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The sparse decomposition based on matching pursuit is an adaptive sparse expression method for signals. This paper proposes an idea concerning a composite dictionary multi-atom matching decomposition and reconstruction algorithm, and the introduction of threshold de-noising in the reconstruction algorithm. Based on the structural characteristics of gear fault signals, a composite dictionary combining the impulse time-frequency dictionary and the Fourier dictionary was constituted, and a genetic algorithm was applied to search for the best matching atom. The analysis results of gear fault simulation signals indicated the effectiveness of the hard threshold, and the impulse or harmonic characteristic components could be separately extracted. Meanwhile, the robustness of the composite dictionary multi-atom matching algorithm at different noise levels was investigated. Aiming at the effects of data lengths on the calculation efficiency of the algorithm, an improved segmented decomposition and reconstruction algorithm was proposed, and the calculation efficiency of the decomposition algorithm was significantly enhanced. In addition it is shown that the multi-atom matching algorithm was superior to the single-atom matching algorithm in both calculation efficiency and algorithm robustness. Finally, the above algorithm was applied to gear fault engineering signals, and achieved good results.

Gears are important components in rotating machinery, and fault detection and diagnosis of gears has been the subject of intensive investigation. Generally, gear fault vibration signals heavily corrupted by noise are non-stationary signals whose fault features are more difficult to successfully extract than stationary signals. Therefore, studies on such signals are of extreme significance to engineering applications.

Fault diagnosis of gears is currently a topic of intensive study, and many time-frequency analysis methods based on vibration signal have been developed, which include Short-time Fourier transform, Wigner-Ville distribution, Wavelet transform and Hilbert-Huang transform,

The proposal of the sparse decomposition based on the MP algorithm aroused extensive interest among researchers who did a lot of work on the optimization and improvement of the algorithm, as well as the expansion of its fields of application. In terms of the atom dictionary construction, atom dictionary construction methods based on the Gabor function [

However, all the references above have focused on the MP algorithms and applications using single-atom dictionaries, and few papers on composite dictionaries with different atom dictionaries being combined were reported. The concept of composite dictionaries was proposed in [

The paper is organized as follows: Section 2 presents the concept of sparse decomposition based on the MP algorithm. Sections 3 and 4 address the specific decomposition and reconstruction algorithm of composite dictionary multi-atom matching and its operational details for gear faults. Section 5 presents simulation signal analysis results and a segmented decomposition algorithm for long signal process is given in Section 6. The algorithm is validated through an application example in Section 7. Finally, Section 8 concludes the paper with some remarks about possible future work.

For a given set _{k}^{N}_{k}

Supposing that _{k}

Thus, signal

It is clear that _{k0} and ^{1} _{k}_{k}_{k}

To minimize the energy ‖_{k0} ∈ _{k0} >_{k}_{k0}, therefore:
_{k}

Then, the same step is performed on the residual ^{1}

Satisfying:

The MP algorithm is an iterative process. Signal residual _{km} satisfies:

Because ^{m+}^{1} _{km} are orthogonal, then:

With the decomposition process above being executed to order

Similarly, energy ‖^{2} can also be decomposed into the following summation form:

According to ^{M}

A key step to implement the MP algorithm is how to construct an atom dictionary, so that signals can better match atoms during the sparse decomposition. The construction of the atom dictionary directly affects the sparse expression of the signals to be analyzed. The atom dictionary construction method based on the parameterized function model is the most frequently adopted. A specific primitive function is discretely parameterized (e.g., time, frequency, contraction, translation, modulation,

The introduction of the above idea is known as the composite dictionary multi-atoms matching. The decomposition algorithm is implemented as follows:

Corresponding characteristic functions are selected to construct atom dictionary _{i}

Primary signal _{0}.

For residual signal _{m}_{mi}_{i}_{mi}_{i}

With the residual signal being subtracted by the overall projection, a new residual signal is obtained.

Steps (3–4) are iterated till the iteration termination condition is satisfied.

After the sparse decomposition, matching coefficient _{mi}_{mi}

The corresponding flow chart is shown in

The reconstruction algorithm is an inverse process of the decomposition algorithm.

The principle of threshold de-noising is introduced in _{mi}_{i}

The calculation equation of the threshold is
_{i}_{i}

The vibration signals of such typical rotating machinery such as gears are basically induced by the meshing effect and the rotation of gears, and the characteristics of shock vibration and transient vibration may also emerge in the vibration signals of fault gears. To achieve the effective matching analysis on the characteristic structures of gear vibration signals, the Fourier dictionary and the impulse time-frequency dictionary [_{fou}_{fou}_{2} = 1, the primitive function of the impulse time-frequency dictionary refers to the exponential decay function, _{imp}

The genetic algorithm (GA), a kind of calculation model of biological evolution simulating of natural selection and genetic mechanism based on Darwin’s theory of evolution, was proposed by Holland in 1975. It is a method of searching the optimal solution by simulating the natural evolutionary process [

In Reference [

The meanings of the various parameters in _{s}_{n}

Composite dictionary multi-atom matching decomposition and reconstruction were performed on the simulation signal corrupted by noise using the algorithm. Experiments show that parameter setting in GA has a great effect on optimization results and computational efficiency. After balancing these two factors, the analysis parameters are as follows: in the Fourier dictionary, the range of _{1} and _{2}, are equal to 0.5. The waveform and the frequency spectrum after the reconstruction are shown in

In the reconstruction algorithm, the weighted coefficients of the impulse time-frequency dictionary and the Fourier dictionary changed. For instance, it is set that _{1} = 1, _{2} = 0 or _{1} = 0, _{2} = 1. Here, the signals reconstructed only through impulse time-frequency atoms are known as impulse components while those reconstructed only through Fourier atoms are known as harmonic components. Thus, the impulse components or the harmonic components can be separately extracted, as shown in

From the equation for calculating the threshold, it can be seen that the soft threshold only has the effect of amplitude attenuation different from the hard threshold. Therefore, the hard threshold is adopted here. The waveform and frequency spectrum of the reconstructed signals after the threshold are shown in _{1} = 1, _{2} = 0 or _{1} = 0, _{2} = 1, then the impulse component and the harmonic component can be extracted respectively, as shown in

For signals corrupted by noise in

To illustrate the influences of multi-atom matching, single-atom matching and threshold or non-threshold on the extraction effects of impulse signals, kurtosis indices and impulse indices are introduced. Their definitions are in

The kurtosis index is defined as:

The impulse index is defined as:
_{x}_{|}_{x}_{|}

Intensified noise is applied to the simulation signals above. When the

The frequency spectrogram in

To solve the problem of the running time of the program, the algorithm is improved. The primary data sequence is evenly segmented by 512, and each section is decomposed to obtain the matching coefficients and matching atoms of all orders for each section of the data sequence. The threshold is also conducted upon sections, respectively. After the reconstruction, all sections are combined to form the reconstructed signal. Thus, the running time of the improved algorithm is significantly reduced.

The waveforms and frequency spectra of the separately extracted impulse component and harmonic component are shown in

Historical data on 1 July, 4 August, 28 August and 18 September were reviewed for analysis, and their waveforms and frequency spectra are shown in

If the same four groups of signals after hard threshold were reconstructed with the impulse time-frequency dictionary single-atom matching, then demodulation spectra could be obtained through demodulation, as shown in

MP is a classic algorithm for sparse decomposition. However, it has certain drawbacks in the sparse expression of complicated and non-stationary signals due to rather singleness in the matching between the atoms in an atom dictionary and signals as well as the enormous amount of computation. Aiming at resolving this problem, a new sparse decomposition and reconstruction algorithm is proposed based on the composite dictionary multi-atom matching pursuit in this paper. The algorithm constituted a composite dictionary combining the impulse time-frequency dictionary and the Fourier dictionary to extract the impulse component on the basis of the structural characteristics of gear fault signals.

With the principle of threshold de-noising introduced in the reconstruction algorithm with composite dictionary multi-atom matching, the threshold was set. A hard threshold was performed on matching coefficients of all orders acquired with the decomposition algorithm,

The data analysis showed that increasing data length would cause the computing amount of the decomposition algorithm and the running time of the program to be significantly increased. Accordingly, the algorithm was improved,

This work is supported by the National Natural Science Foundation of China (Grant No. 50805001 and 51075023), Scientific Research of Beijing Municipal Commission of Education (KM200910005007), Beijing Science & Technology Star Plans (2008A014), National 863 Project (2009AA04Z417), and Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHR20110803).

Flow chart of the decomposition algorithm.

Signal waveform and frequency spectrum.

Signal waveform and frequency spectrum after noising.

Waveform and frequency spectrum of reconstructed signals.

Impulse component (above) and harmonic component (below).

Waveform and frequency spectrum harmonic of reconstructed signals with hard threshold.

Impulse component (above) and component (below) with hard threshold.

Reconstructed signals with single-atom atom matching without threshold.

Reconstructed signals with single- matching and hard threshold.

Waveform and frequency spectrum of primary signals (2,048).

Waveform and frequency spectra of reconstructed signals after hard threshold (2,048).

Waveform and frequency spectrum of signals after filtration with hard threshold.

Impulse component with hard threshold (2,048).

Harmonic component with hard threshold.

Demodulation spectra.

Driving chain of gearbox.

Fault gear Z5 on shaft II of the bevel box.

Waveforms and frequency spectra of historical data.

Demodulation spectra with composite dictionary multi-atoms matching.

Demodulation spectra with single-atom matching.

Comparison results of kurtosis indices and impulse indices.

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

Kurtosis index | 3.61 | 5.11 | 14.81 | 4.30 | 11.36 |

Impulse index | 4.77 | 6.21 | 14.07 | 5.80 | 12.35 |

Influences of noise intensity on the effects of impulse signal extraction with multi-atoms matching or single-atom matching.

Kurtosis index | Impulse index | Kurtosis index | Impulse index | Kurtosis index | Impulse index | |
---|---|---|---|---|---|---|

3.61 | 4.77 | 2.76 | 3.22 | 3.18 | 4.10 | |

5.11 | 6.21 | 4.39 | 5.15 | 3.79 | 4.92 | |

14.81 | 14.07 | 9.99 | 10.82 | 8.30 | 8.92 | |

4.30 | 5.80 | 2.83 | 3.50 | 3.97 | 6.24 | |

11.36 | 12.35 | 6.24 | 6.94 | 4.22 | 5.81 |

Comparison of iteration times and running time of initial algorithm and improved algorithm.

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

| |||||

512 | 30 | 50 | 128 | 54 | 141 |

1,024 | 15 | 106 | 470 | 117 | 294 |

2,048 | 7.5 | 234 | 2,210 | 235 | 587 |

The sampling parameters of historical data in each group.

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

2,048 | 2,048 | 2,048 | 2,048 | |

10 kHz | 4 kHz | 4 kHz | 4 kHz |