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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 nonlinear redundant lifting wavelet packet algorithm was put forward in this study. For the node signals to be decomposed in different layers, predicting operators and updating operators with different orders of vanishing moments were chosen to take norm ^{p}

With the continuous changes and improvements of modern science, technologies and industries, various kinds of mechanical equipments are developing rapidly towards the trend of large scale, high precision, high speed and automation. With the increasingly meticulous design and manufacturing of equipment, the social and economic benefits created by them have also been accumulating. However, the normal equipment operation inevitably causes the dissipation of components, the long-term accumulation of which will eventually cause the failure of the whole equipment. Due to the difference in the production and assembly of equipments as well as the complexity of the operation environment, there are generally many uncertainties during the operation. In order to prevent the occurrence of accidental faults and avoid the resulting severe consequences, the investigation and application of advanced signal processing techniques and the achievement of the effective monitoring of equipment status is of great practical significance.

Equipment in operation generally exhibits nonlinear engineering characteristics, so the wavelet transform has been widely applied to the fault diagnosis of equipment owing to its multi-resolution analysis feature. However, conventional wavelet functions are generally constructed in the field of mathematics and have difficulties in fitting with practical engineering signals; besides, wavelet functions in different scales are acquired from mother wavelets after scaling and translation. Therefore, once a wavelet function is chosen, identical filter groups are employed both within one scale and among different scales, which suggests a lack of flexibility and certain limitation in capturing variable information.

In 1996, Sweldens from Bell Laboratories proposed a lifting framework to construct compactly supported wavelets and dual wavelet functions in the time domain. Wavelet functions with expected characteristics could be obtained with the design on the lifting operator based on prior initial biorthogonal filter groups, e.g., increasing the orders of vanishing moments of wavelets or making wavelet functions to approximate specific waveforms [

The paper is organized as follows: Section 2 presents the implementation of the nonlinear redundant lifting algorithm in detail, with the problems of frequency alias and band interlacing in the algorithm being analyzed and improved; Section 3 presents a characteristics extraction method based on the wavelet packet energy and single-branch reconstruction signal demodulation; in Section 4, the algorithm is validated through the analysis of engineering examples; finally, conclusions are made.

The lifting wavelet transform is implemented in the following three steps [

_{o}_{e}

_{e}(k)_{e}(k)

Since the lifting wavelet transform is performed completely in the time domain, the reconstruction course is very simple, including updating recovery, prediction recovery and merging,

Although the lifting algorithm has been widely used, it still has the following problems:

The first step of the lifting wavelet transform is to perform a subdivision which is actually a down-sampling course, so the lengths of the acquired odd and even samples are both the half of original signals. With the increase of the decomposition scale, the point number of samples decreases constantly, and the amount of the information provided decreases consequently.

Since the split is a down-sampling course, the sampling rate of detail signals may no longer satisfy the Nyquist sampling principle. Accordingly, frequency alias emerges, and false frequency components are created.

Due to the existence of the split course, the output results change when original signals delay for an odd number of sampling points. Therefore, the lifting algorithm does not have the translation invariability.

According to the analysis, all the above problems are induced in the link of split. Accordingly, the split step was considered to be removed. The representation of multi-phase matrix for the lifting wavelet transform was shown in

Expression of multi-phase matrix for lifting wavelets.

With the equivalent translocation transform [

The reconstruction course of the redundant lifting algorithm still included three steps: updating recovery, prediction recovery and merging. The approach to the implementation of updating recovery and prediction recovery was the same as that to the lifting algorithm, but the course of acquiring reconstructed signals by merging was changed into the process of averaging the samples ^{u}(k)^{p}(k)

According to

The construction of wavelet functions based on the lifting algorithm is conducted completely in the time domain rather than on a basic function after scaling and translation, which made it possible to design different predicting operators and updating operators for one same decomposition layer or different decomposition layers. Claypoole

Based on the above idea and prior studies, the idea of a nonlinear transform was introduced in the redundant lifting wavelet packet transform in this study to obtain a nonlinear redundant lifting wavelet packet algorithm. Since the nodes generated by the decomposition of wavelet packets involved different band information, predicting operators and updating operators with different orders of vanishing moments were employed when performing redundant lifting wavelet packet decomposition on the node signals to be decomposed in layer ^{j}

In this study, the initial prediction coefficients and updating coefficients in all layers were designed with interpolating subdivision before ^{j}

After their respective design, the predicting operators and updating operators (for decomposition) with different lengths were chosen according to the time-frequency characteristics of the scale function and wavelet function. The length of the predicting operator was denoted by

Selection of predicting operators and updating operators.

4 | 12 | 12 | 20 | 20 | 20 | |

4 | 4 | 12 | 4 | 12 | 20 |

Therefore, six groups of decomposition results could be obtained from the node signals of each wavelet packet for each decomposition result. The answer as to which pair of predicting operator and updating operator generated by corresponding decomposition result was optimal depended on the established objective function.

After being decomposed with the redundant lifting wavelet packet algorithm, the signals could be characterized by a series of approximation coefficients and wavelet coefficients. In the various application fields of wavelets, such as fault signal analysis, signal denoising and image compression, it is generally preferred that the number of non-zero wavelet coefficients is as small as possible. Because the wavelet transform is flexible in basis selection while the time-domain structure characteristics of wavelets based on the lifting algorithm bring more freedom in selecting predicting operators and updating operators, which wavelet basis is the optimal one matching the characteristics of signals and satisfying analysis requirements? Since the wavelet transform is the inner product operation between signals and wavelet function and the autocorrelation function and cross-correlation function of the signals can be expressed by inner product form, the wavelet transform could be regarded as a measure for the correlation or similarity between the wavelet function and signals [

There are multiple parameters used for the sparsity evaluation. For the case without noise, generally norm ^{0}^{0}^{p}^{p}

Norm ^{0}^{p}^{p}^{0}^{p}^{p}

For computation simplification and the convenience of comparison, the coefficients acquired through the decomposition of wavelet packets were normalized to solve ^{p}_{j−1,m}^{j−1})_{j,n}^{j})^{p}_{j,n}_{j,n,k}_{j−1,m}||_{p}^{p}_{j−1,m} in this study, six ||_{j−1,m}||_{p}_{j−1,m}. The group of predicting operators and updating operators corresponding to the minimal value was selected as the optimal operators.

In conclusion, the nonlinear redundant lifting wavelet packet algorithm could be divided into the following five steps:

The number

Totally six groups of wavelet functions with different vanishing moments were chosen to perform wavelet packet decomposition on _{j−1,m}(1 ≤

The _{j,n}^{p}

The predicting operators and updating operators corresponding to the minimal ||_{j−1,m}||_{p}_{j−1,m};

The above steps (2)–(4) were repeated till layer

There were two important problems in the lifting wavelet transform: frequency alias and band interlacing. The results of signal processing may be affected somewhat if such problems were ignored. Therefore, they were analyzed one by one and solved in this study.

The same as the classic wavelet transform, the frequency alias also existed in the lifting wavelet transform. There were two causes for this [

The subdivision step in the lifting algorithm was a down-sampling course, and the sampling rate of detail signals in the wavelet packet decomposition would no longer satisfy the Nyquist sampling principle. Therefore, the frequency alias occurred with
_{s}

The undesirable cut-off characteristics of the high-pass filter and low-pass filter corresponding to predicting operators and updating operators made the frequency components of other nodes within the transitional zone of the filter to be folded up with the frequency boundary
_{s}^{j}

For the frequency alias induced by cause (1), there were two solutions:

Single-branch reconstruction was performed on node signals, so that the folded frequencies in the decomposition could be folded back during the reconstruction;

The split step in the decomposition course and the induced down-sampling problem were removed, with the redundant lifting wavelet transform brought in.

The solution to the problem of frequency alias induced by cause (2) was as follows:

With the redundant lifting wavelet packet transform being performed on signal _{j, k}^{j}_{j,k}_{j,k}

_{j,k}

The frequency components excluding the band where _{j,k}

_{j,k}

In this study, the problem of frequency alias in the lifting algorithm was solved with the above method.

There was the band interlacing in the lifting algorithm apart from the problem of frequency alias. Although the frequency alias induced by the course of subdivision down-sampling could be overcome with the redundant lifting algorithm, the problem of band interlacing still could not be solved. A simulation signal was given as follows: the redundant lifting wavelet packet decomposition was performed on the above simulation signal, and the result was as follows:

It was seen that the frequency interchange still occurred at nodes (2,3) and (2,4) in

Node sequence in redundant lifting wavelet packet decomposition.

It is clear from

Schematic solution to band interlacing.

According to

Block diagram of improved forward transform of nonlinear redundant lifting wavelet packets.

^{p}

_{new}

_{new}

In order to extract characteristic information from the interested bands, the nonlinear redundant lifting wavelet packet single-branch reconstruction algorithm was applied to node signals. The specific implementation of the reconstruction was as follows:

The node information to be reconstructed was preserved, while all other node information was set to zero;

Since other node information was set to zero, the frequency alias induced by the undesirable cut-off characteristics of the filter could be ignored;

In the redundant lifting wavelet packet decomposition, the information about two nodes obtained from high-frequency signal decomposition was interchanged to solve the problem of band interlacing. Therefore, this course must be taken into account in reconstruction; otherwise, wrong reconstruction results would be obtained. In this study, an approach of recording decomposition paths was employed to record the decomposition paths of nodes in the wavelet packet decomposition, and reverse reconstruction was carried out based on the decomposition paths of nodes in single-branch reconstruction;

The predicting operators and updating operators chosen for the decomposition of nodes were also recorded, and reverse reconstruction was carried out based on the recording results in single-branch reconstruction because a nonlinear algorithm was used in decomposition.

The implementation flow of the above node-signal single-branch reconstruction algorithm was as shown in

Nodes were reconstructed according to recorded decomposition path

In order to successfully extract the faint fault characteristics from strong background noise and achieve an effective fault diagnosis of mechanical equipment, the nodes obtained from the nonlinear redundant lifting wavelet packet decomposition must be processed further combined with the fault mechanisms of corresponding parts.

A spectral peak group with concentrated energy will be formed in a certain high-frequency band because of the modulating characteristics of the fault signals of roller bearings induced by resonance. Generally, great attention is paid to the band where the spectral peak group lies, due to the abundant fault information contained in it. Through the redundant lifting wavelet packet transform, signals are decomposed into different bands. In order to identify the node whose band is involved the spectral peak group, the analysis of the wavelet packet energy should be conducted according its energy concentration characteristics. Suppose _{j,n,k}^{j}

A node with the maximal _{j,n,k}

In the equation,

The improved nonlinear redundant lifting wavelet packet transform was performed on signals;

The wavelet packet energy analysis was performed on the nodes obtained from decomposition;

Single-branch reconstruction, Hilbert modulation and envelop spectrum analysis were conducted on the nodes corresponding to the maximal energy.

The improved algorithm for nonlinear redundant lifting wavelet packet was applied to the case analysis on the step-up boxes in the high-speed finishing mills of a steel mill. The driving chain of the finishing mill is shown in the following figure:

Driving chain of finishing mill in a steel mill.

In the figure, black stripes represented the locations of measurement points. The on-line monitoring system detected that the peaks at the horizontal measurement point at the southern output terminal of the step-up box (marked by a red ellipse in ^{2}. The vibration acceleration signals (with the sampling frequency of 10,000 Hz and the number of sampling points of 2,048) at the measurement point at 3:00, February 22 were selected for time-domain and frequency-domain analysis. The results were as follows:

Basic analysis of vibration acceleration signals at measurement point:

From the time-domain image, it was clear that there were shock components and energy concentration also appeared in the frequency spectrogram. Accordingly, it was deduced preliminarily that the step-up box may have the potential for failure.

The method in this study was applied to the signal for triple-layer wavelet packet decomposition, with the result obtained as follows: the calculated values of norm ^{p}

^{p}^{29}) of all nodes.

(0,1) | (1,1) | (1,2) | (2,1) | (2,2) | (2,3) | (2,4) | |
---|---|---|---|---|---|---|---|

(4,4) | 8.4610 | 10.029 | 9.7141 | 10.466 | 10.430 | 10.288 | 9.9694 |

(12,4) | 8.4799 | 9.9424 | 9.7322 | 10.443 | 10.419 | 10.300 | 9.9515 |

(12,12) | 8.4499 | 9.9080 | 9.7571 | 10.438 | 10.425 | 10.300 | 9.9772 |

(20,4) | 8.4866 | 9.9333 | 9.7375 | 10.438 | 10.413 | 10.314 | 9.9509 |

(20,12) | 8.4482 | 9.9071 | 9.7595 | 10.434 | 10.421 | 10.312 | 9.9774 |

(20,20) | 8.4536 | 9.9088 | 9.7654 | 10.423 | 10.425 | 10.308 | 9.9871 |

According to

Optimal predicting operator and updating operator for nodes.

(0,1) | (1,1) | (1,2) | (2,1) | (2,2) | (2,3) | (2,4) | |

(20,12) | (20,12) | (4,4) | (20,20) | (20,4) | (4,4) | (20,4) |

According to

From the wavelet packet energy shown in

Modulation analysis:

From the analysis of the results in

The above frequencies could not be found in

The method in this study was superior according to the above comparison;

The base frequency of

It was deduced that the bearing had a fault on its outer ring. This analysis result agreed completely with the result of unboxed overhaul in mid-March of 2009. The image in

In this study, an improved algorithm for nonlinear redundant lifting wavelet packets was put forward and applied to the extraction of faint fault characteristics. With the minimal norm ^{p}

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

Node spectrogram about redundant lifting wavelet packet decomposition of simulation signal.

Block diagram about node-signal single-branch reconstruction of improved nonlinear redundant lifting wavelet packets.

Triple-layer nonlinear redundant lifting wavelet packet decomposition of signals.

Wavelet packet energy analysis.

Schematic damage of bearing outer-ring of axis I at the southern output terminal of the step-up box.