Abstract
The first and the second order moment functions of the vectorial periodically correlated random processes—the mathematical models of the vectorial physical phenomena—are analyzed. The properties of the linear and the quadratic invariants of the covariance tensor-function are described. The representations of the covariance tensor-function and its invariants in the form of Fourier series are considered. Fourier coefficient properties of these series are analyzed. Obtained relations are specified for the amplitude and phase-modulated signals. The examples of using the vector PCRP methods for the rolling bearing vibration analysis are given.
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Notes
- 1.
The second-rank tensor, which components are defined by the elements of the matrix \( \left( {\begin{array}{*{20}c} {a_{x} b_{x} } & {a_{x} b_{y} } \\ {a_{y} b_{x} } & {a_{y} b_{y} } \\ \end{array} } \right) \) is called the tensor product \( \vec{a}\,\otimes\,\vec{b} \) of the two vectors \( \vec{a} = a_{x} \vec{i} + a_{y} \vec{j} \) and \( \vec{b} = b_{x} \vec{i} + b_{y} \vec{j} \) (of the first-rank tensor).
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Javorskyj, I., Matsko, I., Yuzefovych, R., Dzeryn, O. (2017). Vectorial Periodically Correlated Random Processes and Their Covariance Invariant Analysis. In: Chaari, F., Leskow, J., Napolitano, A., Zimroz, R., Wylomanska, A. (eds) Cyclostationarity: Theory and Methods III. Applied Condition Monitoring, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-51445-1_8
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