## 1 Introduction  
In order to prevent greater losses from malfunctions, unnecessary overhaul, and human loss, the fault detection of rotating machinery has been studied in various ways [1, 2]. Recently, companies have been investing in automated fault-detection systems using various sensors and equipment to accurately and efficiently improve the subjective judgment of human operators on the factory floor [3, 4]. To accomplish fault-detection systems, signal processing techniques for detecting faults in rotating machinery that generate vibration and noise-depending on the operating speed can transform original signals into useful features [5]. In other words, to design an effective automated fault-detection system using vibration and acoustic signals, appropriate techniques should be selectively used according to the characteristics of the measured signal [6]. The cyclo-stationary (CS) process refers to a non-stationary process in which the statistical properties of the signal have periodicity. CS approach plays a very important role in extracting the faulty signature of a rotating machinery [7, 8]. Envelope analysis has been widely used as a tool to analyze CS signals for a long time, especially as a powerful fault-detection technique for rolling bearings [9, 10, 11]. The fault in rolling elements periodically impacts the mechanical system. This impact causes the system to generate amplitude-modulated vibration signals by the resonance frequency. The envelope of a CS signal is extracted by the Hilbert transform, always as well as a band-pass filter. This procedure is called amplitude demodulation. Through the analysis of the extracted envelope, it is possible to find the fault signatures inherent in the resonant frequency band excited by the faults. Therefore, selecting an appropriate band for demodulation is important. Spectral kurtosis (SK), also known as Kurtogram, is a well-known method in this field. SK helps selecting the optimal frequency band that represents impulsiveness of the signal. Kurtosis is calculated for each band-passed signal with equal bandwidth but different center frequency [12; 13; 14]. [PERSON] et al. [15; 16] proposed the method based on envelope analysis using SK to detect the wheel flat of a railway train. The proposed system was verified by numerical model and experiments in the laboratory. The results showed that the wheel flat can be detected with two accelerometers spaced half the circumference of the wheel on the rail. Another method to estimate the informative frequency band, Prottrugram, is similar to SK, but calculates the kurtosis of the envelope spectrum of the band-passed signal with different band-pass filter [12; 14]. These two methods mentioned above focus on the impulsiveness rather than the periodicity of the fault, such as reflecting the kurtosis that is decreased by the high rate peaks or increased by the single peak. Likewise, they are very vulnerable to impulsive noises generated by random sources [13]. Due to these limitations, many studies have been conducted to determine the suitable frequency band by using the impulsiveness and periodicity of the fault together [14; 17; 18]. [PERSON] et al. [19] proposed the method that applied particle swarm optimization (PSO) to find the envelope window for rolling element bearing fault detection. PSO is an optimization algorithm inspired by the social behavior of birds and fish groups. The authors implemented the iteration of the algorithm with the objective function to maximize the peak of the bearing characteristic defect frequency in the envelope spectrum. Moreover, it is necessary to remove various noise in the measured signals to effectively carry out envelope analysis because it reflects the shape of variations in time signals. [PERSON] et al. [20] developed envelope analysis method for signals with an adaptive filter to remove deterministic signals from gears and shafts that mask the defective signals in planetary bearings. [PERSON] et al.