Shock and Vibration

Volume 2017 (2017), Article ID 6987250, 10 pages

https://doi.org/10.1155/2017/6987250

## An Adaptive Spectral Kurtosis Method Based on Optimal Filter

Tianjin Key Laboratory of Optoelectronic Detection Technology and Systems, Tianjin Polytechnic University, Tianjin 300387, China

Correspondence should be addressed to Yanli Yang

Received 1 June 2017; Revised 29 September 2017; Accepted 8 October 2017; Published 5 November 2017

Academic Editor: Carlo Rainieri

Copyright © 2017 Yanli Yang and Ting Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

As a useful tool to detect protrusion buried in signals, kurtosis has a wide application in engineering, for example, in bearing fault diagnosis. Spectral kurtosis (SK) can further indicate the presence of a series of transients and their locations in the frequency domain. The factors influencing kurtosis values are first analyzed, leading to the conclusion that amplitude, not the frequency of signals, and noise make major contribution to kurtosis values. It is helpful to detect impulsive components if the components with big amplitude are removed from composite signals. Based on this cognition, an adaptive SK algorithm is proposed in this paper. The core steps of the proposed SK algorithm are to find maxima, add window around maxima, merge windows in the frequency domain, and then filter signals according to the merged window in the time domain. The parameters of the proposed SK algorithm are varying adaptively with signals. Some experimental results are presented to demonstrate the effectiveness of the proposed algorithm.

#### 1. Introduction

Kurtosis proposed by Dyer and Stewart in the 1970s [1] is used generally in the statistical field to describe the distribution, or skewness, of observed data around the mean. In engineering, kurtosis can be used to detect machine faults. It is already used for diagnosis of bearing, because it helps to find the crack failure from vibration signals. As a dimensionless parameter, kurtosis index has nothing to do with the bearing size, speed, and load, but it is sensitive to impact signals. It is especially suitable for the surface damage fault detection in the early fault diagnosis.

However, kurtosis is a global method and it only provides a vague suggestion. To solve this problem, spectral kurtosis (SK) is introduced by Dwyer [2] for detecting impulsive events in sonar signals. A formalization of SK is given by Antoni [3], in which the short-time Fourier transform- (STFT-) based SK is proposed. At the same time, Antoni and Randall [4] proposed the concept of kurtogram which uses the SK as a basis for detecting ad hoc detection filters to extract mechanical signature of faults. The SK is a statistical tool that can indicate the presence of series of transients and their locations in the frequency domain [3]. It provides a robust way of detecting incipient faults even in the presence of strong masking noise and offers an almost unique way of designing optimal filters for filtering out the mechanical signature of faults, which served as a useful tool to monitor the running status and diagnosis fault of mechanical machine [4]. The maximum correlation kurtosis deconvolution technique [5], the Morlet wavelet [6], and the probabilistic principal component analysis [7] can be used to enhance the capability of the SK.

Although the SK has attracted much attention due to the kurtogram, it is pointed out in [8] that the kurtogram cannot cope well with signals of composite and frequency of randomly impulsive nature, and it is vulnerable to random extraneous signals. It is further pointed out in [9] that the STFT-based SK technique may not be practical because it is unrealistic to find the optimal bandwidth and frequency of filters by examining all the window lengths. To overcome the limitation of kurtogram, a method for the selection of optimal bandpass filter to calculate SK, termed as protrugram, is proposed in [8]. An improved kurtogram was proposed in [10] based on wavelet packet. It is an intrinsic demand of the SK to separate the impulsive components into subband signals adaptively and then calculate the value of kurtosis. In fact, an adaptive windowed SK is proposed by Wang and Liang [9] through merging windows along the frequency axis in the frequency domain, which can optimize filter bandwidth and locate center frequency when used in fault detection on rotate machine. It is shown in [11] that the adaptive windowed SK more effectively extracts signatures of multiple bearing faults. Nevertheless, the optimal merging window associated with the highest SK does not match better with the signal transient feature when the transient impulse decays slowly [12]. An adaptive SK filtering method based on the Morlet wavelet is proposed in [12] to extract the signal transient buried in noise. A kurtosis-guided adaptive demodulation technique based on the tunable-Q wavelet transform is proposed in [13] for bearing faults detection. An optimized SK is proposed in [14] for selecting the best demodulation band to extract bearing fault-related impulsive content from vibration signals contaminated with strong electromagnetic interference.

The SK method is already proven to be useful in detecting nonstationary components of signals. Empirical mode decomposition (EMD) is a method to deal with nonstationary signals adaptively [15]. Inspired by our previous work [15, 16] and the work done in [8–14], an adaptive SK algorithm is proposed in this paper. In Section 2, kurtosis and SK are reviewed briefly. Section 3 analyzes the factors that have an influence on kurtosis values. In Section 4, the detailed description of the proposed adaptive SK algorithm is presented. Then, Section 5 shows some simulated examples. Finally, the conclusions are drawn in Section 6.

#### 2. Kurtosis and Spectral Kurtosis

Kurtosis is a reflection of the probability distribution of signals. Mathematically, for a time series , kurtosis is defined as [17]where is the mean of , is the standard deviation of , and is the expectation operator. Equation (1) shows that kurtosis is the fourth central moment divided by the square of the variance. The kurtosis of the normal distribution is 3, so some definitions of kurtosis subtract 3 from the computed value. In this paper, (1) is used to calculate kurtosis in the subsequent analysis.

By using the theory of signal and system, Antoni [3] has given the definition of SK as the fourth-order spectral cumulant. For a signal , the SK is written as [3]where denotes the SK of , represents the time/frequency envelope of the signal , and stands for the time averaging operator. For the case of detecting a signal buried in strong additive noise , written asthe SK of can be expressed by [4]where denotes the SK of and is the noise-to-signal ratio (NSR).

The SK is an effective and important tool to locate the frequency bands with a high amount of impulsiveness. However, the biggest value of kurtosis does not locate at the lowest level in kurtogram. Here, a simple example is presented to illustrate it. The signal considered iswhere is a strong signal and is a periodic impulsive signal, as shown in Figure 1. According to [18], the signal shown in Figure 1 is a simulated vibration signal arising from a rolling element bearing which can be written aswhere is the amplitude of the fault impulse, denotes the structural damping characteristic, denotes the time period corresponding to the fault characteristic frequency, represents the effect of random slippage of the rollers, denotes the excited resonance frequency, and is a unit step function. In (5), are some harmonic interferences which can be written as [13]Combining (5), (6), and (7), the signal can be expressed asWe can see from Figure 1 that the center of impulsive signal is located at 1500 Hz. The kurtogram results of the simulated signal are shown in Figure 2. It is clear from Figure 2(a) that the biggest value of kurtosis is located on level 2, and the band is located between 5000 Hz and 7500 Hz. However, Figure 2(b) shows that the biggest value of kurtosis is located on level 5, but it is less than 3.2. Some questions naturally arise. The first question is as follows: why is the value of kurtosis big for a signal but small in its subbands? Another question is why does the SK value drop obviously in the case of strong noise? To answer those questions, we argue that the main reasons of influencing on the value of kurtosis must be clear. In the next section, we discuss the factors that have an influence on kurtosis values.