TY - JOUR

T1 - Instantaneous frequency and wave shape functions (I)

AU - Wu, Hau Tieng

N1 - Funding Information:
The author acknowledges support by FHWA grant DTFH61-08-C-00028 and Award Number FA9550-09-1-0551 from AFOSR; he is also grateful for valuable discussions with Eugene Brevdo, Gaurav Thakur and Professor Ingrid Daubechies, and to Professor Chung-Kang Peng for kindly providing the respiratory signal during sleep and for many discussions. He thanks the anonymous reviewers for their useful recommendations to improve this paper.

PY - 2013/9

Y1 - 2013/9

N2 - Although one can formulate an intuitive notion of instantaneous frequency, generalizing "frequency" as we understand it in e.g. the Fourier transform, a rigorous mathematical definition is lacking. In this paper, we consider a class of functions composed of waveforms that repeat nearly periodically, and for which the instantaneous frequency can be given a rigorous meaning. We show that Synchrosqueezing can be used to determine the instantaneous frequency of functions in this class, even if the waveform is not harmonic, thus generalizing earlier results for cosine wave functions. We also provide real-life examples and discuss the advantages, for these examples, of considering such non-harmonic waveforms.

AB - Although one can formulate an intuitive notion of instantaneous frequency, generalizing "frequency" as we understand it in e.g. the Fourier transform, a rigorous mathematical definition is lacking. In this paper, we consider a class of functions composed of waveforms that repeat nearly periodically, and for which the instantaneous frequency can be given a rigorous meaning. We show that Synchrosqueezing can be used to determine the instantaneous frequency of functions in this class, even if the waveform is not harmonic, thus generalizing earlier results for cosine wave functions. We also provide real-life examples and discuss the advantages, for these examples, of considering such non-harmonic waveforms.

KW - Heart rate variability

KW - Instantaneous frequency

KW - Reassignment

KW - Respiratory rate variability

KW - Sleep stage

KW - Synchrosqueezing transform

KW - Wave shape function

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U2 - 10.1016/j.acha.2012.08.008

DO - 10.1016/j.acha.2012.08.008

M3 - Article

AN - SCOPUS:84878555758

SN - 1063-5203

VL - 35

SP - 181

EP - 199

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

IS - 2

ER -