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      Novel Fourier quadrature transforms and analytic signal representations for nonlinear and non-stationary time-series analysis

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          Abstract

          The Hilbert transform (HT) and associated Gabor analytic signal (GAS) representation are well known and widely used mathematical formulations for modelling and analysis of signals in various applications. In this study, like the HT, to obtain quadrature component of a signal, we propose novel discrete Fourier cosine quadrature transforms (FCQTs) and discrete Fourier sine quadrature transforms (FSQTs), designated as Fourier quadrature transforms (FQTs). Using these FQTs, we propose 16 Fourier quadrature analytic signal (FQAS) representations with following properties: (1) real part of eight FQAS representations is the original signal, and imaginary part of each representation is FCQT of real part; (2) imaginary part of eight FQAS representations is the original signal, and real part of each representation is FSQT of imaginary part; (3) like the GAS, Fourier spectrum of all FQAS representations has only positive frequencies; however, unlike the GAS, real and imaginary parts of FQAS representations are not orthogonal. The Fourier decomposition method (FDM) is an adaptive data analysis approach to decompose a signal into a set Fourier intrinsic band functions. This study also proposes new formulations of the FDM using discrete cosine transform with GAS and FQAS representations, and demonstrates its efficacy for improved time-frequency-energy representation and analysis of many real-life nonlinear and non-stationary signals.

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          ENSEMBLE EMPIRICAL MODE DECOMPOSITION: A NOISE-ASSISTED DATA ANALYSIS METHOD

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            Observation of Gravitational Waves from a Binary Black Hole Merger

            On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards in frequency from 35 to 250 Hz with a peak gravitational-wave strain of \(1.0 \times 10^{-21}\). It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole. The signal was observed with a matched-filter signal-to-noise ratio of 24 and a false alarm rate estimated to be less than 1 event per 203 000 years, equivalent to a significance greater than 5.1 {\sigma}. The source lies at a luminosity distance of \(410^{+160}_{-180}\) Mpc corresponding to a redshift \(z = 0.09^{+0.03}_{-0.04}\). In the source frame, the initial black hole masses are \(36^{+5}_{-4} M_\odot\) and \(29^{+4}_{-4} M_\odot\), and the final black hole mass is \(62^{+4}_{-4} M_\odot\), with \(3.0^{+0.5}_{-0.5} M_\odot c^2\) radiated in gravitational waves. All uncertainties define 90% credible intervals.These observations demonstrate the existence of binary stellar-mass black hole systems. This is the first direct detection of gravitational waves and the first observation of a binary black hole merger.
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              Variational Mode Decomposition

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                Author and article information

                Journal
                R Soc Open Sci
                R Soc Open Sci
                RSOS
                royopensci
                Royal Society Open Science
                The Royal Society
                2054-5703
                November 2018
                28 November 2018
                28 November 2018
                : 5
                : 11
                : 181131
                Affiliations
                School of Engineering and Applied Sciences, Bennett University , Greater Noida, India
                Author notes
                Author for correspondence: Pushpendra Singh e-mail: spushp@ 123456gmail.com , pushpendrasingh@ 123456iitkalumni.org
                Author information
                http://orcid.org/0000-0001-5615-519X
                Article
                rsos181131
                10.1098/rsos.181131
                6281917
                92329f01-7fc1-400f-83d0-b742e7b94555
                © 2018 The Authors.

                Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.

                History
                : 11 July 2018
                : 1 November 2018
                Categories
                1003
                3
                1006
                62
                1008
                119
                Engineering/Mathematics
                Research Article
                Custom metadata
                November, 2018

                hilbert transform,gabor analytic signal representation,instantaneous frequency,fourier quadrature transform,fourier quadrature analytic signal representations/fourier–singh analytic signal representations,zero-phase filtering

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