A New Understanding of the Envelope and Resolving the Mode-M

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 3, MARCH 20121075EMD Revisited: A New Understanding of the Envelope and Resolving the Mode-Mixing Problem in AM-FM SignalsXiyuan Hu, Silong Peng, and Wen-Liang HwangAbstract—Empirical mode decomposition (EMD) is an adaptive and data-driven approach for analyzing multicomponent nonlinear and nonstationary signals. The stop criterion, envelope technique, and mode-mixing problem are the most important topics that need to be addressed in order to improve the EMD algorithm. In this paper, we study the envelope technique and the mode-mixing problem caused by separating multicomponent AM-FM signals with the EMD algorithm. We present a new necessary condition on the envelope that questions the current assumption that the envelope passes through the extreme points of an intrinsic mode function (IMF). Then, we present a solution to the mode-mixing problem that occurs when multicomponent AM-FM signals are separated. We experiment on several signals, including simulated signals and real-life signals, to demonstrate the ef cacy of the proposed method in resolving the mode-mixing problem. Index Terms—Empirical mode decomposition (EMD), envelope algorithm, mode mixing, signal decomposition.I. INTRODUCTION INGLE-CHANNEL signal separation and estimation has attracted a great deal of attention in recent years because it affects many applications. Typical single-channel signal separation approaches model a signal as a superposition of additive coherent basic signals. For instance, a nonlinear and nonstationary signal can be modeled as a multicomponent AM-FM signal. The methods used to separate signals vary because different subcomponents are used to construct the signals. Recently, the Empirical Mode Decomposition (EMD) approach has generated a lot of interest because it has a number of useful features [1]. The EMD algorithm is a fully data-driven and self-adaptive process that models the target signal as a series of intrinsic modeSManuscript received October 03, 2010; revised August 09, 2011 and November 26, 2011; accepted December 05, 2011. Date of publication December 14, 2011; date of current version February 10, 2012. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ta-Hsin Li. This work was supported by the National Natural Science Foundation of China (Grant 60972126), the Joint Funds of the National Natural Science Foundation of China (Grant U0935002/L05), the Beijing Natural Science Foundation (Grant 4102060), the State Key Program of National Natural Science Foundation of China (Grant 61032007), and the National Science Council of Taiwan (Grant 100-2221-E-001-017-). X. Hu and S. Peng are with the Institute of Automation, The Chinese Academy of Sciences, Beijing 100190, China (e-mail: xiyuan.hu@http://www.77cn.com.cn; silong.peng@http://www.77cn.com.cn). W.-L. Hwang is with the Institute of Information Science, Academia Sinica, Taipei 11529, Taiwan, and is also with the Department of Information Management, Kainan University, Taiwan (e-mail: whwang@iis.sinica.edu.tw). Digital Object Identi er 10.1109/TSP.2011.2179650functions (IMFs) plus a residual signal. An IMF satis es two conditions: 1) in the whole data set, the number of extrema and the number of zero-crossings must be equal, or differ by one at most; and 2) at any point, the local average of the upper and lower envelopes must be zero. To obtain an IMF, the EMD uses a sifting process based on the estimated upper and lower envelopes, interpolated from the extrema, of the input signal. The theoretical analysis and experiments on envelope interpolation are discussed in [2] and [3]. However, because of the obscure nature of the envelope, mathematically explicit and physically meaningful answers to questions like “What is an envelope?” and “What constitutes a good envelope?” are still elusive. We do not provide a de nition of the envelope in this paper; however, we derive the necessary conditions that an envelope must satisfy in EMD. EMD extracts the highest frequency component locally as the current IMF, which is derived from the extrema and the envelopes during the sifting process. Hence, it is inevitable that the IMF will be affected by the mode mixing problem caused by the intermittency, as noted by Huang [4]. The intermittency was originally referred to as the alternation of phases of apparently periodic and chaotic dynamics in turbulence. In this paper, we use the de nition in [5], where the intermittency is referred to as a component that comes into existence or disappears from a signal entirely at a particular time scale. The mode mixing problem occurs when the frequency tracks of an IMF jump as an intermittent component arrives or departs. In this paper, we analyze the properties of the envelope and the mode-mixing problem caused by separating AM-FM signals with the EMD algorithm. First, we analyze the properties of the envelope. Then, we propose an AM-FM demodulation algorithm, which is used later to solve the mode-mixing problem of an IMF. Normally, a multicomponent AM-FM signal is used to model a nonstationary signal and applied in the analysis of a broad range of signals in various elds, such as mechanics and vibrations [4], [6]. The IMF of a signal is always a monocomponent AM-FM signal, which facilitates meaningful instantaneous frequency (IF) estimation via the Hilbert transform, Teager-Kaiser energy operator [7] or other techniques [8]–[11]. We study the envelope of a monocomponent AM-FM signal and observe that its upper (resp. lower) envelope does not necessarily pass through its maximum (resp. minimum) points; however, it does pass by, and is tangential to, the points with a phase of . This observation motivates us to angle1053-587X/$26.00 © 2011 IEEE

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