## 2 Materials and Methods
### Phase-Separation Technique
\[\mathbf{f_{D}}=[f_{Dm},f_{Db},f_{Dc},\dots f_{Dm}],\ \mathbf{t}=[t_{a},t_{b},t_{c}, \dots t_{m}] \tag{10}\]Figure 5: Diagram of a time-frequency analysis of Doppler signal. Figure 6: \(Cont.\)  
Furthermore, the amplitude of the Doppler signal at each point is ascertained by making the envelope of the Doppler signal using the Hilbert transform as shown in Equation (11). The data at each point are arranged in array form (\(\mathbf{a}\)) as Equation (12), and these data are selected by time location index (\(\mathbf{a_{s}}\)) as in Equation (13). \[D(n)=\left(s^{2}(n)+\hat{s}^{2}(n)\right)^{1/2},\;\hat{s}(n)\stackrel{{ def}}{{=}}H[s(n)] \tag{11}\]  
\[\mathbf{a}=[a_{n=0},a_{n=1},a_{n=2},\ldots a_{n=N_{REP}-1}] \tag{12}\]  
\[\mathbf{a_{s}}=[a_{a},a_{b},a_{c},\ldots a_{m}] \tag{13}\]  
The selected data is then compared with a threshold value. The value is defined as being higher than the maximum Doppler amplitude of the particle and lower than the Doppler amplitude obtained from the bubble (the threshold value is determined by the experimental data in Section 4.2). The amplitude data index (\(i\)) is separated into two categories as expressed in Equation (14). Firstly, when the amplitude value is higher than the threshold, its index is defined as a bubble index. Secondly, when the value is lower than the threshold, its index is expressed as a particle index. These are called \"separation indexes,\" and are defined as index selectors. Within these indexes, Doppler frequency data analyzed by peak detector is classified as a Doppler frequency of bubble group and particle group according to Equations (15) and (16). The average Doppler frequency in each group is then shown as Equations (17) and (18). Therefore, the Doppler frequency of bubble and particle in the same \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Readability** & **Resolution** & **Computational Time** \\ \hline Short-Time Fourier Transform & Excellent & Good & Excellent \\ Gabor & Excellent & Poor & Poor \\ Wigner–Ville distribution & Poor & Excellent & Good \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of time-frequency estimator. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Computational Time (ms)** \\ \hline Short-Time Fourier Transform & 0.2657 \\ Gabor & 0.2807 \\ Wigner–Ville distribution & 0.5865 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Computational time of time-frequency estimators. Figure 6: The result of Time-frequency spectrogram: (**a**) Short-Time Fourier Transform (STFT), (**b**) Gabor, and (**c**) Wigner–Ville distribution (WVD). measurement channel is classified apparently. Consequently, the velocity of both liquid and bubble can be computed simultaneously. \[\text{index}(i) \stackrel{{\text{def}}}{{=}} \left\{ \begin{array}{l}\text{i}_{\text{bubble}}\left[i_{b},i_{b-1}, \ldots,i_{b-(x-1)}\right]\textbf{if }\textbf{a}_{\textbf{\emph{si}}}\geq\textbf{Threshold}\\ \text{i}_{\text{particle}}\left[i_{p},i_{p-1},\ldots,i_{p-(y-1)}\right]\textbf {if }\textbf{a}_{\textbf{\emph{si}}}<\textbf{Threshold}\end{array} \right. \tag{14}\] \[\textbf{f}_{\textbf{D}(\textbf{bubble})} =\left[f_{Dbr},f_{Db-1},\ldots,f_{Db-(x-1)}\right]\] (15) \[\textbf{f}_{\textbf{D}(\textbf{particle})} =\left[f_{Dpr},f_{Dp-1},\ldots,f_{Dp-(y-1)}\right]\] (16) \[\overline{\textbf{f}}_{\textbf{D}(\textbf{bubble})} =\frac{f_{Db}+f_{Db-1}+,\ldots,+f_{Db-(x-1)}}{x}\] (17) \[\overline{\textbf{f}}_{\textbf{D}(\textbf{particle})} =\frac{f_{Dp}+f_{Dp-1}+,\ldots,+f_{Dp-(y-1)}}{y} \tag{18}\]  
Figure 7: Block diagram of the proposed technique in the UVP system. Figure 9a represents an example of the Doppler frequency classification for the proposed technique in one measurement channel, showing the processing system on single-phase flow. The Doppler frequencies observed from the peak on the spectrogram are only influenced by particles because the Doppler amplitude of each time location point is lower than the threshold value. Figure 9b shows the spectrogram in two-phase bubbly flow conditions.