## 2 Presentation of Proposed FPO-Guided FMLT Method
### FMLT for TFR Enhancement  
To illustrate the influence of matching degree between base frequencies and signal frequencies on TFR readability, a simulated signal \(x(t)\) is constructed as:  
\[x(t)=\sin\left(2\pi\int_{0}^{t}f_{s}(\tau)\mathrm{d}\tau\right) \tag{12}\]  
where \(f_{s}(\tau)\) denotes the shaft IF and is set as:  
\[f_{s}(t)=40e^{-0.4(t-6)^{2}}+15 \tag{13}\]  
The signal is displayed in Figure 2a and the shaft IF \(f_{s}(\tau)\) is plotted in Figure 2b. The conventional LT methods, STFT and CT, are applied to process the signal defined by Equations (12) and (13). Resulting TFR is presented in Figure 2c,d, respectively. It can be observed in Figure 2c that the TFR in the first 3 s, where the IF is almost time invariant, has no sme  
Figure 1: Illustration of fast path optimization (FPO): (**a**) amplitude peaks, (**b**) all possible paths, (**c**) the optimization path (reprinted from ref. [26]). Figure 2: Time frequency representation (TFR) of the simulated signal: (**a**) simulated signal, (**b**) Instantaneous frequency (IF) of the simulated signal, (**c**) short time Fourier transform (STFT)-resulting TFR and (**d**) chirplet transform (CT)-resulting TFR. IF ridge in the range of 3-6 s is not concentrated and suffers from the smearing. On the contrary, TFR in 4-6 s in Figure 2d is energy-concentrated and has a sharp IF ridge; but the TFR in the first 4 s has the smear problem. The reason for this is that bases of STFT, which have time-invariant frequencies, match the frequency of the signal in the first 3 s, while bases of CT, which linearly varies with time, match the frequency of the signal in the last 2 s, as shown in Figure 3a,b. To address the TFR smearing problem caused by frequency un-matching between bases and signals, the FMLT approach is proposed. The key step of FMLT is to construct the bases with frequency matching the signal components of interests. Since the FPO algorithm can be used for IF extraction, it facilitates the establishment of such transforming bases. The derivation of FMLT is presented in the following. Recall the LT of the signal \(x(t)\):  
\[ \begin{array}{rl}X(\tau,f)&=\int_{-\infty}^{+\infty}x_{win(\tau)}(t)P^{*}_{( \tau,f)}(t)\mathrm{d}t\\ &=\int_{-\infty}^{+\infty}x_{win(\tau)}(t)e^{j2\pi\int_{-\infty}^{+\infty}f_{ \mathrm{f}(\tau,f)}(t)\mathrm{d}t}\mathrm{d}t\end{array}  \tag{14}\]  
where the base frequencies \(f_{\mathrm{f}(\tau,f)}(\tau)\) equals \(f\) and transforming bases \(P_{(\tau,f)}(t)\) and \(P^{*}_{(\tau,f)}(t)\) are orthogonal pairs. Since the frequencies of signal components are proportional to shaft IF denoted by \(f_{\mathrm{s}}(t)\), \(f_{\mathrm{b}(\tau,f)}(t)\) can be defined as follows to realize that the base frequencies match the frequencies of signal components:  
\[f_{\mathrm{b}(\tau,f)}(t)=C_{(\tau,f)}f_{\mathrm{s}}(t) \tag{15}\]  
where \(C_{(\tau,f)}\) is the coefficient. In a small time window \(t\in[\tau-\Delta t,\ \tau+\Delta t]\), the frequency ridge can be considered as linear; therefore, the shaft IF in such a window can be approximated using Taylor series:  
\[f_{\mathrm{s}}(t)\big{|}_{t\in[\tau-\Delta t,\ \tau+\Delta t]}=f_{\mathrm{s}}( \tau)+{f^{\prime}}_{\mathrm{s}}(\tau)(t-\tau) \tag{16}\]  
The base frequencies \(f_{\mathrm{b}(\tau,f)}(t)\) can then be calculated as:  
\[f_{\mathrm{b}(\tau,f)}(t)=f\!\!\left(\frac{{f^{\prime}}_{\mathrm{s}}(\tau)(t- \tau)}{f_{\mathrm{s}}(\tau)}+1\right) \tag{17}\]  
Figure 3: Base frequencies of different linear transform (LT) methods: (**a**) base frequencies of STFT, (**b**) base frequencies of CT, (**c**) bases frequencies of the proposed frequency matching linear transform (FMLT), and (**d**) FMLT-resulting TFR. From Equation (17), one can observe that the base frequencies are determined by shaft IF\(f_{\text{s}}(t)\), which can be pre-estimated by FPO algorithm from STFT-resulting TFR.