## 4 Methodology
Based on this idea, the proposed method is employed as presented in the summary chart in Figure 1. Hence, the sidebands next to the meshing frequency can be efficiently explored by employing the Generalized Fourier Transform in conjunction with the Wavelet Synchrosqueezed Transform. This is a fault symptom of a faulty gearbox. Figure 1: Flowchart of the proposed method. In summary, the proposed method, which can be divided into three parts, is implemented step by step as follows:  
1. Execute CWT: * Step 1 (CWT): Apply FFT to the mother wavelet function to obtain the base wavelet in the frequency domain. Then, the time-frequency representations from the raw signal and the base wavelet function in the frequency domain are calculated. 2. Execute WSST: * Step 2 (Estimate frequency): Apply derivative to the TFR to obtain the instantaneous rotation frequency. * Step 3 (Execute WSST): with the instantaneous rotation frequency, it is possible to squeeze the CWT's TFR. As a result, the resolution is much better. 3. Execute the proposed method: * Step 4 (analytic signal): select const frequency \(f_{0}\) to calculate phase \(x_{0}(t)\). Then, the Hilbert transform is applied to obtain an analytic signal. * Step 5 (execute GFT): apply GFT by multiplying the analytic signal with a reference phase \(e^{-\beta\tau x_{0}(t)}\). Apply the Hilbert transform again to obtain a new analytic signal. * Step 6 (execute CWT): convert the new signal to TFR. * Step 7 (execute iGFT): apply inverse GFT to reconstruct the signal with separated frequency component \(f_{0}\). * Step 8 (execute WSST): apply WSST to obtain the high-resolution GMF and its sidebands. Then, conclude the gear condition based on the obtained. In order to clarify the method's concept, a numerical simulation example is demonstrated. This example involves creating a multi-frequency signal and exploring techniques to filter and extract various frequency components, even when they overlap. Figure 2 displays the outcomes of examining a signal with three distinct frequency components that fluctuate over time within a limited frequency region. \[ \begin{array}{l}f_{1}(t)=10\big{(}-25.8t^{2}+40t+24\big{)}-0.24 sin(0.6\pi t) \\ f_{2}(t)=11\big{(}-25.8t^{2}+40t+24\big{)}-0.24 sin(0.6\pi t)\\ f_{3}(t)=9\big{(}-25.8t^{2}+40t+24\big{)}-0.24 sin(0.6\pi t)\end{array}  \tag{11}\]  
This multi-component signal is represented by the CWT transformation, as shown in Figure 2 on the left and by the WSST transformation, as shown in Figure 2 on the right. It is evident that the CWT transformation yields a high time resolution but needs to clearly distinguish the three frequency components \(f_{1}\), \(f_{2}\), and \(f_{3}\). With the WSST transformation, having frequency concentration around the central frequency is better, as  
Figure 2: Comparing the TFR of the signal using CWT, WSST. shown in Figure 3. However, determining the boundary frequencies around the central frequency is challenging. The results of the simulation verification demonstrate that applying the updated Wavelet analysis in the case of multi-component signals yields better results than traditional Wavelet analysis, especially for signals with time-varying frequencies.