On the distribution of the modulus of Gabor wavelet coefficients and the upper bound of the dimensionless smoothness index in the case of additive Gaussian noises: Revisited

In previous work by Bozchalooi and Liang (Journal of Sound and Vibration 308 (2007) 246–267), the dimensionless smoothness index was defined as the ratio of the geometric mean to the arithmetic mean of the modulus of Gabor wavelet coefficients. Moreover, it was proven that the modulus of Gabor wavelet coefficients follows the Rician distribution and the upper bound of the smoothness index converges to a constant of 0.8455… in the case of a very low signal-to-noise ratio. However, there are two problems in the work of Bozchalooi and Liang. The first problem is that an underlying assumption was made for the Rician distribution, namely, that only one harmonic is retained by the Gabor wavelet transform. For bearing fault diagnosis by envelope analysis, the frequency of the bearing fault and several of its harmonics are required to assess the severity of the fault, because the number of harmonics in the envelope is directly related to information on the geometrics of the fault. Consequently, there is a contradiction between the Rician distribution and the envelope analysis. To solve the first problem, we have mathematically proven that the ratio of the modulus/squared modulus of Gabor wavelet coefficients to the noise standard deviation/variance follows the noncentral chi/chi-square distribution, which does not require the aforementioned underlying assumption. The second problem is that Bozchalooi and Liang assumed that a bearing fault signal is periodic when they calculated the upper bound of the dimensionless smoothness index. In theory, because of slippage of rollers, a bearing fault signal is not purely periodic but slightly random. To solve the second problem, we have incorporated cyclostationary analysis into our proof procedure for calculating the upper bound of the dimensionless smoothness index. Moreover, we have redefined the smoothness index as the ratio of the geometric mean to the arithmetic mean of the squared modulus of the Gabor wavelet coefficients, and we have proven that the upper bound of the smoothness index converges to a constant of 0.5614… in the case of a very low signal-to-noise ratio.