## 1 Introduction
Studies on weak feature detections of incipient faults are rare and these weak features are always immersed in noises generated by the equipment and the surrounding environment. It is significant to detect weak fault features as early as possible, which is a complicated and challenging task that requests advanced analytical methods with high reliability, high accuracy and high efficiency. The emerging notion of multiwavelet transform (MWT), which uses vector-valued scaling and wavelet functions, is an important development of the wavelet theory. Multiwavelets possess excellent properties of orthogonality, symmetry, compact support and high vanishing moments simultaneously [14, 15]. Since 1994, Geronimo-Hardin-Massopust (GHM) multiwavelet [16, 17], Chui-Lian (CL) multiwavelet [18] and Hermite multiwavelet [19] have been proposed successively and received considerable attention from wavelet research communities both in theory and in applications. [PERSON] and [PERSON] [20] applied GHM multiwavelet to detect the gearing system faults. [PERSON] and [PERSON] [21] proposed multiwavelet sliding window denoising to detect the gearbox fault features of the hot strip finishing mills. Although these methods showed their advantages over scalar wavelets, prior researches always selected mother multiwavelets from a library of previously designed multiwavelets. However, the chosen standard and fixed multiwavelets were usually not the suitable ones for specified applications [22]. To overcome the limitations of standard or fixed MWTs, integrating multiwavelets with lifting schemes (LS) is an exciting motivation to construct customized multiwavelets with desired properties. LS, introduced by [PERSON] [23, 24], is a powerful tool to construct biorthogonal wavelets. It provides a great deal of flexibility and freedom to construct adaptive wavelets by the design of prediction operators and update operators. [PERSON] and [PERSON] [25] proposed the customized multiwavelets originatedfrom Hermite splines via symmetric lifting schemes. [PERSON] and [PERSON] and [PERSON] (2010) proposed a method incorporating customized multiwavelet with sliding window denoising, which was an effective and promising tool for gear fault detection. It is a challenging task to detect weak features of incipient faults, which are always immersed in heavy noises generated by the surrounding environment or the equipment. Multiwavelet denoising plays an important role in eliminating noise as much as possible. Its effect mainly depends on the feature separation by using multiwavelets and the threshold denoising. A redundant multiwavelet possesses the time invariant property [PERSON] and [PERSON] (2010) and provides abundant information for feature detection of periodical impulses. Symmetry is another important property which avoids the phase error in MWT. To ensure the time invariant and symmetry property of multiwavelets, a method integrating the symmetric lifting scheme and redundant multiwavelet is proposed to construct customized multiwavelets. Then a critical problem is how to evaluate the obtained multiwavelets and to select the optimal ones for specific applications. The quotient of kurtosis and entropy is proposed to select the optimal multiwavelets because kurtosis is sensitive to sharp impulses of incipient faults and entropy is effective for periodic impulses of moderate or severe faults. Furthermore, based on the correlation of neighboring coefficients, the improved neighboring coefficients (INC) Yuan and He (2010) is adopted to eliminate noises from the decomposed signals. In this paper, a method which incorporates the customized multiwavelets and INC is proposed for fault detections of planetary gearboxes. The experimental results show that the proposed method is effective and promising to detect these weak impulse features. The rest of the paper is organized as follows: The theory of multiwavelets and the symmetric lifting schemes are briefly introduced in Section 2.