## 2 The Proposed SK Diagnosis Technology
### The Procedure for Extraction of Diagnostic Feature  
Figure 1 shows the steps of the proposed technology, which concludes with extraction of the diagnostic features, used for the final diagnosis purposes. (a) Angular resampling  
Systems operating under fluctuating speed (low level of fluctuation) can inaccurately track shaft instantaneous speed and cause spectral smearing of shaft harmonic frequencies. Sampling vibrations signals at constant angular intervals by using optical encoders can alleviate this problem. This is done by interpolating the space between tachometer pulses to obtain the new sample times. (b) Segmentation into segments  
The novel technology proposes segmentation of the vibration signal into specially selected short segments; segment duration is defined as \(\frac{1}{(f_{r}-FTF)}\) seconds for inner race defect, and \(\frac{1}{f_{r}}\) for gear defects, where \(f_{r}\) is the rotational frequency and FTF the cage fault frequency. Duration of each bearing segment is selected in a way, that faulty area is coming into contact with all N rolling elements, and, thus, producing exactly N impulse responses per segment in case of a defective bearing. In case of gears, a gear segment should cover one shaft revolution when all teeth once come into meshing. (c) Time synchronous averaging (TSA) of gear vibration signal and removal of gear-mesh componentsPrior to the estimation of the SK and filtering, the raw vibration signal needs to be pre-processed to isolate vibrations of a single component and remove interfering periodic components. Therefore, in case of gears, the SK should be estimated on a gear residual signal, which is obtained by time synchronous averaging and cleaning of periodic gear mesh components. The time synchronous average (TSA) mt is estimated as the signal average (first order cyclostationary part) on the selected cycle [6] as:  
\[m(t)=\frac{1}{N_{r}}\sum_{k=0}^{N_{r}-1}x(t-kT_{r}) \tag{5}\]  
where \(T_{r}\) is the period of rotation of the shaft of interest and \(N_{r}\) the number of rotations for the averaging. The effect of incipient gear tooth fault will be to create a low intensity impact, which may not be visible in the TSA signal itself. Since high intensity mesh harmonics are masking these low intensity impacts, the common practice is to extract these mesh harmonics to form the classical residual signal \(r(t)\)[26]. It is shown that the process of removing mesh  
Figure 1: Schematic of the technology for extraction of gearbox/bearing diagnostic features. harmonics is in fact equivalent to the subtraction of an averaged tooth meshing vibration (the regular signal) from the original TSA signal [27], which reads  
\[r(t)=m(t)-\frac{1}{N_{t}}\sum_{k=0}^{N_{t}-1}m(t-kT_{m}) \tag{6}\]  
where \(T_{m}\) is the mesh period and \(N_{t}\) the number of teeth of the gear of interest. Estimation of the spectral kurtosis (SK) and denoising filter  
Each signal segment is subjected to the estimation of the SK using Equation (1). The Hamming window is used for STFT estimation, in which the main parameter is the window length. The length should be smaller than the distance between two impulses and larger than the length of a single impulse response [3]. By following these guidelines, the length can be chosen based on BPFI, or \(F_{gmf}\), as they are related to the frequency of the excitation impulses. In this study, the window length of \(1/10\) times the duration between consequent impulses is used for processing bearing vibration data, which corresponds to the frequency resolution of 10 times of the bearing characteristic defect frequency. A window size equal to half meshing period is used for estimation of the SK of gear simulated vibrations. The SK is used to define the Wiener filter by Equation (4). Before filtering, the SK should be compared to so-called the \"significance threshold\" s that indicates the values of SK greater than the SK produced by the Gaussian noise.