## 2 Presentation of Proposed FPO-Guided FMLT Method
### Pre-IF Estimation via FPO Algorithm  
As stated previously, the construction of bases of the proposed FMLT requires the information of shaft IF information to make sure that the frequencies of bases match those of the signal components. The FPO algorithm is used for the shaft IF extraction from the original TFR generated by LT (i.e., STFT in this paper). For a signal \(x(t)\), the LT can be formulated as:  
\[X(\tau,f)=\int_{-\infty}^{+\infty}x_{win(\tau)}(t)P^{*}_{(\tau,f)}(t)\mathrm{d}t \tag{1}\]  
where \(x_{win(\tau)}(t)\) stands for windowed signal, \(P_{(\tau,f)}(t)\) represents the set of the transforming base function and the asterisk denotes the complex conjugate. \(SP(\tau,f)=\left|X(\tau,f)\right|^{2}\) is the spectrum of the signal. Transforming bases \(P_{(\tau,f)}(t)\) can be expressed as:  
\[P_{(\tau,f)}(t)=e^{j\theta(t)}=e^{j2\pi\int_{-\infty}^{+\infty}f_{b(\tau,f)}(t )\mathrm{d}t} \tag{2}\]  
where \(f_{b(\tau,f)}(t)\) denotes the frequencies of bases. For STFT, base frequencies \(f_{b(\tau,f)}(t)\) equal a time-invariant frequency \(f\). Equation (1) then is reduced to:  
\[ \begin{array}{ll}X(\tau,f)&=\int_{-\infty}^{+\infty}x_{win(\tau)}(t)e^{-j2 \pi\int_{-\infty}^{+\infty}f\mathrm{d}t}\mathrm{d}t\\ &=\int_{-\infty}^{+\infty}x_{win(\tau)}(t)e^{-j2\pi f\mathrm{d}t}\mathrm{d}t \end{array}  \tag{3}\]Based on the TFR from STFT, local maximum values of amplitudes (i.e., local peaks) at time point \(\tau_{n}\) on the TFR can be searched and the frequencies corresponding to the local peaks at the time point \(\tau_{n}\) can then be obtained, which can be formulated as:  
\[f_{m}(\tau_{n})=f,\quad s.t.,\left\{ \begin{array}{l}d[X(\tau_{n},f)]/df=0\\ d^{2}[X(\tau_{n},f)]/df^{2}<0\end{array} \right.\Big{(}m=1,2,\ldots,N_{p}\Big{)}, \tag{4}\]  
where \(\tau_{n}\) (\(n\) = 1,2,....\(N\)) is time bins throughout the whole TFR, \(N_{p}(\tau_{n})\) is the number of local peaks at \(\tau_{n}\), \(f_{m}(\tau_{n})\) represents the frequency of the \(m\)th local peak at \(\tau_{n}\), and \(N\) stands for the number of time bins (i.e., the number of windows used for STFT). All the local peaks of amplitudes in the whole TFR are denoted by circles in the peak map, as shown in Figure 1a. In order to extract an accurate IF ridge, the FPO algorithm is proposed to determine which peak should be identified at every time bin. Frequencies corresponding to extracted peaks constitute the estimated IF. The FPO algorithm can be expressed as [25]:  
\[\left\{m_{c}(\tau_{1}),\ldots,m_{c}(\tau_{N})\right\}=\underset{[m_{1},\ldots,m _{N}]}{\text{argmax}}\sum_{n=1}^{N}F\Big{[}\tau_{n},SP_{m_{n}}(\tau_{n}),f_{m_ {n}}(\tau_{n}),\left\{f_{m_{1}}(\tau_{n}),\ldots,f_{m_{n}}(\tau_{n})\right\} \Big{]} \tag{5}\]  
where \(m_{c}(\tau_{n})\) denotes the extracted peak at \(\tau_{n}\), \(F[\ ]\) represents a support function for optimization, and \(SP_{m}(\tau_{n})\) denotes the amplitude of the \(m\)th local peak. The IF ridge can then be connected as \(f_{p}(\tau_{n})=f_{m_{c}(\tau_{n})}(\tau_{n})\). The support function can be formulated as [24; 25]:  
\[F[\ ]=\left\{ \begin{array}{l}\log SP_{m}(\tau_{n})+\omega_{2}(f_{m}(\tau_{n} ),m[f_{d}],IQR[f_{d}],1)\\ \quad+\omega_{1}(f_{m}(\tau_{n})-f_{d}(\tau_{n-1}),m[\Delta f_{d}],IQR[\Delta f _{d}],1)\,n\geq 3\\ \log SP_{m}(\tau_{n})+\omega_{2}(f_{m}(\tau_{n}),m[f_{d}],IQR[f_{d}],1),n=2 \\ \log SP_{m}(\tau_{n}),n=1\end{array} \right.