## 2 The Cyclic Modulation Spectral Based on Continuous Wavelet Transform  
The CWT is widely used in signal processing as an effective time-frequency analysis tool. It is able to construct a time-frequency representation of a signal that shows better time and frequency localization. CWT analysis also allows for choosing a longer time interval when a more precise low-frequency feature is needed, and shorter ones when high-frequency information is desired. It is operated by breaking up a signal into shifted and scaled versions of the mother wavelet, so that the local features can be described better with wavelets that are concentrated on a localized region [45]. The CWT of a signal \(x\)(\(t\)) is defined as a convolution integral of \(x\)(\(t\)) with scaled versions of a mother wavelet function \(\Psi\tau\), \(c(t)\) and is given by Equation (1):  
\[W_{x}(\tau,c)=\frac{1}{\sqrt{c}}\int_{-\infty}^{+\infty}x(t)\Psi^{*}(\frac{t- \tau}{c})dt \tag{1}\]  
where \(\Psi^{*}(t)\) is the complex conjugate of \(\Psi(t)\), \(c\) is the scale parameter for changing the oscillating frequency, and \(\tau\) is the translation parameter. Because the CWT is performed through convolution between wavelet function and original signal, the \(\tau\) = 0, 1, 2... \(L\)-1 (\(L\) denotes the length of original signal). There are a number of different wavelet functions that can be utilized for CWT analysis. In this work, the Coif4 wavelet is determined as the mother wavelet due to its similarity to a BRB fault component that is a fault characteristic in an IMs. Theoretically, the actual signal frequency corresponding to the scale parameter \(c\) is \(f=Fc\)-_Fs_/_c_ (\(Fc\) is the wavelet center frequency, and _Fs_ is the sampling frequency). According to the equivalence between \(f\) and \(c\), Equation (1) can be turned into Equation (2) as follows:  
\[W_{x}(\tau,f)=\frac{1}{\sqrt{c}}\int_{-\infty}^{+\infty}x(t)\Psi^{*}(\frac{t- \tau}{c})dt \tag{2}\]  
Then, it is noted that \(\left|W_{x}(\tau,f)\right|^{2}\) can reflect the intensity as well as the distribution of signal energy, which is an important distinction between fault components and noise components. At last,the CMS-CWT algorithm can be efficiently computed as the discrete Fourier transform (DFT), which causes time variable \(\tau\) to be converted into cyclic frequency variable \(\alpha\).  
\[S_{x}^{\text{CMS-CWT}}(\alpha,f)=\frac{1}{LFs}{\sum_{\tau=0}^{L-1}}{\left|{W_{x} (\tau,f)}\right|^{2}}e^{-{\rho}^{2\pi\tau}\frac{\alpha}{Fs}} \tag{3}\]  
In the case of the CMS and FSC based on STFT, the number used for DFT is the total number of \(Nw\)-long window shifted by \(R\) samples in a \(L\)-long signal. In contrast, the proposed CMS-CWT method uses the \(L\)-length signal to do DFT every time. This is the reason why CMS-CWT method can flexibly handle the signal with the maximum cycle frequency associated with the faults in the range of 0-\(Fs\)/2. Theoretical analysis shows that the proposed method can provide the same cyclic frequency resolution as CMS and FSC. All in all, this method can provide a larger cycle frequency analysis range and more accurate carrier frequency resolution than CMS and FSC. The flowchart of the proposed method based on CMS-CWT is shown in Figure 1.