## 1 Introduction  
Rolling bearing is the most important component of machines in both industrial machinery and household equipment. The fault diagnosis of rolling bearing is necessary for the operation of machinery and to prevent property loss and dangerous situations [1; 2; 3; 4]. Many scholars have studied various fault identification and classification methods [5; 6; 7]. However, signal acquisition, noise separation, and signal parameter estimation are the most critical segments for fault diagnosis due to the feature signal being typically weak and polluted by noise. Weak signal detection and parameter estimation methods in complex noise environments have been widely used in other fields, such as scientific research [8; 9], bioscience [10], chemistry [11], and so on. To extract enough useful information from the low signal-to-noise ratio (SNR) signals, researchers proposed a variety of detection methods from the aspect of linear theory, such as time-frequency analysis and adaptive filters [12], to the aspect of nonlinear theory, such as the chaos system [13], stochastic resonance [14], and machine learning methods [15]. The lock-in amplifier (LIA) is a mature weak signal detection method extensively used in spectroscopy signal analysis [16] and atom spin gyroscope signal processing [17]. The LIA uses the principle that the measured signal is not correlated with noise. The measured low SNR signal with known frequency is multiplied by a reference signal with the same frequency to achieve frequency shifting. Then, a lowpass filter is used to filter the noise and high-frequency component to obtain the DC part which is proportional to the amplitude of the detected low SNR signal. Due to the LIA having a strong noise inhibition ability and high reliability, companies and scholars proposed many LIA applications and modified structures. [PERSON] et al. [18] proposed a matrix version of a digital LIA (DLIA) which is used to simultaneously estimate the amplitudes and phases of a multiple frequencies signal of the optical tomography. [PERSON] et al. [19] proposed a DLIA structure to process the multiple frequencies signal from the contact-resonance force microscopy. [PERSON] et al. [20] designed a high-speed digital LIA structure based on an FPGA. [PERSON] et al. [21] used the integral average filter as the lowpass filter to compress the filter bandwidth and improve the amplitude measurement accuracy of the DLIA. [PERSON] et al. [22] designed a microcontroller-based dual-channel DLIA which enhanced the frequency resolution of the reference signal. [PERSON] et al. [23] proposed a single-channel LIA using a phase shifter to eliminate the phase difference between the reference signal and the measured signal which is used to process the signal of the capacitance or resistance output sensor under the low power consumption condition. Due to the analog LIA being easily affected by the temperature drift and time drift of the analog device, many researchers mainly focus on the specific digital LIA realized by a DSP, FPGA, MCU, and other digital chips, which is applied as a signal processing module in a dedicated research field or application [21]. One of the key factors affecting the amplitude measurement accuracy is the frequency error between the reference signal and the measured signal. [PERSON] et al. [24] analyzed the phase error of the DLIA output and proposed a reference signal frequency error tracking method that enhanced the amplitude measurement accuracy. However, this proposed frequency tracking method can only satisfy the small frequency variation condition and had a slow dynamic response. [PERSON] et al. [25] proposed an automatic frequency tracking structure based on Fast Fourier transform (FFT) and digital phase-locked loop (DPLL) on an FPGA chip to reduce the frequency error.