## 1 Introduction  
Currently, aircrafts have an increasing number of applications in the transportation industry, as well as elsewhere. Furthermore, the safe operation of aircraft is always an important research interest. As one of the most important components of an aircraft, the condition of the aero-engine greatly impacts on the safe operation of the aircraft. Assembly looseness in the aero-engine rotor leads to local stress beyond safe limits on the pressure holes, which leads to a rupture accident of the labyrinth seal toothed disk in the high pressure compressor rotor of an aero-engine, or to catastrophic accidents directly [1]. Thus, fault diagnosis is a significant requirement when scheduling proper maintenance and for avoiding serious accidents. Thus far, many aircrafts have been equipped with condition monitoring and fault diagnosis systems. Vibration analysis might continue to be the one of the most popular and useful approaches employed in the fault detection of mechanical equipment [2, 3]. The corresponding important problem of fault detection is how to process the vibration data in order to get the diagnostic feature information. One solution is to describe the vibration data as a process that can be parameterized based on simple statistical analysis (mean, minimum, maximum) or advanced higher order statistics (kurtosis, _etc._) [4]. One more implementing method is to express the vibration signal in the frequency domain (fast-Fourier transform, _etc._) [4]. However, this method would make no sense when the vibration data are acquired under a non-stationary operation regime. However, unfortunately, a great number of engineering practices indicate that vibration data gathered from machine-integrated sensors always appears non-stationary. This is why a rupture accident of the labyrinth seal toothed disk in the high pressure compressor rotor still often occurs in many different types of aircrafts, even if they have been equipped with condition monitoring and fault diagnosis systems [1]. Thus, an advanced and effective method should be developed and studied so that it may be introduced for this task. Many vibration signal-processing methods have been developed for mechanical equipment fault detection, such as envelope analysis method [5], empirical mode decomposition (EMD) [6], stochastic resonance (SR) [7] and spectral kurtosis (SK) [8]. Furthermore, these methods have also been applied in fault detection of mechanical equipment in recent years. However, all these signal processing method have some special limitations relevant to this task. Envelope analysis can effectively detect and extract periodic features, but it also suffers great difficulties in detecting transient features. One of the most typical features of EMD is that intrinsic mode functions (IMFs) are computed on the basis of the cubic spline function. Thus, EMD is sensitive to detect the harmonic feature on the vibration signal but invalid for transient features. SR will be helpless in this type of vibration signal because the signal-to-noise (SNR) ratio is too low. By computing the kurtosis value at \"each frequency line\", SK can effectively detect the presence of the hidden non-stationarities in the vibration signal. Thus, SK is ideally suitable for vibration signal processing in long records and possesses special properties: frequency-concentration and sensitive to kurtosis. Therefore, mechanical equipment fault detection still calls for a more effective vibration signal processing method. As a powerful tool for describing the non-stationary signal, wavelet transform (WT) [9, 10] has already shown its tremendous effectiveness in mechanical equipment condition monitoring and fault diagnosis because of its property of multi-resolution analysis [11, 12, 13]. Different from Fourier transform, a specific fault symptom can be matched and extracted by WT on the basis of selecting the basis function from the basis function library, which is greatly beneficial to fault feature extraction.