## 1 Introduction
Moreover, CS2 signal is a type of special non-stationary signal that includes periodic modulation contents related to mechanical failures [38]. For example, the Fourier transform (FT) is effective in processing first order cyclostationary (CS1) signals, and it has been proven theoretically and practically not applicable to the identification of CS2 contents [39]. The BRB fault vibration signals issued from IMs are CS2 signals, which can be effectively processed via cyclostationary analysis methods. The cyclic modulation spectral (CMS) and fast spectral correlation (FSC) based on the short-time Fourier transform (STFT) proposed by [PERSON] in 2007 [40] and 2017 [41], respectively, are two computationally efficient cyclostationary analysis approaches, and their effectiveness and high efficiency were validated on bearing faults signals. Besides, they have the advantages of high cycle frequency resolution and the resolution is _Fs/L_ (_Fs_ denotes the sampling frequency and \(L\) is sampling points). On the one hand, although CMS can be performed efficiently, it still has the limitation that the maximum cyclic frequency \(\alpha_{max}\) that can be analyzed is not more than \(\Delta f\) (\(\Delta f\) = _Fs/Nw_ and _Nw_ is the window length used in STFT). Hence, the introduce of FSC compensates the deficiency of CMS, and it not only enlarges the range \(\alpha_{max}\) to _Fs/2R_ (_R_ is the step size of window used in STFT), but also still guarantees high computational efficiency. However, the existence of \(R\) limits the range of \(\alpha_{max}\) for FSC. Meanwhile the \(\alpha_{max}\) can be normally extended to _Fs/2_ by setting the \(R\) to 1, which makes the analysis of FSC method increase the computational cost and the redundancy of the signal. At the same time, these two methods cannot provide a satisfactory carrier frequency resolution because the STFT has a well-known limitation on time-frequency resolution, mainly due to the use of a fixed window size. However, the STFT can only offer the time-frequency information with lower precision, and this precision is determined by just the window size. In the application of STFT, the time domain and frequency domain resolution cannot be optimized simultaneously. However, it requires a more flexible method, which can adaptively change the window size to receive more accurate time or frequency [42]. In the case of good time-frequency localization performance of wavelet transform, a new approach was proposed in this paper that optimized the CMS algorithm based on the use of the continuous wavelet transform (CWT) to overcome the deficiencies from simple CMS and FSC and provide more accurate analysis results. The CWT adopts multi-scale analysis by analyzing lower frequencies with larger window size and analyzing higher frequencies with smaller window size. This makes CWT adaptively choose finer frequency resolution and coarser time resolution when processing lower frequencies, while choose coarse frequency resolution and finer time resolution at higher frequencies. Positioning the local features of the sample can improve recognition accuracy [43]. The CWT is a useful method to characterize local time-frequency characteristic of non-stationary signals [44]. As a result, the proposed method combining CMS and CWT (CMS-CWT) approaches has the ability to enlarge the range of detectable modulation frequencies \(\alpha_{max}\) to _Fs_/2 and make sure the cyclic frequency resolution is still _Fs_/_L_. Meanwhile, it can receive higher carrier frequency resolution than CMS and FSC based on the STFT method. In order to verify the effectiveness and performance, the proposed approach is validated using CS2 simulation signals and vibration signals collected from healthy IM and faulty motor with one BRB fault under various load conditions. The rest of the paper is organized as follows: the brief description of CMS-CWT method is presented in Section 2. Subsequently, the Section 3 shows the simulation results of synthetic signals.