## 2 Vibration Data and Early Data Processing
### Fast Fourier Transform  
In the first phase, we extract significant characteristics of the input signal by performing feature extraction. These characteristics vary from signal to signal and are statistical, domain-specific features, or both. In Figure 2, the vibration data are collected and stored as the time series, and we transform it into the frequency domain using Fast Fourier Transform (FFT) (Figure 3). The main purpose of this study is to generate and evaluate broken data that is limited in the experiments. Therefore, we need to analyze both the original and generated data with different approaches and AI models that can affect final results.  
The Fourier Transform is a function of frequency, which has the magnitude to represent the frequency in the time domain. The Fourier Transform can vary in the domains, but the original function is mostly considered as the time domain. The definition of the Fourier Transform of a Discrete Function is as follows.  
Figure 3: Fast Fourier Transform (FFT) transform of a (**a**) normal machine; (**b**) broken tooth machine in the _g_x_ direction.  
Figure 2: Vibration signal of normal machine (**a**); broken machine (**b**) in the _g_x_ direction.  
Let \(x_{0}\),..., \(x_{N-1}\) be complex numbers. We can calculate the Discrete Fourier Transform (DFT) in the time domain by using the following formula:  
\[x_{k}\ =\ \sum_{n=0}^{N-1}x_{n}e^{-i2\pi kn\ /N}, \tag{1}\]  
where \(x_{0}\),..., \(x_{N-1}\) are complex numbers, and \(e^{2\pi/N}\) is a primitive N-th root of 1 [23].  
For FFT, the formula changes into the following expression:  
\[x_{k}\ =\ \sum_{m=0}^{N/2-1}x_{2m}e^{-i2\pi k(2m)\ /N}+\ \sum_{m=0}^{N/2-1}x_{2m+1}e^{-i2\pi k (2m+1)\ /N}, \tag{2}\]  
\[x_{k}\ =\ \sum_{m=0}^{N/2-1}x_{2m}e^{-i2\pi k(m)\ /(N/2)}+\ \sum_{m=0}^{N/2-1}x_{2m+1}e^{-i2\pi k (m)\ /(N/2)}. \tag{3}\]  
By transforming Formula (1) into Formula (2) and (3), the Fourier Transform is split into two smaller transforms with odd-numbered values and even-numbered values. At this point, we did not decrease the computational complexity, which consists of \(2\times[(N/2)\times N]\), for a total of \(N^{2}\).  
With the symmetrical characteristics of Formulas (2) and (3), we can reduce the number of computations. The value of k is defined as \(0\leq k<N\), whereas the value of \(n\) is \(0\leq n<M\equiv N/2\); each sub-problem required half the computation of the original one. The total computation is reduced from \(O[N^{2}]\) to \(O[(N/2)^{2}]\)[24].  
The FFT requires lower computation as compared with the original FFT, which is suitable for real-time applications. In this study, we focus on the commercial application in the industrial environment, which requires both high accuracy and real scenario response. Due to these reasons, FFT would be a better choice compared to the DFT. The advantage of FFT is that we can process more significant features in the frequency domain classification between the normal and broken machine signals. Another advantage of the FFT transform is that the generated signal is evaluated indirectly, which leads to better performance analysis. We can consider FFT as one of the most effective methods used to extract the vibration input pattern [5]. Therefore, we use FFT as the basic processing method for further research. Based on the FFT input signal, we propose three approaches: full analysis, PCA transform, and statistical analysis. In the next section, we will discuss different approaches and AI models that can be applied to the fault diagnosis results.