## 3 Experiment Setup
\[x[n]=\sum_{i\in I}a_{i}s[n-iT-\tau_{i}]+\eta[n], \tag{7}\]  
where \(\{a_{i}\}_{i=1}^{I}\) is the set of real-value amplitudes, \(\{\eta[n]\}_{n\in\mathbb{N}}\) is additive white Gaussian noise, \(s[n]\) is a function that models an impact over time domain \(n\), and \(\{\tau_{i}\}\) is a random variable that adds small variations within fault-frequency period \(T\), which remained constant between impacts. Table 2 presents the REB faults modeled by a frequency that depends on REB geometry, including contact angle, pitch diameter, and ball diameter. The top row of Figure 2 presents an REB failure in the ball-pass frequency inner race (BPFI), modeled by impulse function \(s[n]=\sin(2\pi 5000t)\exp(-0.22\pi 5000t)\), having a resonance frequency of 5 kHz and a fault frequency of 5.875 times the rotational speed fixed at 50 Hz. Figure 2a,b display an example of clean and noisy records, respectively, showing that the resonance band was at 5 kHz on both representations of power spectral density (PSD; marked in orange). However, the failure is not visible on either spectrogram. \begin{table}
\begin{tabular}{l c} \hline
**Failure** & **Model** \\ \hline Ball-pass frequency inner race (BPFI) & \(Nf_{r}\left(1+\frac{d}{D}\cos\[PERSON]\right)/2\) \\ Ball-pass frequency outer race (BPFO) & \(Nf_{r}\left(1-\frac{d}{D}\cos\[PERSON]\right)/2\) \\ Fundamental train frequency (FTF) & \(f_{r}\left(1-\frac{d}{D}\cos\[PERSON]\right)/2\) \\ Ball spin frequency (BSF) & \(Df_{r}\left(1-\left(\frac{d}{D}\cos\[PERSON]\right)^{2}\right)/d\) \\ \hline \end{tabular}
\end{table} 
Table 2: Adjusted frequencies of rolling-element-bearing (REB) failures. Note: \(f_{r}\), instantaneous angular speed (IAS); \(d\), REB diameter; \(D\), pitch circle diameter; \(N\), number of REBs; and \(\Phi\), contact angle. Figure 2: Modeling of REB failures. Figures in top row (**a,b**) show the spectrograms estimated for constant IAS and the figures bottom row (**c,d**) show the spectrogram of time-varying IAS. Left-hand side, corresponding plots of the clean signal; right-hand side, signal perturbed by noise at 3 dB. To simulate a record that presents time-varying IAS, an additional function \(\phi[n]\) was introduced to rule the time-varying fault period, resulting in the following framework to model an REB angle-varying failure signal:  
\[y[n]=\sum_{i\in I}a_{i}s[\phi[n]-iT-\tau_{i}]+\eta[n], \tag{8}\]  
With the aim of removing the influence of variable speed on failure detection, the ruling function must fulfil composition operation \(n=\phi[n]\circ b_{1}^{-1}[n]\), such that the uniform angular resampling holds (or computed order tracking): \(y[\phi^{-1}[n]]=y_{\gamma}[n]\), where \(y_{\gamma}[n]\) is a version of \(y[n]\) with unitary speed, and \(\gamma\) represents the angular domain. The bottom row in Figure 2 shows an example of simulated REB failure with variable speed, linearly increasing from 25 to 150 Hz. Thus, the period between impulses becomes time-varying, but impulse function's resonance frequency \(s[n]\) remains at 5 kHz. Transforming the signal into the angular domain removes the influence of speed, yielding a vibration signal at a constant speed of 1 Hz. The use of impulse-based models in Equations (7) and (8) led to spectral representation with an extended frequency content. Instead, failure analysis was performed on the fluctuating spectral magnitude (i.e., envelope spectrum). However, the impulsive signal behavior had to be filtered to remove spurious components such as IAS harmonics to extract the envelope spectrum. In practice, envelope-spectrum analysis for fault detection is presented through visual examination in the absence of its standard scoring. Lastly, Figure 3 displays envelope spectra computed for the evaluating scenarios. As seen in Figure 3a, even though the first SK harmonic is the most powerful, some higher fault harmonics vanished (marked with dashed red lines), especially the third.