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Shock and Vibration
Volume 2017 (2017), Article ID 1963769, 9 pages
Research Article

Measures of Dependence for -Stable Distributed Processes and Its Application to Diagnostics of Local Damage in Presence of Impulsive Noise

1Diagnostics and Vibro-Acoustics Science Laboratory, Wrocław University of Science and Technology, Na Grobli 15, 50-421 Wrocław, Poland
2Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, Wrocław, Poland
3KGHM Cuprum Research & Development Center, Wrocław, Poland

Correspondence should be addressed to Grzegorz ┼╗ak

Received 12 June 2017; Accepted 1 August 2017; Published 6 September 2017

Academic Editor: Andrzej Katunin

Copyright © 2017 Grzegorz Żak 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.


Local damage detection in rotating machinery is simply searching for cyclic impulsive signal in noisy observation. Such raw signal is mixture of various components with specific properties (deterministic, random, cyclic, impulsive, etc.). The problem appears when the investigated process is based on one of the heavy-tailed distributions. In this case the classical measure can not be considered. Therefore, alternative measures of dependence adequate for such processes should be considered. In this paper we examine the structure of dependence of alpha-stable based systems expressed by means of two measures, namely, codifference and covariation. The reason for using alpha-stable distribution is simple and intuitive: signal of interest is impulsive so its distribution is heavy-tailed. The main goal is to introduce a new technique for estimation of covariation. Due to the complex nature of such vibration signals applying novel methods instead of classical ones is recommended. Classical algorithms usually are based on the assumption that theoretical second moment is finite, which is not true in case of the data acquired on the faulty components. Main advantage of our proposed algorithm is independence from second moment assumption.