*2.1. Squared Envelope Spectrum*

Demodulation analysis methods can extract and identify fault characteristic frequencies from the resonance band. Envelope analysis is widely applied to acquire the harmonics of characteristic frequencies, such as the squared envelope spectrum (SES). Since the faulty rotating machinery vibration signals usually contain second-order cyclostationary (CS2) components, they are often used to extract fault features. CS2 is commonly calculated using SES, and the formulas can be expressed as follows:

$$\text{SES}(n) = \left| \frac{1}{L} \sum\_{n=0}^{L-1} |\tilde{\mathbf{x}}(n)|^2 e^{-j2\pi n \mathbf{x} / F\_s} \right|^2 = \left| DFT \left( |\tilde{\mathbf{x}}(n)|^2 \right) \right|^2 \tag{1}$$

where <sup>α</sup> represents cyclic frequency, and *Fs* denotes the sample frequency. *<sup>x</sup>*(*n*) is converted by the Hilbert transform from time-domain vibration signals. *DFT*(•) represents the discrete flourier transform, and it is formulated as follows:

$$DFT[\mathbf{x}(n)] = \sum\_{n=0}^{L-1} \mathbf{x}(n) \ e^{-j \cdot k \cdot n \frac{2\pi}{L}} \tag{2}$$

where *x* (*n*) represents the signal sequence [0 *L*−1]. Thus, CS2 components is acquired with cyclic frequency α.
