## 1 Introduction
However, the method may be ineffective for analyzing signals with strong noise from mechanical systems because the frequencies of bases are easily mis-estimated under the guidance of the kurtosis index for the signal with corruptive noise. Since the estimated frequencies of bases do not match the target signal components, TFR readability cannot be enhanced at all. Then, IF paths are neither clearly observed on TFR nor accurately extracted from such TFR. LT often acts as a pre-process of TFA, from which a string of TFR reassignment methods have been developed. WT or STFT is integrated with reassignment strategy in order to improve the readability of TFR and highlight weak features from the recorded signals. For example, [PERSON] and [PERSON] use standard TFA methods with amplitude normalization and reassignment to observe smaller variations in earthquakes [19]. As reported in ref. [20], wavelet scalogram is combined with the reassignment method to improve the concentration of TFR and alleviate the interference terms. STFT is incorporated with generalized demodulation to post-process the STFT-resulting TFR to improve the energy concentration [9; 21]. Another popular TFR reassignment method is synchrosqueezing transform (SST) [22] and its improved version, the synchroextracting method (SET) [23]. These reassignment techniques can trim TFRs and have enriched the literature of TFA for nonstationary signal processing. However, these TFA reassignment methods are devised to post-process TFRs, instead of addressing the root cause that LT cannot produce a smear-free TFR. Furthermore, IF estimation in advance is required for performing post-processing. Accurate IF estimation from vibration signals from mechanical systems is always challenging. With the literature reviewed above, the motivations of this paper are as follows: (1) IF estimation is vital for bearing fault diagnosis under time-varying speeds without a tachometer being involved;(2) accuracy of extracted IF depends on the readability of TFR which often suffers from the smearing problem due to base frequencies not matching the frequencies of signal components; (3) TFA is a promising tool for bearing fault diagnosis under variable speeds in addition to order tracking. Motivated by this, this paper aims at solving the smearing problems of TFR and exploiting the TFA-based fault diagnosis method for bearing condition monitoring under time-varying operations, from which multiple IF ridges are attempted to be extracted for the purpose of fault diagnosis. To achieve the purpose mentioned above, this paper proposes the frequency matching linear transform (FMLT) with the guidance of fast path optimization (FPO) for TFR enhancement and then IF path extraction for bearing fault diagnosis under time-varying speed operations. The FPO algorithm, which is effective in IF estimation [24; 25], is used for IF extraction to tackle the problem of improper IF estimation. It is worth mentioning that the most straightforward IF extraction method is to select a start point and locate the ridges forwards or backwards point-by-point with the guidance of optimization functions. The disadvantage of this method is that its accuracy largely depends on the start point. An improperly selected start point may result in a completely incorrect IF ridge [24]. Therefore, the FPO algorithm is adopted. With the extracted IF (pre-IF), the transforming bases for FMLT can be subsequently constructed and the TFR free from smearing can be obtained. Then, more accurate multiple IF ridges can be extracted using FPO again from the improved TFR for bearing health condition monitoring under variable speeds. The paper is organized as follows. Section 2 presents the proposed FPO-guided FMLT method, including the FPO approach for the pre-IF estimation and details of FMLT for TFR readability enhancement. Section 3 gives the experimental examination. The paper is summarized in Section 4.