## 2 Selection Method of Wavelet Parameters
### The Essence of CMOR Wavelet Transform  
Wavelet function CMOR \(f_{b}-f_{c}\) is generally expressed as follows [23]:  
\[\psi(t)=\frac{1}{\sqrt{\pi f_{b}}}e^{-\frac{t^{2}}{f_{b}}}e^{j2\pi f_{c}t}=g( t)[\cos(2\pi f_{c}t)+j\sin(2\pi f_{c}t)] \tag{4}\]  
It can be found from Equation (4) that the real part of CMOR \(f_{b}-f_{c}\) is the product of the Gaussian function \(g(t)\) and the cosine function, which has a center frequency \(f_{c}\). As shown in Figure 1a, the support region of \(\psi(t)\) is defined by the \"3\(\sigma\)\" limits of Gaussian Distribution \(X\)-\(N\) (\(\mu\), \(\sigma^{2}\)) [25]. The \"3\(\sigma\)\" limits mean that the area under the distribution curve between \(t=\mu-3\sigma\) and \(t=\mu+3\sigma\) is 0.9973 [26, 27]. According to Equation (4), one can note the \(\sigma^{2}\) and \(\mu\) of Gaussian Distribution are \(f_{b}/2\) and 0, respectively. Thus, the support region of \(\psi(t)\) is \(\left(-3\sqrt{f_{b}/2},3\sqrt{f_{b}/2}\right)\). The real part of CMOR2-1 is shown in Figure 1a. Obviously, the cosine function (blue dotted line) is enveloped by the Gaussian function (green line) to form the real part of CMPR2-1 (red line) and there are \(6T_{c}\) in the support region.  
The CWT of \(u(t)\) is generally expressed as [28]:  
\[CWT(\alpha,\beta)=\left\langle\frac{1}{\sqrt{\alpha}}\psi\left(\frac{t-\beta}{ \alpha}\right),u(t)\right\rangle=\left\langle\psi_{\alpha,\beta}(t),u(t)\right\rangle \tag{5}\]  
where \(\alpha\) and \(\beta\) represent the dimensionless scale factor and the time translation factor, respectively. Plugging Equation (4) into Equation (5), the CMOR wavelet transform of \(u(t)\) is:  
\[CMOR(\alpha,\beta)=\left\langle e^{i2\pi(\frac{\xi}{\alpha})(t-\beta)},f(t) \right\rangle=F\left(\frac{f_{c}}{\alpha}\right) \tag{6}\]  
where \(f(t)=\frac{1}{\sqrt{\alpha\pi f_{b}}}e^{-\frac{(t-\beta)^{2}}{f_{b}\alpha^{2} }}u(t)=w(t)u(t)\). It can be found from Equation (6) that the essence for the CMOR wavelet transform of the signal \(u(t)\) is the FT of \(f(t)\). Moreover, \(f(t)\) is derived from the multiplication of \(u(t)\) and the window function \(w(t)\). Let the target frequency \(f_{k}=f_{c}/\alpha\), then the \(w(t)\) can be expressed as:  
\[w(t)=\sqrt{\frac{f_{k}}{\pi f_{c}f_{b}}}e^{-\frac{f_{k}^{2}}{f_{c}^{2}f_{b}}(t -\beta)^{2}} \tag{7}\]  
Obviously, \(w(t)\) is a Gaussian function and is dependent on the target frequency \(f_{k}\). In Figure 1b, we illustrated how the window function \(w(t)\) with different support regions is applied to a signal. For the signal with lower frequency, the window function having a large support region (orange line) is applied. Otherwise, the window function having a small support region (green line) is applied to the signal with higher frequency.  
By analogy with Equation (3), the CMOR wavelet transform of \(x(n)\) can be expressed as:  
\[WT(s,\tau)=\sum_{n=0}^{N-1}\frac{1}{\sqrt{s\pi f_{b}}}e^{-\frac{(n-\tau)^{2}} {a^{2}f_{b}}}x(n)e^{-j2\pi\frac{f_{c}}{\alpha}(n-\tau)} \tag{8}\]  
where \(s\) and \(\tau\) represent the scale factor in Hz and the time translation factor, respectively. It can be found from Equations (3) and (8) that \(f_{c}/s=k/N=f_{b}/f_{s}\), namely \(s=f_{c}f_{b}/f_{k}\). Therefore, the relationship between \(s\) and \(\alpha\) is \(s=af_{s}\).  
Figure 1: The essence of CMOR wavelet transform. (**a**) The real part of CMOR2-1, (**b**) The window functions corresponding to different target frequencies (\(f_{1},f_{2}\)) in \(u(t)\), (**c**) Time resolution and edge effect of CMOR wavelet.