Short communication The sum of weighted normalized square envelope: A unified framework for kurtosis, negative entropy, Gini index and smoothness index for machine health monitoring Dong Wang ⇑ , Zhike Peng, Lifeng Xi The State Key Laboratory of Mechanical Systems and Vibration, Shanghai Jiao Tong University, Shanghai 200240, PR China article info Article history: Received 19 July 2019 Received in revised form 25 November 2019 Accepted 6 February 2020 Keywords: Health monitoring Gini index Smoothness index Kurtosis Negative entropy Unified framework abstract Machine health monitoring aims to use monitoring data collected from a machine to assess machine degradation and prevent unexpected machine failures. Spectral kurtosis, spectral negative entropy, spectral Gini index and spectral smoothness index are well-known indices for characterizing the impulsiveness of repetitive transients caused by early machine faults. In this paper, it is discovered that all of these indices fall into the sum of weighted normalized square envelope. The main difference among these indices is that dif- ferent weights are respectively applied to normalized square envelope. Further, weight designs by domain knowledge and a data-driven method are respectively presented. Then, the proposed unified framework is applied to analyze bearing run to failure degrada- tion data. The proposed framework can be easily extended to other health monitoring sit- uations in which repetitive transients and fault frequencies are of concern. Ó 2020 Elsevier Ltd. All rights reserved. 1. The sum of weighted normalized square envelope for machine health monitoring Machine health monitoring is to use monitoring data collected from a machine to assess machine degradation over time. Repetitive transients are common symptoms of incipient faults in rotating machines. Characterizing repetitive transients is beneficial to constructing some indices for evaluating machine health conditions [1]. The main idea of spectral kurtosis [2], spectral negative entropy [3], spectral Gini index [4] and the reciprocal of spectral smoothness index [4] (an initial use of smoothness index for bearing health monitoring can be found in reference [5]) is to respectively use kurtosis [6], negative entropy [7], Gini index [8] and the reciprocal of smoothness index [9] to quantify a signal x l;h n ½; n ¼ 1; 2; ... ; N preprocessed by a band-pass filter with a non-dimensional pass-band l 6 k < h. The larger spectral kurtosis, spectral negative entropy, spectral Gini index and the reciprocal of spectral smoothness index are, the more impulsive the filtered signal x l;h n ½ is. There- fore, maximization of these indices can be used to characterize the impulsiveness of repetitive transients and then adaptively find optimal filtering parameters. Moreover, these indices over time can be used to describe machine degradation. To dis- cover a unified framework for spectral kurtosis, spectral negative entropy, spectral Gini index and the reciprocal of spectral smoothness index, these indices are reformulated in the following. https://doi.org/10.1016/j.ymssp.2020.106725 0888-3270/Ó 2020 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: dongwang4-c@sjtu.edu.cn (D. Wang), z.peng@sjtu.edu.cn (Z. Peng), lfxi@sjtu.edu.cn (L. Xi). Mechanical Systems and Signal Processing 140 (2020) 106725 Contents lists available at ScienceDirect Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp