## 1 Introduction  
An accurate measurement of rotational speed is important in condition monitoring, fault diagnosis and torsional vibration analysis for rotating machinery, e.g., gearbox and rotor-bearing systems [1, 2, 3]. Many common fault diagnosis methods, such as time synchronous average (TSA) and order analysis (OA), all have high requirements for accurate rotational speed. Therefore, it is of great significance to measure the rotational speed stably and accurately [4, 5]. At present, encoder, toothed or notched encoder discs are the most wildly used methods for rotational speed measurements. The underlying mechanism of the encoder-based approach is that as the shaft rotates, a tooth or notch in the encoder disc will generate a series of pulse signals, and the rotational speed can then be obtained by counting the number of pulses [6]. The measured instantaneous rotational speed through the encoder-based method has been successfully applied in fault diagnosis and condition monitoring of rotating machinery. For instance, [PERSON] et al. proposed a kurtosis-guided local polynomial differentiator to estimate the instantaneous angular speed (IAS) of rotating machines based on the encoder signal; the results demonstrated that the proposed method could not only detect fault signatures but also identify defective components of a planetary gearbox [7]. However, sometimes the kurtosis is sensitive to the impulsive noise component, which is not beneficial for the estimation. [PERSON] et al. introduced a new concept of root mean square (RMS) value calculations using angle domain signals within small angular ranges; a new diagnosis algorithm based on the time pulses of an encoder was developed to overcome the difficulty of fault diagnosis for helical gears at low rotational speeds [8]. But the angular ranges are difficult to determine, it has a direct influence on the diagnosis accuracy. [PERSON] et al. presented a novel approach to conducting the rephasing of encoder signals for absolute dynamic transmission error (DTE) measurements of a gearbox; not only deviations from an involute profile but also the average wear depths of all teeth were well estimated [9]. However, the DTE measurement usually needs a higher sampling rate of the encoder, especially in high rotational speed situations. [PERSON] et al. proposed a bearing fault diagnosis method based on angular domain resampling; the non-stationary vibration signals in the time domain are transformed into the angular domain [10]. Although encoders have many success cases in rotational speed measurements, DTE estimation and IAS estimation may not be suitable in some cases, e.g., high rotational speed situations, limited space, and in an oil-contaminated environment [11]. Therefore, many scholars have tried to estimate the rotational frequency from the vibration signal [12], current [13], vision [14], etc. Among these rotational frequency estimation methods, the vibration-based method is the most popular one because of its convenient installation and rich information [15]. The basic principle of the vibration-based estimation method is that when a machine rotates in varying speed conditions, the rotating frequency and characteristic frequency also vary synchronously. Hence, the rotating information can be obtained by extracting the instantaneous frequency or instantaneous phase [16]. For instance, [PERSON] et al. introduced a rolling element bearing defect detection method using the generalized synchrosqueezing transform guided by TF ridge enhancement to settle the problem of the time-varying shaft speed [17]. However, the TF ridge fusion strategy sometimes may degrade, especially when there is a big difference between TF ridge 1 and 2. [PERSON] et al. investigated a two-step procedure for the estimation of the IAS with large fluctuations which combines the advantages of phase demodulation and TFR and overcomes the obstacles of the harmonic overlapping phenomenon [18].