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 -