## 3 Simulation Study  
As shown in Equation (4), a synthetic fault signal equivalent to the BRB fault signal that is also a kind of CS2 signal is established to verify the performance of the proposed method. Where \(\alpha\) is the main cyclic frequency caused by BRB faults and \(f_{r}\) denotes the carrier frequency of IMs. When the BRB fault occurs, the main cyclic frequency \(\alpha\) will modulate on the rotation frequency \(f_{r}\) or its harmonics. Where \(A_{1}\) and \(A_{2}\) represents the amplitudes of \(\alpha\) and \(f_{r}\), and the simulated signal was corrupted by Gaussian white noise \(n(t)\) with the signal-to-noise ratio (SNR) of \(-2\) dB. The simulation analysis was done by processing the signal expressed in Equation (4) with the amplitudes \(A_{1}=A_{2}=1\) and two different sets of cyclic frequencies and carrier frequencies. The frequency settings were \(\alpha_{1}=4\) Hz,\(f_{r1}\) = 40 Hz and \(\alpha_{2}=100\) Hz,\(f_{r2}\) = 400 Hz. In addition, the sampling frequency \(Fs\) and sampling points  
Figure 1: The flowchart of the proposed method. \(L\) of simulation signal was setup as 1000 Hz and 10\({}^{5}\), respectively. It needs to be noted that all data processing is performed on a computer with an i5-7400 CPU processor 3.00 GHz. \[x_{BRB}(t)=(1+A_{1}\cdot\sin(2\cdot\pi\cdot\alpha\cdot t))\cdot A_{2}\cdot\sin(2 \cdot\pi\cdot f_{r}\cdot t)+n(t) \tag{4}\]  
Firstly, the advantage of larger cyclic frequency range of CMS-CWT method will be verified. In the processing of the CMS and FSC methods based on STFT, a Hamming window of length \(Nw=2^{7}\) was used for STFT calculation and the step size \(R\) is set to 25% of the window length (\(R=32\)). The total number of windows shifted by \(R\) samples in a \(L\)-long signal is \(K=3122\). Table 1 presents the cyclic frequency resolution and the maximum cycle frequency values that can be analyzed through three approaches based on the simulated signals. As shown in Figure 2, the CMS method can only analyze the signal with the cyclic frequency in the range of 0-_Fs/\(Nw\)_. When the cycle frequency to be analyzed is greater than 7.8125, the signal begins to drift downward, which greatly limits the signal analysis ability of the CMS algorithm. Although the FSC method extends the analysis range to _Fs/\(2R\)_ as illustrated in Figure 3, the cyclic frequencies of 15.625-500 Hz cannot be analyzed and processed effectively, which significantly limits the application of the FSC method in condition monitoring and fault diagnosis fields. According to the analysis results presented in Figure 4, the proposed CMS-CWT approach received a better analysis in the range of 0-500 Hz. Furthermore, the FSC can enlarge the cyclic frequency range to 500 Hz by setting \(R\) to 1, but it can be seen from Figure 5 the signal generates drift, which increased both signal redundancy and computational cost due to excessive signal overlap in the calculation of STFT. Moreover, the FSC took 109.053 s to get analysis result as shown in Figure 5 compared to 3.5920 s utilized by CMS-CWT to process the same data as shown in Figure 4. In short, the proposed CMS-CWT approach can provide larger analysis range of cyclic frequency and more accurate diagnosis results than the CMS and FSC in the cyclic frequency axis. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameters** & **CMS** & **FSC** & **CMS-CWT** \\ \hline \(\Delta\alpha\) & _Fs/L_ & _Fs/L_ & _Fs/L_ \\ Value & 0.01 Hz & 0.01 Hz & 0.01 Hz \\ \(a_{max}\) & _Fs/Nw_ & _Fs/2R_ & _Fs/2_ \\ Value & 7.8125 Hz & 15.625 Hz & 500 Hz \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters setting. Figure 2: Vibration spectrum analyzed by cyclic modulation spectrum (CMS) with \(R=32\). Moreover, the strength of higher carrier frequency resolution of CMS-CWT algorithm was verified. According to the Equation (4), two simulation signals with sampling rates of 1000 Hz and 10,000 Hz  
Figure 4: Vibration spectrum analyzed by cyclic modulation spectral-continuous wavelet transform (CMS-CWT).