###### Abstract
combined AR inverse filtration and squared envelope to estimate the pall size within the bearing [16]. [PERSON] et al. established an autoregressive model based on discrete wavelet transform to estimate the defects of tapered roller bearings [17]. [PERSON] et al., addressing the stator/rotor winding fault, established the model, analyzed the characteristics, and proposed a new type of robust diagnosis design for stator/rotor winding early fault [18]. [PERSON] et al. studied the signal model of rolling bearings and evaluated the stationarity of real and pseudo cycles in the rolling bearing signal, as well as the first and second order (pseudo) cycle stationary fault symptoms [19]. The autoregressive models based on time series analysis have a wide range of applications in signal processing, and have certain applications in bearing fault detection and bearing life-prediction. It establishes a statistical model based on a finite-length sample, and then uses the model to achieve prediction, filtering or control purposes. The model parameters contain important state characteristics of the system and are very sensitive to system state changes. The autoregressive model can also be transformed with the state space model, in which it is convenient to use Kalman filter for noise reduction. The Kalman filter is a time domain method. The recursive algorithm is used, which is convenient for real-time implementation on the computer, with small computation and storage, and it can handle multi-variable, time-varying, and non-stationary time series filtering problems. With the Kalman filter, the system state is specified and defined in combination with specific problems, which can reflect the characteristics and conditions of the system, which is conducive to solving practical problems [20]. Considering the limitations of the methods in the literature cited above, combined with the advantages of the autoregressive model and the Kalman filter, this article is based on the analysis of bearing vibration signals detected by fiber Bragg grating resonant sensors. First, it establishes a state space model of rolling bearing based on the autoregressive model of time series. The recursive least squares estimation algorithm is used to determine the model parameters, the final error prediction criterion is used to determine the model order, and the autocorrelation function is used to verify the tailing of the autoregressive model. Then, the Kalman filter is applied to the vibration signal to realize signal noise reduction processing. The noise-reduced signal is first analyzed by power spectrum to detect whether the bearing has fault. If a fault occurs, improved autocorrelation envelope spectrum analysis is used to determine the location of the bearing fault. Finally, it is compared with the analysis result of wavelet transform and empirical mode decomposition to verify the accuracy and effectiveness of the method in this article. The combination of Kalman filter and spectrum analysis can effectively extract the evidence of the faulty bearing and realize the fault detection and identification of the rolling bearing. The main contributions of this article are as follows:  
1. Based on the vibration signal detected by the fiber Bragg grating resonant sensor, an autoregressive model is established and transformed into a state space model of bearing vibration. 2. Based on the established state equation and observation equation of bearing vibration, a Kalman filter is used to realize state estimation and noise reduction. 3. Bearing fault diagnosis is realized by the improved autocorrelation envelope spectrum analysis method. The first method proposed is the improved autocorrelation envelope power spectrum, which can extract the fault frequencies and their multipliers. The second method proposed is the autocorrelation envelope maximum entropy spectrum, which can directly extract the bearing failure frequency. The two envelope spectrum lines are pure and the noise interference is small. The bearing fault detection and fault identification are realized.