## 2 Envelope Analysis to Detect Defects in Rotating Machinery
### Signal Modeling for Simulation  
In this study, the characteristics of the envelope through the Hilbert transform are analyzed, and signals generated in rotating machinery are modeled to find improvements. In general, it is known that acoustic and vibration signals measured in rotating machinery are CS process [42; 43], so the signal of the rotating machinery is divided into first-order CS, second-order CS, and high-order CS processes. The first-order CS process \(m_{x}(t)\) with period \(T\) is expressed by Equation (4), and the second-order CS process \(R_{x}(t_{1},t_{2})\) with period \(T\) is expressed by Equation (5). Here the expected value \(E\{\cdot\}\) means ensemble average. The first-order CS process is deterministic, and the second-order CS process is stochastic. It is known that the signals caused by defects comply with the second-order CS process [7; 23; 42]. \[m_{x}(t)\triangleq E\{x(t)\}=m_{x}(t+T) \tag{4}\]  
\[R_{x}(t_{1},t_{2})\triangleq E\{x(t_{1})x(t_{2})\}=R_{x}(t_{1}+T,t_{2}+T) \tag{5}\]  
It is modeled as follows, assuming a fault with the rotating machinery. Equations (6) and (7) are models complied with the deterministic process and the stochastic process, respectively. The synthetic signals expressed by these equations are shown in Figure 1a,b. As mentioned above, these two signals simulate the first-order CS and second-order CS processes generated from rotating machinery, respectively. Equation (6) expresses the deterministic part, which is expressed by convolving an arbitrary three-degree-of-freedom transfer function \(h_{d}(t)\) on the superposition of the cosine wave with the harmonic frequency of the rotation frequency. Equation (7) expresses the stochastic part by multiplying white noise by the superposition of cosine signals with the harmonic frequencies of the rotation frequency. \(A_{l}\) and \(A_{m}\) represent the arbitrary amplitude for each cosine wave. \[d(t)=\left[\sum_{l}A_{l}cos(2\pi.l.f_{rotat.}t-\varphi_{l})\right]*h_{d}(t) \tag{6}\]  
\[s(t)=\left[W(t)\sum_{m}A_{m}cos(2\pi.m.f_{rotat.}t-\varphi_{m})\right]*h_{s}(t) \tag{7}\]  
Equations (8) and (9) are for the normal and defective signals composed of a combination of Equations (6) and (7), respectively. Equation (8) assumes that the signal can be measured in normal rotating machinery without the defective signal. It is the sum of the deterministic part shown in Equation (6) and white noise \(w(t)\). Equation (9) is the sum of the deterministic part \(d(t)\), the stochastic part \(s(t)\), and white noise \(w(t)\), assuming that the signal occurs in faulty rotating machinery. \[x_{normal}(t)=d(t)+w(t) \tag{8}\]  
\[x_{fault}(t)=d(t)+s(t)+w(t) \tag{9}\]  
Figure 2a shows the synthetic signal models of normal and defect expressed by Equations (8) and (9), respectively. It is difficult to distinguish between normal and defective signals in the time domain. However, comparing the spectrum in Figure 2b, the  
Figure 1: Cyclo-stationary signal model: (**a**) deterministic part and (**b**) stochastic part. signal with defect shows a distinct difference in the band of 3.4 to 3.7 kHz compared to the normal signal. This band excited by the defect can be clearly differentiated by conducting envelope analysis according to the following method. Figure 2(a) shows the filtered signals with aforementioned frequency band and envelopes of them. The top and bottom represent the normal signal model and the faulty signal model, respectively. When comparing the two, it can be seen that the faulty model shows a difference as much as the stochastic signal in Figure 0(b). Figure 2(b) shows the spectrums obtained by Fourier transform on the two envelope signals, and are significantly different at the frequency corresponding to the period of the stochastic signal. In Figure 2(b), a large magnitude means that the degree of envelope fluctuation and amplitude modulation are large. In other words, the signature of defect can be well observed.