## 1 Introduction  
As one of the most important power transmission systems, gearboxes are widely used in many areas to provide a controlled application of power; e.g., in industrial machinery, ships, helicopters, wind turbines and automobiles [1; 2; 3]. A failure of the gearbox may cause the breakdown of the equipment as a whole and result in major economic losses or even disasters. Therefore, it is of great significance to detect gearbox failures as early as possible to prevent accidents from happening. In order to detect gearbox failures, various fault diagnosis methods have been developed. Among them, vibration-based methods are attractive since vibration signals are easy to acquire and contain a great deal of gearbox health information within the signals, and they have shown good performance for diagnosing diverse gearbox failures to date [4; 5]. According to theoretical studies [6; 3; 7; 8], gearbox failures--e.g., crack, spalling and broken teeth--tend to result in abnormal vibration symptoms and create modulation phenomena in the higher-frequency range in the vibration signal, as sidebands caused by these gearbox failures are equally spaced around the modulation frequencies. Therefore, there is a need for a frequency analysis and demodulation technique to detect fault-related components from vibration signals [9]. Envelope analysis is one of the most useful demodulation techniques, and it has been widely used for the fault diagnosis of rotating machinery; e.g., bearings and gearboxes. For instance, [PERSON] et al. applied envelope and wavelet transform for the diagnosis of incipient faults in ball bearings; the fault diagnosis performances of different pitting failures on the outer or inner race or a rolling element were investigated [10]. [PERSON] presented a digital signal processing technique for calculating the amplitude and phase modulation of the tooth meshing vibration of a gear, which was applied to the early detection of local defects [11]. [PERSON] introduced a new resonance demodulation technique to extract features related to sudden crack-induced changes in a complete revolution of a gear, which is an effective tool for the early detection of gear tooth cracks [12]. [PERSON] et al. compared some traditional methods for ball bearing damage detection; the envelope analysis method was proven to be effective for localized faults on both inner and outer races [13]. [PERSON] et al. proposed a new fault detection method that combined envelope analysis and wavelet packet transform, and it was applied to extract the modulating signal and help to detect gear faults in gearboxes at an early stage [14]. However, the traditional envelope analysis methods require band-pass filtering to capture the fault-related components, and the determination of the filter parameters requires considerable experience and expertise [15; 16]. Therefore, some scholars have tried to develop improved envelope analysis methods to determine the filter parameters automatically. The fast Kurtogram method, first proposed by [PERSON], has been proven to be a very powerful and practical tool for the determination of filter parameters in envelope analysis methods [17]. For instance, [PERSON] et al. presented a combination method based on the fast Kurtogram method and the genetic algorithm, which was successfully applied to the fault diagnosis of bearings [18]. [PERSON] et al. proposed a time-frequency analysis method based on ensemble local mean decomposition (ELMD) and the fast Kurtogram method for rotating machinery fault diagnosis; it was successfully implemented in the fault diagnosis of both bearings and a gearbox [19]. [PERSON] et al. introduced an improved Kurtogram method that adopted wavelet packet transformation (WPT) as the filter banks of the Kurtogram method, which was verified to be an effective method for diagnosing faults in rolling element bearings [20]. [PERSON] et al.