## 1 Introduction  
Bearings are one of key components in rotating machinery; thus, their fault detection and diagnosis have long been investigated to prevent severe equipment damage and unscheduled downtime [1]. When a local fault occurs on the surface of the outer race, inner race or rolling elements, the measured vibration signals present repetitive fault impulses with exponential decay [2]. These fault-induced impulses are the main features to be detected to perform bearing fault diagnosis for both constant and variable speed conditions. There are state of the art methods for bearing fault diagnosis [3; 4; 5]. For constant speed conditions, once these impulses are extracted, the frequency and its harmonics of impulse repetition can be obtained by frequency analysis, which is related to the fault existence and fault category. However, when the rotational speed is non-stationary, the speed fluctuationsmay cause \"smearing\" of the discrete frequencies in the frequency representation, indicating that these frequencies are no longer detectable [6]. As a result, approaches developed for bearing fault diagnosis under constant speed operations would be ineffective. Order tracking has proven powerful in bearing fault detection under variable speeds [7]. Nevertheless, order tracking may propagate errors to the result because the resampling is achieved via polynomial interpolations, while vibration signals are generated by cyclic phenomena, and are thus sinusoidal rather than polynomial in nature [8; 9]. Moreover, order tracking might cause the carrier frequencies of the transient responses to extend to a wider scope as the natural characteristics of bearing systems rarely vary, which is not effective in bearing fault feature extraction [10]. Undoubtedly, order tracking is one of the most useful tools for bearing health condition monitoring under variable speed conditions. Nevertheless, as stated, it still has drawbacks. Ref. [11] analyzes different factors that incur errors and influence the accuracy of order tracking. More importantly, order tracking requires the shaft instantaneous frequency (IF). However, shaft IF cannot always be measured by tachometers, which may be not allowed to be mounted due to design reasons. In this case, extracting shaft IF from time frequency analysis (TFA) of vibration signals is preferred. TFA is also an effective approach for vibration-based bearing fault diagnosis under time-varying speeds as it has strong potential to characterize both time and frequency features of vibration signals, in addition to order tracking [12]. LT is the extensively used technique for TFA of nonstationary signals. Most common LT approaches include short time Fourier transform (STFT) [13], wavelet transform (WT) [14], and chirplet transform (CT) [15] and its variants, such as polynomial chirplet transform (PCT) [16; 17]. However, as revealed by [PERSON] et. al. [18], the frequencies of bases of both STFT and WT are time-invariant, which cannot match the frequencies of target signal components. Then, the smearing problem may be incurred. The frequencies of transforming bases of CT and its variants vary with time; however, the chirp rate of bases is invariant and this kind of LT method can only eliminate the smearing effects of the signal with the same chirp rate as the pre-set one. In reality, the chirp rate of frequencies of vibration signals from the bearing is often time-varying and unknown in advance. The CT and its advanced versions, hence, cannot generate a smearing-free time frequency representation (TFR). To address the problem that the frequencies of transforming bases do not match the frequencies of signal components, [PERSON] et al. propose a method named velocity synchronous linear chirplet transform, which can relieve the smearing of TFR and result in a desirable time frequency resolution [18].