Probability distributions and typical sparsity measures of Hilbert transform-based generalized envelopes and their application to machine condition monitoring

The establishment of probability distributions of machine vibration signals is crucial for calculating theoretical baselines of machine health indicators. Health indicators based on the envelope and squared envelope are an important family for condition monitoring. Under the assumption that the vibration signals of a good machine are Gaussian distributed, the envelope of a normal machine signal with zero mean is proven to follow a Rayleigh distribution with one parameter that depends on the noise variance, and its squared envelope follows an exponential distribution with one parameter, while the exact distribution parameter is undefined. The recently introduced log-envelope (i.e. the logarithm of the envelope) and generalized envelope (GE) exhibit attractive properties against interfering noise, however, their probability distributions have not yet been established. In this paper, the probability distributions of the squared envelope, log-squared envelope (i.e. the logarithm of the squared envelope), log-envelope and GE with parameter greater than 0 of Gaussian noise and corresponding distribution parameters are derived and established theoretically, and the important characteristic that their distribution parameters vary with the noise variance is clarified. On this basis, typical sparsity measures of GE of Gaussian noise are theoretically calculated, including kurtosis, skewness, Li/Lj norm, Hoyer measure, modified smoothness index, negentropy, Gini index, Gini index II and Gini index III. These typical sparsity measures of GE with parameter greater than 0 of Gaussian noise and the skewness and kurtosis of the log-envelope of Gaussian noise are proven to be independent of the noise variance, which enables them to serve as baselines for machine condition monitoring. Numerical simulations verify the correctness of the probability distributions and theoretical values of typical sparsity measures of GE with different parameters of Gaussian noise. The analysis results of four bearing run-to-failure experiments verify the feasibility and effectiveness of the sparsity measure of Gaussian noise as a condition monitoring baseline and demonstrate the efficacy and performance of GE-based sparsity measures for machine condition monitoring.