## 1 Introduction
In 2015, to deal with the centroid selection of local neighborhoods in the local projective noise reduction algorithm, [PERSON] [27] used higher order polynomials to estimate the neighborhood centroids more accurately, and proposed an improved high-order and multiscale local projection algorithm, which was successfully applied to medical signal processing. For decades, first-order statistics [28] (e.g., mean, variance) and second-order statistics [29] (e.g., autocorrelation functions, power spectra) have been widely used in the analysis of mechanical vibration signals, but these methods are only theoretically applicable to the analysis of linear and Gaussian signals, which requires higher orders to reflect the signal characteristics. Compared with the first and second-order statistics, the higher-order statistics have the advantages of suppressing Gaussian noise, recognizing a non-minimum phase system, and checking signal nonlinearity [30]. However, the most obvious drawback of the higher-order statistics analysis is that it requires a large amount of computation, and the amount of computation increases with the increase of order, which makes it difficult to realize the real-time calculation of a large amount of data. Thus, there is a cumulative method of slicing, in which the most used feature is the third-order cumulant diagonal slice. Such a method can not only significantly reduce the computational complexity, but also preserve the phase information and the system's non-linear information [31]. When the gear and rolling bearing generate faults, the vibration signals are usually accompanied by large numbers of modulation phenomenon, namely the two frequency components that interact with each other [32; 33]. This phenomenon produces the sum of two frequencies and the difference between the two frequencies, and the fault signal then contains significant secondary phase coupling characteristics. When the multiscale local projection algorithm is used separately, it is not easy to extract the modulation frequency component in the fault signal, while the diagonal slice spectrum can identify the frequency components of quadratic phase coupling and enlarge the coupled frequency component in the nonlinear signal. Therefore, according to the characteristics of gear and rolling bearing fault signals in mechanical equipment, a method of mechanical fault feature extraction based on the multiscale local projective algorithm and diagonal slice spectrum is proposed. Firstly, the multiscale local projective algorithm can filter out noise while maximally preserving characteristics related to the fault. Secondly, the diagonal slice spectrum can identify the frequency components of quadratic phase coupling and enlarge the coupled frequency component in the nonlinear signal. Thus, the combination of the two algorithms can achieve the fault feature extraction of gears and rolling bearings more accurately. This paper is organized as follows. The basic principle and characteristics of the proposed feature extraction method of a mechanical fault based on the multiscale local projection algorithm and diagonal slice spectrum are introduced in the Section 2. Section 3 introduces the numerical experiments of multiscale local projection, and simulation researches of diagonal slice spectrum, which can both prove the effectiveness of the algorithm. Section 4 presents the processing of Case Western Reserve University Bearing Data [34] and applications to Drivetrain Diagnostics Simulator to verify the validity of the proposed approach described in this paper. The conclusions of the studies and necessary discussions are given in Section 5.