Mechanic signal analysis based on the Haar-type orthogonal matrix

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Abstract

Between Haar and Walsh, there exist other Haar-type orthogonal matrixes (HTOMs), which are rarely utilized in practice. In this paper, we introduce HTOMs, which have fast algorithm, to the mechanic signal analysis. Concretely speaking, the mechanic signals are transformed by various HTOMs, which can be generated easily by varying any one of two parameters in the same program, then the performance of the transform results is compared by viewing FDC as the evaluation criterion, and the most optimal HTOM is achieved, which provides guidance and reference for the HTOMs applied in the signal analysis.

Introduction

Mechanic signal analysis, which can bring the potential of mechanical equipment into full play and reduce expenditure, plays an important role in the industrial application (Xiang et al., in press). Presently, there are numerous analysis methods, among which the matrix transform-based methods, such as DCT, FFT, wavelet transform, etc., are frequently used since that representing and processing signal in the spectrum domain is more robust than that in the time domain (Cheon and Shader, 2000, Elliott and Rao, 1982, Rai and Mohanty, 2007, Rao, 1985, Uyar et al., in press, Wu and Liu, 2008).

Recently, the recursive generation and fast algorithm about Haar-type orthogonal matrixes (HTOMs) have been achieved (Shi et al., 1998, Shi and Wang, 2003). According to its idea, different HTOMs, which include Haar, Walsh and other matrixes intervenient former two, can be produced conveniently only by varying any one of two parameters s and r in the same program. All these advantages make HTOMs very suitable for the engineering application (Glabisz, 2004, Hsiao and Wu, 2007, Mali and Majumder, 1989, Ukil and Ivanovic, 2008). Nevertheless, up to now, HTOMs have attracted little attention that they are rarely utilized in practice.

For the reason mentioned above, HTOMs are applied to the mechanic signal analysis in this paper. To be concrete, various mechanic state signals are transformed by various HTOMs, respectively. Meanwhile, FDC is viewed as the evaluation criterion, the comparison analysis of the transform results is performed, and the most optimal HTOM in the signal analysis is determined. The rest of this paper is organized as follows: in Section 2, the mechanic signals are transformed and analyzed by taking the unique advantages of HTOMs. In Section 3, the optimum selection of HTOMs is carried out based on the FDC criterion. In Section 4, a conclusion is briefly drawn.

Section snippets

Mechanic signal analysis based on HTOM

Before the mechanic signal analysis, the characteristics of HTOMs are described in this section.

Application of HTOM in the mechanic signal analysis

As mentioned in above subsection, HTOMs, which have unique advantages, contain numerous matrixes. However, HTOMs are rarely utilized in practice presently. On this background, we introduce HTOMs into the mechanical signal analysis in this paper.

First, four kinds of mechanical state signals shown in Fig. 1, obtained from data set of the rolling element bearings (Loparo, 2008), are transformed by the same HTOM, respectively (without loss of generality, corresponding parameters s and r are set 3

Optimum selection of HTOM in the mechanic signal analysis

In Section 2, different transform results are obtained even the same signal is transformed by different HTOMs. This brings us a question: among the numerous HTOMs, which one should be utilized in the mechanic signal analysis. To deal with this issue, we need know how to evaluate the performance of the transform results. For this purpose, FDC, which is the most intuitive classification criterion presently, is introduced to the optimum selection of HTOMs.

Conclusions

In this paper, HTOMs, which have not only simple operation but also fast algorithm, are utilized to the mechanic signal analysis, and the most optimal HTOM is obtained under the FDC criterion. The final results provide guidance and reference for the HTOMs applied in the signal analysis. Meanwhile, any one of two parameters is varied in the same program, different HTOMs may be generated conveniently, which exhibits great flexibility and operability. Additionally, from this case study, we may

Acknowledgements

We would like to thank Professor Baochang Shi (Department of Mathematics at the Huazhong University of Science and Technology, China), which definitely helped to improve the quality of this paper. In addition, this investigation is supported by the Special Research Foundation for the Public Welfare Industry of the Ministry of Science and Technology and the Ministry of Water Resources (200701008), the Project of National Natural Science Foundation of China (50539140).

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