Performance of Envelope Demodulation for Bearing Damage Detection on CWRU Accelerometric Data: Kurtogram and Traditional Indicators vs. Targeted a Posteriori Band Indicators  
[PERSON]  
1 Dipartimento di Ingegneria Meccanica e Aerospaziale, Politecnico di Torino, C.so Duca degli Abruzzi, 24, 10129 Torino, Italy; [EMAIL_ADDRESS] (G.L. ); [EMAIL_ADDRESS] (F.A. ); [EMAIL_ADDRESS] (F.A. ); [EMAIL_ADDRESS] (M.S.) 1 Dipartimento di Ingegneria Meccanica e Aerospaziale, Politecnico di Torino, C.  
The selection of an appropriate demodulation band, usually called the informative frequency band (IFB), is then a fundamental issue for finding the best squared envelope spectrum (SES). From a mathematical perspective, the selection of such an IFB can be regarded as an optimization problem involving a utility function to assess the demodulation performance in a particular band and a scheme to move within the search space of all the possible frequency bands (f,\(\Delta\)f) (center frequency and band size) towards the optimal one. Milestones for the band selection task are the spectral kurtosis (SK) [4] and the kurtogram [5], which uses as indicator (i.e., utility function) the kurtosis of the coefficients at the output of quasi-analytic filters with different central frequencies CF and bandwidths BF. This idea led to the fast kurtogram (FK) algorithm [6], based on the computation of the utility function over a discrete paving of the (f,\(\Delta\)f) plane built with a multirate filter bank. The consequent fixed discrete partitioning is computationally very efficient, but could sometimes be limiting. Hence, a more flexible band selection was proposed in [7] using a genetic algorithm to directly optimize the Finite Impulse Response (FIR) filter parameters against a kurtosis-based utility function. To enhance band separation on the contrary, wavelet packet transform (WPT) was proposed in [8]. The main limit of the kurtosis, in any event, was recognized to be its vulnerability to non-Gaussian noise and its tendency to decrease when the transient's repetition rate increases [9]. Several different variations were then proposed. In [10], for example, a frequency-domain \"protrusion\" was introduced in the protrugram. In this case the utility function corresponds to the kurtosis of the amplitudes of the envelope spectrum of the demodulated signal. Likewise, the kurtosis of the power spectrum of the envelope of the wavelet packet transform (WPT) filtered signal was proposed in [11]. Nevertheless, the protrugram, as the kurtogram, is the maximum for a single harmonic in the SES, so it is still not ideal for the detection of repetitive transients. The sparsogram defined in [12] proposes, on the contrary, a measure of spectral sparsity (i.e., the ratio of \(L2\) and \(L1\) norm of the squared envelope spectrum (SES)) was used to define the sparsogram. An in-depth study of spectral \(L2/L1\) norm can be found in [13] to explain the closeness of SK with spectral squared \(L2/L1\) norm and with spectral correlation (SC) [14], a key approach of cyclic spectral analysis (CSA) [15] together with the cyclic modulation spectrum (CMS) [16], later generalized into the Fast-SC [17], a fast estimator of SC. Other indices, derived from different fields, can be found in the literature to improve the kurtogram. The Gini index, for example, originally meant to produce a measure of wealth inequality (i.e., sparseness, similarly to the idea of spectral \(L2/L1\) norm [18]), was derived from econometrics by [19] to improve the resistance to noisy random impulses. Derived from physics, a characterization of both time- and frequency-domain impulsiveness can be found in the infogram [20], based on the negentropy of the Squared Envelope and on the negentropy of the Squared Envelope Spectrum, averaged to characterize both the impulsive and cyclostationary signatures of the repetitive transients.