Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Rectangular and Triangular Pulse Function01:19

Rectangular and Triangular Pulse Function

1.3K
The unit rectangular pulse function is mathematically represented by a rectangular function centered at the origin with a height of one unit. This function is defined by two parameters: T, which specifies the center location of the pulse along the time axis, and τ, which determines the pulse duration.
For example, consider a rectangular pulse with a 5V amplitude, a 3-second duration, and centered at t=2 seconds. This pulse can be expressed using the rectangular function, written as,
1.3K
Reconstruction of Signal using Interpolation01:10

Reconstruction of Signal using Interpolation

417
Signal processing techniques are essential for accurately converting continuous signals to digital formats and vice versa. When a continuous signal is sampled with a period T, the resulting sampled signal exhibits replicas of the original spectrum in the frequency domain, spaced at intervals equal to the sampling frequency. To handle this sampled signal, a zero-order hold method can be applied, which creates a piecewise constant signal by retaining each sample's value until the next...
417
Graphical and Analytic Representation of Sinusoids01:20

Graphical and Analytic Representation of Sinusoids

605
Analyzing two sinusoidal voltages with equal amplitude and period but different phases on an oscilloscope, an instrument used to display and analyze waveforms, involves a three-step process.
The first step is measuring the peak-to-peak value, which is twice the amplitude of the sinusoid. This provides information about the maximum voltage swing of the waveform.
Secondly, the period and angular frequency are determined. The period is the time taken for one complete cycle of the waveform, while...
605
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

169
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
169
Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

196
Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
196
Convolution Properties I01:20

Convolution Properties I

297
Convolution computations can be simplified by utilizing their inherent properties.
The commutative property reveals that the input and the impulse response of an LTI (Linear Time-Invariant) system can be interchanged without affecting the output:
297

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Direct Determination of Protein Rotational Diffusion Tensors and Generalized Order Parameters from Multifield <sup>15</sup>N NMR Spin Relaxation.

Journal of the American Chemical Society·2026
Same author

A steady-state approach for analysis of high-resolution relaxometry.

Journal of magnetic resonance (San Diego, Calif. : 1997)·2025
Same author

Parsing Dynamics of Protein Backbone NH and Side-Chain Methyl Groups using Molecular Dynamics Simulations.

Journal of chemical theory and computation·2024
Same author

Deuterium spin relaxation of fractionally deuterated ribonuclease H using paired 475 and 950 MHz NMR spectrometers.

Journal of biomolecular NMR·2024
Same author

Breaking down walls: Continuous potential models for internal motions in NMR spin relaxation.

Journal of magnetic resonance (San Diego, Calif. : 1997)·2024
Same author

LC-Photo-CIDNP hyperpolarization of biomolecules bearing a quasi-isolated spin pair: Magnetic-Field dependence via a rapid-shuttling device.

Journal of magnetic resonance (San Diego, Calif. : 1997)·2024

Related Experiment Video

Updated: Oct 23, 2025

Experimental Methods to Study Human Postural Control
08:12

Experimental Methods to Study Human Postural Control

Published on: September 11, 2019

9.7K

Approximate Representations of Shaped Pulses Using the Homotopy Analysis Method.

Timothy Crawley1, Arthur G Palmer1

  • 1Department of Biochemistry and Molecular Biophysics, Columbia University, 630 West 168th Street, New York, NY 10032, United States.

Magnetic Resonance (Gottingen, Germany)
|August 20, 2021
PubMed
Summary

The Homotopy Analysis Method provides accurate, efficient approximate solutions for shaped radiofrequency pulses in nuclear magnetic resonance (NMR) spectroscopy. This method also effectively models relaxation effects in NMR, showing broad applicability in magnetic resonance.

More Related Videos

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

43.1K
Paramagnetic Relaxation Enhancement for Detecting and Characterizing Self-Associations of Intrinsically Disordered Proteins
07:24

Paramagnetic Relaxation Enhancement for Detecting and Characterizing Self-Associations of Intrinsically Disordered Proteins

Published on: September 23, 2021

1.9K

Related Experiment Videos

Last Updated: Oct 23, 2025

Experimental Methods to Study Human Postural Control
08:12

Experimental Methods to Study Human Postural Control

Published on: September 11, 2019

9.7K
Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

43.1K
Paramagnetic Relaxation Enhancement for Detecting and Characterizing Self-Associations of Intrinsically Disordered Proteins
07:24

Paramagnetic Relaxation Enhancement for Detecting and Characterizing Self-Associations of Intrinsically Disordered Proteins

Published on: September 23, 2021

1.9K

Area of Science:

  • Magnetic Resonance
  • Spectroscopy
  • Applied Mathematics

Background:

  • Nuclear spin magnetization evolution under radiofrequency pulses is described by Euler angles.
  • These angles are derived from the Riccati differential equation, but analytical solutions are limited to simple pulse shapes.
  • Complex pulse shaping in NMR spectroscopy necessitates advanced solution methods.

Purpose of the Study:

  • To apply the Homotopy Analysis Method for approximate solutions to the Riccati equation governing magnetization evolution.
  • To extend the method to incorporate relaxation effects as a perturbation.
  • To demonstrate the efficiency and accuracy of the Homotopy Analysis Method in NMR spectroscopy.

Main Methods:

  • Utilized the Homotopy Analysis Method to solve the Riccati differential equation for shaped radiofrequency pulses.
  • Applied the method to derive approximate solutions for nuclear spin magnetization trajectories.
  • Extended the Homotopy Analysis Method to model relaxation phenomena in magnetic resonance.

Main Results:

  • Achieved highly accurate approximate solutions to the Riccati equation even at low approximation orders.
  • Demonstrated efficient computation of magnetization trajectories for shaped radiofrequency pulses.
  • Successfully represented relaxation effects by perturbing the non-relaxation magnetization trajectory.

Conclusions:

  • The Homotopy Analysis Method offers a powerful and flexible approach for analyzing shaped radiofrequency pulses in NMR.
  • The method provides accurate and efficient solutions, overcoming limitations of analytical methods.
  • The Homotopy Analysis Method shows significant potential for various applications within magnetic resonance research.