Understanding kurtogram parameters

Question

I am about to understand a kurtogram, and don't understand what means the value of "K" (presented in table 1),or especially why takes values of 1.6 ; 2.6 ; 3.6 etc. Other question is how do interpret the kurtogram (figure 2)

Answer 1

[Given figures are from A new improved Kurtogram and its application to planetary gearbox degradation feature analysis, Xianglong Ni et al.]

This $k$ in the graph (and the $K$ in the Table) corresponds to an equivalent "dyadic-power", providing the number of subsequences at each level, alternating dyadic splits with filters on subbands $[0,1/4]/[1/4,1/2]$ and $[0,1/6]/[1/6,1/3]/[1/3,1/2]$. The number of such subsequences in the 1/3-binary tree are $2,3,4,6,8,12\ldots$, approximately $2^1,2^{1.6},2^{2},2^{2.6},2^3,2^{3.6}\ldots$, since $\log(2)/\log(3)\sim 0.6$.

Additional sources:

• Calcul Rapide du Kurtogramme et Applications, Jérôme Antoni (short, in french)
• Fast computation of the kurtogram for the detection of transient faults, Mechanical Systems and Signal Processing, 2007

I unfortunately cannot help totally on the interpretation of this particular paper. I remember that the original spectral kurtogram was meant to detect, localize or characterize nonstationarities from signals. The kurtogram was a fast version, reusing principles from wavelet packets and multiband filter banks and quasi-analytic filters. When looking at maximum values in a kurtogram, if they are concentrated, one may suspect non-stationary components in a noisy signal. Then, it is possible to extract or invert the aforementioned component to check or denoise the original data. I would suggest you to read several papers, especially the early ones, showing interpretations of different kurtograms.

Answer 2

This will be helpful for You to interpret kurtogram https://www.mathworks.com/help/predmaint/ug/Rolling-Element-Bearing-Fault-Diagnosis.html

If we look on the results the most important elements are:

Kmax = 2.7083 - is maximum value or kurtosis

Optimal Window Length = 128 - this value we can use to calculate spectral kurtosis

Center Frequency = 2.6703 kHz and Bandwidttch = 0.76294 kHz - this two parmateres we can use do design band pass filter

The kurtogram is used in bearing fault diagnosis to extract the most impulsive signal from the signal. Base on this example we can say: if we select center frequency = 2.6703 kHz and bandwidth = 0.76294 kHz we can obtain a signal after filtration that gives the most impulsive characteristic. Using these parmaters we can calculate the envelope spectrum for the final diagnostic.