## 2 Basic Theory
### Spectral Kurtosis  
Kurtosis refers to a wave peak statistic that describes the distribution characteristics of vibration signals. Because it is sensitive to impact signals, it can be used to diagnose the faults of damaged surfaces, and is effective in the early detection of faults. As global characteristics, kurtosis values are affected by weak fault features and the interference of noise or other unknown components in the signal. SK, which was proposed by [PERSON] [8], represents the kurtosis of the original data at a specific frequency. It can be calculated as follows:  
\[K(f)=\frac{E\Big{[}|X(t,f)|^{4}\Big{]}}{\Big{\{}E\Big{[}|X(t,f)|^{2}\Big{]} \Big{\}}^{2}}-2 \tag{11}\]  
where \(|\cdot|\) and \(E[\cdot]\) represent the absolute value and expectation, respectively, and \(X(t,f)\) refers to the complex envelope of signal \(x(t)\) in frequency \(f\). \(X(t,f)\) can be computed using the short-time Fourier transform (STFT) or fast computation of SK [24]. Based on the STFT, nonstationary signals will be processed as local stationary signals; thus, it is not applicable to impact signals. When an inappropriate frequency resolution \(\Delta f\) is chosen, the computation of SK by STFT will be affected. Therefore, this paper adopts the latter for computation. By building a series of band-pass filters with a pyramid algorithm, the SK in each frequency band inside the plane composed of \(f\) and \(\Delta f\) can be obtained through fast computation of SK [11]. Thus, the maximum value of SK and the optimal frequency band can be determined. The computation steps are as follows:  
1. Design a low-pass filter \(h(n)\) with a cut-off frequency of \(1/8\), and set up a quasi-analytical filter based on \(h(n)\). The normalized analysis frequency band of the quasi-analytical low-pass filter \(h_{0}(n)\) is [0,1/4]. Correspondingly, the normalized analysis frequency band of the quasi-analytical high-pass filter \(h_{1}(n)\) is [1/4,1/2]. The equations are as follows: \[\left\{ \begin{array}{l}h_{0}(n)=h(n)\cdot e^{j\pi n/4}\\ h_{1}(n)=h(n)\cdot e^{j3\pi/4}\end{array} \right.\] (12)
2. Apply \(h_{0}(n)\) and \(h_{1}(n)\) to filtering the signal. Then, the results after being filtered are reducing, sampled twice to ensure that each layer resolved by the filter is the same as the original data in terms of length. According to Equation (11), the original signal will be resolved with the \(k\) layer. The decomposition algorithm can be realized as follows: \[\left. \begin{array}{l}c_{k+1}^{2i}(m)=h_{0}(n)*c_{k}^{i}(2m)\\ c_{k+1}^{2i+1}(m)=h_{1}(n)*c_{k}^{i}(2m)\end{array} \right\}\] (13)  
The output \(c_{k}^{i}(n)\) will be obtained after the original signal is filtered through the \(i\)th filter at the \(k\)th layer, and the value of \(i\) ranges from 0 to \(2^{k}-1\). \(c_{k}^{i}(n)\) can also be regarded as the complex envelope of the band-pass signal whose center frequency is \(f_{i}=(i+2^{-1})2^{k-1}\) and bandwidth is \(\Delta f_{k}=2^{k-1}\). To improve the decomposition accuracy and avoid similarity between the SK values of frequency bands [0,1/4] and [1/4,1/2], additional decompositions of [0,1/6], [1/6,1/3], and [1/3,1/2] are conducted at each layer, and three signals are obtained. The principle here is the same as that of the two decomposition signals previously discussed. Calculate the kurtosis of each output signal \(c_{k}^{i}(n)\) and obtain the kurtosis of the practical signal according to Equation (14): \[K(f_{i},\Delta f_{k})=\frac{E\Big{\langle}\big{|}c_{k}^{i}\big{|}^{4}\Big{\rangle} }{\Big{\{}E\Big{\langle}\big{|}c_{k}^{i}(n)\big{|}^{2}\Big{\rangle}\Big{\}} ^{2}}-2\] (14) Perform quantization of the kurtosis with color and draw a fast spectral kurtosis. 4.