## 1 Introduction  
It was reported that the non-stationary signal generally occurs when there is a nonlinear interaction between the energy source and the elastic system [1; 2]. Since the Fourier transform cannot show how the transient frequency of the signal changes over time, it is essential to study non-stationary signals using time-frequency analysis [3; 4]. The traditional methods of time-frequency analysis include two categories, namely linear time-frequency representations and bilinear time-frequency distributions. The former mainly includes short-time Fourier transform and wavelet transforms, and the latter includes Cohen and affine class distributions based on Wigner-Ville distribution [5; 6; 7; 8]. Considering the nonlinearity and non-Gaussianity of signals, Wigner higher-order spectrum, L-class, and S-class were proposed to suppress the cross-terms and improve time-frequency resolution [9; 10; 11]. In order to accurately estimate the instantaneous frequency in the analysis of non-stationary signals, empirical mode decomposition (EMD) is used to decompose signals [12]. The non-stationary signal can be presented in the time-frequency-intensity 3D space with these methods, and the variation of frequency over time can be analyzed qualitatively by the 3D space [5]. For the non-stationary signal generated by molecular vibration, however, studying the intensity variation over time for one specific frequency, namely quantitative time-frequency (QTF) analysis of molecular vibration, is particularly important when frequency coupling occurs in molecular vibration. Experimentally, the vibration of water molecules is often studied with two-dimensional infrared vibrational echo spectroscopy (2D-IR) [13; 14; 15].  
The wavelet transform is widely applied in various fields such as geology, mechanical engineering, neuroelectrics, and analytical chemistry since it has the capability of multi-resolution analysis and excellent time-frequency resolution [16; 17; 18]. The Continuouswavelet transform (CWT) requires optimal wavelet parameters, which mainly include bandwidth (\(f_{b}\)) and center frequency (\(f_{c}\)) when the complex Morlet function (CMOR) is used as the wavelet basis. Furthermore, some studies have shown that the time-frequency resolution and edge effect of the complex Morlet wavelet transform are related to the value of \(f_{c}\sqrt{f_{b}}\)[19, 20]. [PERSON] and [PERSON] [21] found that in order to fully extract the impulsive feature, \(f_{b}\) must be adequately wide. [PERSON] et al. [22] demonstrated that \(f_{c}\sqrt{f_{b}}\) should be chosen within an interval so that an adjustable compromise can be achieved between the frequency and time resolutions. However, to the best of our knowledge, the available studies on the optimal parameters of \(f_{b}\),\(f_{c}\) are rare, and how to feasibly and reliably select the two optimal parameters is still unclear.  
In this work, we first deduce the essence of the CMOR wavelet transform based on the principle of Fourier Transform (FT). Then, we illustrate clearly the meaning of the time-frequency resolution and edge effect of CMOR wavelet transform. Thirdly, the selection method for the optimal CMOR wavelet parameters of \(f_{b}\) and \(f_{c}\) is proposed. Finally, based on the proposed method, we further perform the QTF analysis of water molecule vibration with the simulated H-atom trajectories. The results demonstrate that our method can accurately obtain the optimal parameters of CMOR wavelet for the target frequency. Furthermore, and as far as we know, the research about the QTF analysis of molecule vibration with the simulated atom trajectories has not appeared in the available literature.