# Effects of Phase Shift Errors in Recurrence Plot for Rotating Machinery Fault Diagnosis
## 1 Introduction
[25] to improve the accuracy. Another method based on recursive least squares was proposed [26]. Through multi-round baseline correction, filtering, and integration, the authors developed an online and real-time acceleration integral scheme to monitor displacement of a civil structure. [PERSON] et al. [27] introduced a recursive Simpon integral method that combines Newton-Leibniz to obtain the integral recurrence calculation of each point, and a polynomial fitting to detrend the resulting waveform. The recursive least square and least means square equalizers optimization algorithms in Rayleigh fading were improved [28] with the use of a stochastic gradient descent method in the filter related to the signal error. To solve low-resolution and diffused energy problems, time-frequency analysis of nonstationary signals using variation mode decomposition and a synchro squeezing technique was proposed by [PERSON] et al. [29]. [PERSON] et al. [30] introduced a Variational Mode Decomposition to identify electromechanical oscillatory modes in power systems. The identification process was based on the time-frequency analysis of nonlinear signals which arise after a large disturbance. Some new techniques to improve the FFT-DDI method efficiency were proposed by [PERSON] et al. [31] to remove the zero shifts from unknown initial conditions, which induce errors in structural integrity evaluations and path control. The displacement reconstruction from measured accelerations and accuracy control of integration based on an attenuation algorithm was presented by [PERSON] et al. [32], who controlled the integration of acceleration in the frequency domain through an accuracy control factor for low frequencies. [PERSON] et al. [33] proposed a method to solve the difficulties of weak and time-varying fault signatures in motor stator current signal, using adaptative demodulation, Fourier transform, and Hilbert spectra. Thanks to the current availability of large amounts of data that can be modeled as signals with complex and irregular structures, graph signal processing has become an emerging field in the few last years. In this field, many engineering and research areas have used wavelet theory for data compression and signal processing. However, some works showed that when the discrete wavelet transform and its inverse were applied, the result was a combination of dilatation and translation of the scaling and wavelet functions due to the elimination of low and high frequencies [34]. Other researchers expanded the wavelet for positive definite distributions over the real line and defined a tractional derivative operator for complex functions [35]. More recently, investigations found that combining the Shannon-entropy and decomposing some low and high frequencies by filters decreased the unbalanced downsampling and achieved a better performance of the framework [36]. In recent years, entropy has been used more frequently to analyze physical and thermodynamics problems. Iterative maps can describe the chaotic motion and attractors as in the fractal geometry method. In combination with entropy, they were used to characterize fractal antennas according to the radiating structure analysis [37; 38]. Moreover, recent research has been applying the link between fractal sets, attractors, and entropy to analyze the binary image for the correlation matrix [37]. According to the literature, most methodologies used to estimate displacement from double integration of acceleration are based on one vibration source. Those studies usually condition the signal and filter the noise at high and low frequencies for analysis and fault diagnosis, but this process loses important information and produces errors in the final signal. Several techniques have been implemented to solve the problems from acceleration integration in the time and frequency domain or from signal processing in other fields.