A real variable definition of the hilbert transform on d' and s'

1988 ◽  
Vol 29 (3-4) ◽  
pp. 235-251 ◽  
Author(s):  
C. Carton-Lebrun
Keyword(s):  
Hilbert Transform ◽  
Real Variable ◽  
Variable Definition ◽  
Definition Of ◽  
The Hilbert Transform

scholarly journals A Consistent Definition of Phase Resetting Using Hilbert Transform

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Sorinel A. Oprisan
Keyword(s):  
Membrane Potential ◽  
Hilbert Transform ◽  
External Perturbation ◽  
Accurate Estimate ◽  
Neural Oscillator ◽  
Dynamical Response ◽  
Phase Resetting ◽  
Potential Oscillations ◽  
Definition Of ◽  
The Hilbert Transform

A phase resetting curve (PRC) measures the transient change in the phase of a neural oscillator subject to an external perturbation. The PRC encapsulates the dynamical response of a neural oscillator and, as a result, it is often used for predicting phase-locked modes in neural networks. While phase is a fundamental concept, it has multiple definitions that may lead to contradictory results. We used the Hilbert Transform (HT) to define the phase of the membrane potential oscillations and HT amplitude to estimate the PRC of a single neural oscillator. We found that HT’s amplitude and its corresponding instantaneous frequency are very sensitive to membrane potential perturbations. We also found that the phase shift of HT amplitude between the pre- and poststimulus cycles gives an accurate estimate of the PRC. Moreover, HT phase does not suffer from the shortcomings of voltage threshold or isochrone methods and, as a result, gives accurate and reliable estimations of phase resetting.

scholarly journals Two-weight inequality for the Hilbert transform: A real variable characterization, I

2014 ◽  
Vol 163 (15) ◽  
pp. 2795-2820 ◽  
Author(s):  
Michael T. Lacey ◽  
Eric T. Sawyer ◽  
Chun-Yen Shen ◽  
Ignacio Uriarte-Tuero
Keyword(s):  
Hilbert Transform ◽  
Real Variable ◽  
Weight Inequality ◽  
The Hilbert Transform

Model of multicomponent narrow-band periodical non-stationary random signal

2020 ◽  
Vol 2020 (48) ◽  
pp. 17-24
Author(s):  
I.M. Javorskyj ◽  
◽  
R.M. Yuzefovych ◽  
P.R. Kurapov ◽  
◽  
...  
Keyword(s):  
Time Series ◽  
Real Time ◽  
Hilbert Transform ◽  
Spectral Properties ◽  
Narrow Band ◽  
Analytic Signal ◽  
Random Signal ◽  
Hilbert Transformation ◽  
The Hilbert Transform

The correlation and spectral properties of a multicomponent narrowband periodical non-stationary random signal (PNRS) and its Hilbert transformation are considered. It is shown that multicomponent narrowband PNRS differ from the monocomponent signal. This difference is caused by correlation of the quadratures for the different carrier harmonics. Such features of the analytic signal must be taken into account when we use the Hilbert transform for the analysis of real time series.

A Bayesian View on the Hilbert Transform and the Kramers-Kronig Transform of Electrochemical Impedance Data: Probabilistic Estimates and Quality Scores

2020 ◽  
Author(s):  
Jiapeng Liu ◽  
Ting Hei Wan ◽  
Francesco Ciucci
Keyword(s):  
Power Generation ◽  
Electrochemical Impedance Spectroscopy ◽  
Hilbert Transform ◽  
Electrochemical Impedance ◽  
The Self ◽  
Impedance Data ◽  
The Hilbert Transform ◽  
Validation Tests ◽  
Self Consistency ◽  
Mathematical Relations

<p>Electrochemical impedance spectroscopy (EIS) is one of the most widely used experimental tools in electrochemistry and has applications ranging from energy storage and power generation to medicine. Considering the broad applicability of the EIS technique, it is critical to validate the EIS data against the Hilbert transform (HT) or, equivalently, the Kramers–Kronig relations. These mathematical relations allow one to assess the self-consistency of obtained spectra. However, the use of validation tests is still uncommon. In the present article, we aim at bridging this gap by reformulating the HT under a Bayesian framework. In particular, we developed the Bayesian Hilbert transform (BHT) method that interprets the HT probabilistic. Leveraging the BHT, we proposed several scores that provide quick metrics for the evaluation of the EIS data quality.<br></p>

scholarly journals A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 65
Author(s):  
Benjamin Akers ◽  
Tony Liu ◽  
Jonah Reeger
Keyword(s):  
Radial Basis Function ◽  
Hilbert Transform ◽  
Basis Function ◽  
Differential Operators ◽  
Convergence Rates ◽  
Traveling Wave Solutions ◽  
Algebraic Decay ◽  
Pseudo Differential Operators ◽  
Radial Basis ◽  
The Hilbert Transform

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

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