by National High Technology Research and Development Program 863 (Grant No. 2009AA04Z417), National Natural Science Foundation of China (Grant No.51075023), Beijing Natural Science Foundation (Grant No. 3112004), and Beijing Key Laboratory of Advanced Manufacturing Technology and SRF for ROCS, SEM.

Wavelet constructed with basis function (1) and different selections of (

Wavelet constructed with basis function (2) and different selection of (

Wavelet constructed with basis function (3) and different selection of (

Diagram of the bearing test-bed.

Analysis of signals of vibration acceleration in the case of rolling element failure: (

Comparative analysis on feature extraction of vibration acceleration signal in the case of rolling element failure: (

Diagram of driving chain of high speed wire finishing mill.

Analysis of vibration acceleration signals at the selected measurement point. (

Comparative analysis on feature extraction of vibration acceleration signals at the measurement points (

Schematic diagram of the damaged bearing on axle I at the north output end of the speed increasing box: (

Normalized ^{p}^{35}).

| |||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2.2867 | 2.2915 | 2.2862 | 2.2803 | 2.2952 | 2.2733 | 2.2769 | 2.2879 | 2.2805 |

2 | 2.3003 | 2.3011 | 2.2891 | 2.2983 | 2.2902 | 2.2663 | 2.2941 | 2.2896 | 2.2894 |

3 | 2.2915 | 2.2925 | 2.2935 | 2.2773 | 2.2980 | 2.2886 | 2.2762 | 2.2958 | 2.2926 |

Subsection power spectrum estimation of experimental signals (×10^{4}).

Power | 13.763 | 19.180 | 35.346 | 3.0143 | 2.9236 | 3.1050 |

Normalized ^{P}^{29}).

| |||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 8.7626 | 8.7422 | 8.6927 | 8.7837 | 8.8049 | 8.7175 | 8.7944 | 8.7780 | 8.8093 |

2 | 8.7863 | 8.8287 | 8.8075 | 8.8101 | 8.8390 | 8.7842 | 8.7894 | 8.8398 | 8.8048 |

3 | 8.7674 | 8.8260 | 8.7363 | 8.7620 | 8.7321 | 8.6784 | 8.7261 | 8.7538 | 8.7583 |

Subsection power spectrum estimation of engineering signals (×10^{4}).

Power | 2.3492 | 17.130 | 9.7421 | 24.518 | 32.812 | 16.223 |