## 1 Introduction  
Gearboxes are common mechanical equipment that is used to change the rotation speed and transmit power. Since the working conditions of gearboxes are very poor, they are vulnerable to damage and failure. The fault diagnosis of a gearbox is one of the most important issues since people are concerned with maintaining equipment [1]. The vibration signals of a gearbox caused by gear impact, resonation, or mutual interference are very complex in actual working conditions. It is difficult to apply the conventional test and diagnosis methods effectively. The faulty characteristics for the diagnosis of a gearbox is different for different sizes of the cracking tooth, rotation speed, misalignment of gears, and so on [2; 3; 4]. Many researchers have established dynamical simulation models to analyze gear faults [5]. [PERSON] et al. [6] established an 18-degree-of-freedom vibration system to study the different degrees of tooth-surface wear. However, their analytical equations are very complex and inconvenient to use. A helical gearbox system is widely used in industrial machinery, and its faults directly cause large mechanical equipment to stop service and increase productivity costs. Early identification of faults in a helical gearbox is helpful for preventing equipment failure to avoid catastrophic losses. The types of gear fault include tooth surface pitting and wear [7; 8; 9; 10; 11; 12], fatigue and crack [13; 14; 15; 16], which can be described by measuring the vibration acceleration response of the gearbox. The spectrum of the response is characterized by the different levels of amplitude modulation and frequency modulation. When a certain degree of shaft deflection or gear wear occurs, the impacted component may also be visible in the gear's vibration signal. The meshing frequency, modulation signal, and side-band information are very important features for the gear vibration signal [17]. Therefore, during the fault diagnosis, the impacted component and the frequency interval of the side-band can be used as the main basis for judging whether a fault happens in the gear system. When a gear failure that has the frequency bandwidth characteristics is measured by an acceleration sensor, high noise is distributed throughout the frequency spectrum. As a result, the frequency bands overlap with each other. Vibration signals also contain high-energy background signals related to the rotational speed. When a gearbox fails, the actual measured signals are generally non-stationary and non-Gaussian signals, which usually contain strong noise. Therefore, it is difficult to eliminate the background signals and noise signals retaining fault information when using only conventional filtering methods. [PERSON] et al. [18] presented an optimal demodulation subband selection method that can detect a gear fault by using envelope analysis, but the subband frequency features are not obvious in the environment with different noise levels. Feature extraction of the vibration signal plays important roles in the health monitoring and fault diagnosis of rotating machinery. Empirical mode decomposition (EMD) was originally proposed by [PERSON] [19] as an adaptive signal processing method that decomposes a complicated signal into a set of IMFs and a final residue. The IMFs only depend on the signal itself and have physical significance. The EMD can separate the nonlinear and non-stationary signal into several mode components. [PERSON] [20] used a dynamic model of a spur gear for the early detection of a localized tooth with the EMD method. EMD has been successfully used in the fields of health monitoring, fault diagnosis, and detection [21]. However, there are some problems with the application of EMD. The EMD technique lacks the support of a theoretical foundation and has drawbacks of boundary effects, mode mixing, and stop effects in the signal processing. The local mean decomposition (LMD) method proposed by [PERSON] [22] uses a similar processing technique to EMD.