Systematic and Statistical Uncertainties of the Hilbert-Transform Based High-precision FID Frequency Extraction Method

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.

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

10.26434/chemrxiv.12152529 ◽

2020 ◽

Cited By ~ 1

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>

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

Mathematics ◽

10.3390/math9010065 ◽

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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Characterization of the peak value behavior of the Hilbert transform of bounded bandlimited signals

Problems of Information Transmission ◽

10.1134/s0032946013030010 ◽

2013 ◽

Vol 49 (3) ◽

pp. 197-223

Author(s):

H. Boche ◽

U. J. Mönich

Keyword(s):

Hilbert Transform ◽

Bandlimited Signals ◽

The Hilbert Transform ◽

Peak Value

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Simple proofs of the Bedrosian equality for the Hilbert transform

Science in China Series A Mathematics ◽

10.1007/s11425-009-0001-2 ◽

2009 ◽

Vol 52 (3) ◽

pp. 507-510 ◽

Cited By ~ 5

Author(s):

SiLei Wang

Keyword(s):

Hilbert Transform ◽

The Hilbert Transform

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Phase Synchronization Is the Amplified Result by the Hilbert Transform

Mathematical Problems in Engineering ◽

10.1155/2015/640107 ◽

2015 ◽

Vol 2015 ◽

pp. 1-3 ◽

Cited By ~ 1

Author(s):

Ming-Chi Lu ◽

Hsing-Chung Ho ◽

Chen-An Chan ◽

Chia-Ju Liu ◽

Jiann-Shing Lih ◽

...

Keyword(s):

Time Series ◽

Dynamical Systems ◽

Hilbert Transform ◽

Phase Synchronization ◽

Nonlinear Dynamical Systems ◽

The State ◽

Nonlinear Dynamical ◽

Large Correlation ◽

Current Usage ◽

The Hilbert Transform

We investigate the interplay between phase synchronization and amplitude synchronization in nonlinear dynamical systems. It is numerically found that phase synchronization intends to be established earlier than amplitude synchronization. Nevertheless, amplitude synchronization (or the state with large correlation between the amplitudes) is crucial for the maintenance of a high correlation between two time series. A breakdown of high correlation in amplitudes will lead to a desynchronization of two time series. It is shown that these unique features are caused essentially by the Hilbert transform. This leads to a deep concern and criticism on the current usage of phase synchronization.

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