## 1 Introduction  
Planetary gearboxes are widely employed in many engineering systems, such as helicopters and wind turbines, for their unique merits of high load bearing capacity and large transmission ratio in a compact structure [1; 2; 3]. Since planetary gearboxes are usually key units in drivetrains, it is vital to maintain them working reliably and steadily. Once a fault occurs to planetary gearboxes, it may result in transmission deficiency, or even lead to shutdown of entire machinery. Therefore, planetary gearbox fault diagnosis plays an important role. To date, researchers have made important contributions to planetary gearbox fault diagnosis. This includes fault vibration mechanism, fault feature extraction, fault pattern identification, and fault prognosis, reported in [4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20], to name a few examples. Through dynamics modeling, [PERSON] et al. [4; 5] analyzed the effect of gear faults (tooth pitting and crack) on the meshing stiffness, and further examined the effects of gear eccentricity and tooth profile error on vibration responses. [PERSON] and [PERSON] [6; 7] studied the effects of non-uniform planet loading due to gear imperfections and the modulation effects of planet carrier torque on vibration spectral sidebands. [PERSON] and [PERSON] [8; 9] investigated the gearbox configuration (such as the number of planets, the planet position phasing, and the number of gear teeth) and the gear manufacturing errors on vibration spectral sidebands. In order to extract fault signatures from vibration signals, [PERSON] [10; 11] studied the sideband asymmetry of planetary gearbox vibrations, and extended the time domain averaging method to planetary gearbox vibration signal analysis. [PERSON] and [PERSON] [12; 13] proposed a method to separate the planet and sun gear vibration signals, and a constrained adaptive wavelet lifting method to analyze individual tooth mesh waveforms. [PERSON] and [PERSON] [14] applied the spectral kurtosis method to detect the ring gear tooth crack in a planetary gearbox. [PERSON] et al. [15] improved the adaptive stochastic resonance method and applied it to extract the weak fault symptoms of a planetary gearbox. [PERSON] et al. [16] analyzed vibration and acoustic emission data using adaptive filter, spectral kurtosis and envelope spectrum, and found that acoustic emission more susceptible to planetary bearing defects through a comparison study. In order to classify planetary gearbox fault patterns, [PERSON] et al. [17] presented a support vector machine based feature selection method, and applied it to planet gear pitting degree classification. [PERSON] et al. [18] proposed to preserve the ordinal information by ordinal ranking, and improved the accuracy of ordinal ranking by a correlation coefficients based feature selection method. [PERSON] et al. [19] identified the running condition of a planetary gearbox using the multiclass relevance vector machine, with accumulative amplitudes of the planet carrier orders and energy ratio based on difference spectra as indices. As for planetary gearbox fault prognosis, [PERSON] et al. [20] proposed a fault prognosis and remaining useful life prediction method for wind turbine systems based on a physical model, data clustering and geolocation principal. These research results have enriched the literature on planetary gearbox fault diagnosis. Planetary gearbox vibration signals have strong modulation features, due to the amplitude modulation (AM) and frequency modulation (FM) effects of gear fault, as well as the AM effect of time-varying vibration transfer paths, on the gear meshing vibration. In the time domain, they are a product of the multiple AM terms and the FM signal. According to the convolution property of Fourier transform, in the frequency domain, their Fourier spectra amount to the convolution between the respective Fourier spectrum of each AM term and the FM signal. Meanwhile, the Fourier spectrum of an FM signal involves a Bessel series expansion of infinite orders.