The Effects of Data Padding Techniques on Continuous Relative-Phase Analysis Using the Hilbert Transform

2019 ◽  
Vol 35 (4) ◽  
pp. 247-255 ◽  
Author(s):  
Patrick Ippersiel ◽  
Richard Preuss ◽  
Shawn M. Robbins
Keyword(s):  
Hilbert Transform ◽  
Relative Phase ◽  
Mean Difference ◽  
End Effects ◽  
Kinematic Data ◽  
Full Period ◽  
Continuous Relative Phase ◽  
Mean Square Difference ◽  
Extraneous Data ◽  
The Hilbert Transform

Continuous relative phase (CRP) analysis using the Hilbert transform is prone to end effects. The purpose was to investigate the impact of padding techniques (reflection, spline extrapolation, extraneous data, and unpadded) on end effects following Hilbert-transformed CRP calculations, using sinusoidal, nonsinusoidal, and kinematic data from a repeated sit-to-stand-to-sit task in adults with low back pain (n = 16, mean age = 30 y). CRP angles were determined using a Hilbert transform of sinusoidal and nonsinusoidal signals with set phase shifts, and for the left thigh/sacrum segments. Root mean square difference and true error compared test signals with a gold standard, for the start, end, and full periods, for all data. Mean difference and 95% bootstrapped confidence intervals were calculated to compare padding techniques using kinematic data. The unpadded approach showed near-negligible error using sinusoidal data across all periods. No approach was clearly superior for nonsinusoidal data. Spline extrapolation showed significantly less root mean square difference (all periods) when compared with double reflection (full period: mean difference = 2.11; 95% confidence interval, 1.41 to 2.79) and unpadded approaches (full period: mean difference = −15.8; 95% confidence interval, −18.9 to −12.8). Padding sinusoidal data when performing CRP analyses are unnecessary. When extraneous data have not been collected, our findings recommend padding using a spline to minimize data distortion following Hilbert-transformed CRP analyses.

scholarly journals PORTRAIT AND HILBERT TRANSFORM METHODS FOR EVALUATING CONTINUOUS RELATIVE PHASE BETWEEN LOWER-LIMB JOINTS IN THE ELDERLY DURING WALKING

2021 ◽  
pp. 2140038
Author(s):  
HYUK-JAE CHOI ◽  
GYOOSUK KIM ◽  
CHANG-YONG KO
Keyword(s):  
Angular Velocity ◽  
Hilbert Transform ◽  
Lower Limb ◽  
Cross Correlation ◽  
Relative Phase ◽  
The Elderly ◽  
Transform Method ◽  
Transform Methods ◽  
Continuous Relative Phase ◽  
The Hilbert Transform

In order to calculate the continuous relative phase (CRP) between joints, the portrait method based on the joint angle and angular velocity and the Hilbert transform method based on the analytical signal have been widely used. However, there are few comparisons of these methods. Therefore, the aim of this study is to quantitatively compare these methods by calculating the CRP in the lower-limb joints of the elderly during level free walking. Eighteen elderly female adults ([Formula: see text] year-old, [Formula: see text][Formula: see text]cm, [Formula: see text][Formula: see text]kg) wearing a Helen Hayes full-body marker set walked 10[Formula: see text]m on level ground at a self-selected velocity. The angles of the hip, knee, and ankle were measured. To calculate the CRP using the portrait method, the angular velocities were measured. Then, the phases between the angle and the angular velocity were calculated. To calculate the CRP using the Hilbert transform method, analytical signals were acquired. Then, the phases between the real and imaginary parts were calculated. A CRP was calculated as the difference between the phase in the proximal joint and the phase in the distal joint. To evaluate the similarity in the shape between the portrait and Hilbert transform methods, the cross-correlation was calculated. Bland–Altman plot analyses were performed to assess the agreement between these methods. For the root mean squares (RMSs) and standard deviations (SDs), a paired [Formula: see text]-test and the Pearson correlation between methods were evaluated. There were similarities in the in-phase or out-of-phase features and in the RMS and SD between the methods. Additionally, a higher cross-correlation and agreement between them were found. These results indicated the similarity between the portrait and Hilbert transform methods for the calculation of the CRP. Therefore, either method can be used to evaluate joint coordination.

scholarly journals Order error in the calculation of continuous relative phase

2018 ◽  
Author(s):  
Sina Mehdizadeh ◽  
Paul Glazier
Keyword(s):  
Hilbert Transform ◽  
Degrees Of Freedom ◽  
Phase Plane ◽  
Relative Phase ◽  
Human Movement ◽  
Alternative Methods ◽  
Order Error ◽  
Movement Data ◽  
Continuous Relative Phase ◽  
The Hilbert Transform

The aims of this study were to demonstrate “order error” in the calculation of continuous relative phase (CRP) and to suggest two alternative methods—(i) constructing phase-plane portraits by plotting position over velocity; and (ii), the Hilbert transform—to rectify it. Order error is the change of CRP order between two degrees of freedom (e.g., body segments) when using the conventional method of constructing phase-plane portraits (i.e., velocity over position). Both sinusoidal and non-sinusoidal simulated signals as well as signals from human movement kinematics were used to investigate order error and the performance of the two alternative methods. Both methods have been shown to lead to correct results for simulated sinusoidal and non-sinusoidal signals. For human movement data, however, the Hilbert transform is superior for calculating CRP.

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.

scholarly journals Phase Synchronization Is the Amplified Result by the Hilbert Transform

2015 ◽  
Vol 2015 ◽  
pp. 1-3 ◽  
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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