## 1 Introduction  
Rolling bearings and gears have been widely used in varieties of rotating machineries. The phenomenon of the whole equipment downtime due to the failure of key components such as rolling bearings and gears is obvious. It is thus of great significance to study the fault diagnosis technology of them [1; 2; 3; 4]. At present, the fault diagnosis of rotating machinery is mainly aimed at their vibration signals [5; 6; 7]. While they fail, the mechanical system will be accompanied by strong non-linear vibrations, and vibration signals are usually affected by the measurement error and ambient noise. Under the combined action of various factors, the dynamic behavior of gears and rolling bearings may present chaos and bifurcation. The measured vibration signal is essentially a one-dimensional nonlinear chaotic time series. The chaotic signals have a similar broad spectrum with the noisy signals in the frequency domain. For nonlinear time series, a part of the useful signal will be mistaken for the noise being filtered out if traditional linear methods are still adopted for noise reduction. This may result in distortion and deformation of the signal [8; 9]. Some methods of nonlinear signal processing such as time-frequency analysis [10], wavelet transform [11; 12], and multi-resolution singular value decomposition [13] have improved traditional methods, which are more dependent on the characteristic frequency components of the fault signal. If the fault characteristic frequency is weak or submerged by other signal components, it is difficult to extract effective information. At present, the problem of noise suppression is still more of a concern in the field of mechanical equipment fault diagnosis, and the main purpose is to maximize the fault characteristics. With the development of nonlinear science, the method of noise reduction based on phase space reconstruction has gradually become one important tool to study nonlinear systems. A one-dimensional time series is reconstructed into high-dimensional phase space through delay embedding. The dynamic characteristics hidden behind original one-dimensional nonlinear time series are revealed by studying the motion characteristics and distribution of the attractor in the high-dimensional phase space [14; 15; 16; 17; 18]. The attractor of the useful signal in phase space is usually confined to a limited area, and the noise signal is randomly distributed in the phase space, thus the noise components in the one-dimensional nonlinear time series can be filtered out according to different distribution rules of the useful signal and the noise in the phase space. The noise reduction methods based on singular spectrum analysis [19; 20] and the local projection algorithm [21; 22; 23] belong to this kind of algorithm, and the local projection algorithm has been widely used in the fields of speech signal processing [21] and biomedicine [23], etc. The main idea of the local projection algorithm was originally proposed by [PERSON] et al. [15] in the early 1990s. After analyzing the advantages and disadvantages of various algorithms, [PERSON] and [PERSON] [24; 25] proposed an improved algorithm. In 2004, [PERSON] and [PERSON] [16] arranged the local projection algorithm, and successfully applied it to many fields, such as speech processing and biomedical signal analysis. In 2014, [PERSON] [26] proposed smooth orthogonal decomposition, a method which projects strands according to the relative position of neighboring strands, and by simultaneously exploiting temporal and spatial correlations. During smooth orthogonal decomposition, the time series is reconstructed to high dimensional phase space, and noise can be reduced during the process of projection to tangent subspace.