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

Properties of Fourier series II01:21

Properties of Fourier series II

390
Time scaling of signals is a crucial concept in signal processing that affects the Fourier series representation without altering its coefficients. The process modifies the fundamental frequency, thereby changing how the series represents the signal over time. This principle is essential in various applications, including audio and image processing, where signal manipulation is frequent. Understanding function symmetries is fundamental to simplifying the Fourier series.
A function f(t) is...
390
Trigonometric Fourier series01:17

Trigonometric Fourier series

576
Fourier series is a foundational mathematical technique that decomposes periodic functions into an infinite series of sinusoidal harmonics. This method enables the representation of complex periodic signals as sums of simple sine and cosine functions, facilitating their analysis and interpretation in various fields, including signal processing, acoustics, and electrical engineering.
The trigonometric Fourier series specifically expresses a periodic function with a defined period T using sine...
576
Basic signals of Fourier Transform01:07

Basic signals of Fourier Transform

747
The Fourier Transform is a pivotal mathematical tool in signal processing, enabling the transformation of time-domain signals into their frequency-domain representations. Among the numerous elements within this domain, certain functions like the sinc function, delta function, and exponential signals hold significant importance due to their unique properties and implications.
The sinc function, defined as sinc(x) = sin(πx)/(πx), is particularly notable for its symmetry and behavior at...
747
Properties of Fourier series I01:20

Properties of Fourier series I

589
The Fourier series is a powerful tool in signal processing and communications, allowing periodic signals to be expressed as sums of sine and cosine functions. A foundational property of the Fourier series is linearity. If we consider two periodic signals, their linear combination results in a new signal whose Fourier coefficients are simply the corresponding linear combinations of the original signals' coefficients. This property is crucial in applications like frequency modulation (FM) radio,...
589
Parseval's Theorem for Fourier transform01:15

Parseval's Theorem for Fourier transform

1.8K
Parseval's theorem is a fundamental principle in signal processing that enables the calculation of a signal's energy in either the time domain or the frequency domain. This theorem is pivotal in demonstrating energy conservation between these two domains, ensuring that the computed energy value remains consistent regardless of the domain of analysis.
To understand Parseval's theorem, it is essential to first comprehend how signal energy is typically calculated. When considering a...
1.8K
Properties of Fourier Transform II01:24

Properties of Fourier Transform II

509
The Fourier Transform (FT) is an essential mathematical tool in signal processing, transforming a time-domain signal into its frequency-domain representation. This transformation elucidates the relationship between time and frequency domains through several properties, each revealing unique aspects of signal behavior.
The Frequency Shifting property of Fourier Transforms highlights that a shift in the frequency domain corresponds to a phase shift in the time domain. Mathematically, if x(t) has...
509

You might also read

Related Articles

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

Sort by
Same author

Mathematical Models for Unstable Quantum Systems and Gamow States.

Entropy (Basel, Switzerland)·2022
Same author

The Definition of Entropy for Quantum Unstable Systems: A View-Point Based on the Properties of Gamow States.

Entropy (Basel, Switzerland)·2020
See all related articles

Related Experiment Video

Updated: Nov 27, 2025

Quantifying Microorganisms at Low Concentrations Using Digital Holographic Microscopy DHM
07:27

Quantifying Microorganisms at Low Concentrations Using Digital Holographic Microscopy DHM

Published on: November 1, 2017

10.7K

Hermite Functions, Lie Groups and Fourier Analysis.

Enrico Celeghini1,2, Manuel Gadella2,3, Mariano A Del Olmo2,3

  • 1Dipartimento di Fisica, Università di Firenze and INFN-Sezione di Firenze, 50019 Sesto Fiorentino, Firenze, Italy.

Entropy (Basel, Switzerland)
|December 3, 2020
PubMed
Summary
This summary is machine-generated.

This study reveals a connection between Hermite/Laguerre functions and Fourier analysis using rigged Hilbert spaces. These findings impact quantum mechanics and signal processing through fractional Fourier transforms and new circle functions.

Keywords:
Fourier analysisquantum mechanicsrigged Hilbert spacessignal processingspecial functions

More Related Videos

Morphology-Based Distinction Between Healthy and Pathological Cells Utilizing Fourier Transforms and Self-Organizing Maps
08:59

Morphology-Based Distinction Between Healthy and Pathological Cells Utilizing Fourier Transforms and Self-Organizing Maps

Published on: October 28, 2018

7.4K
Network Analysis of Foramen Ovale Electrode Recordings in Drug-resistant Temporal Lobe Epilepsy Patients
09:32

Network Analysis of Foramen Ovale Electrode Recordings in Drug-resistant Temporal Lobe Epilepsy Patients

Published on: December 18, 2016

12.7K

Related Experiment Videos

Last Updated: Nov 27, 2025

Quantifying Microorganisms at Low Concentrations Using Digital Holographic Microscopy DHM
07:27

Quantifying Microorganisms at Low Concentrations Using Digital Holographic Microscopy DHM

Published on: November 1, 2017

10.7K
Morphology-Based Distinction Between Healthy and Pathological Cells Utilizing Fourier Transforms and Self-Organizing Maps
08:59

Morphology-Based Distinction Between Healthy and Pathological Cells Utilizing Fourier Transforms and Self-Organizing Maps

Published on: October 28, 2018

7.4K
Network Analysis of Foramen Ovale Electrode Recordings in Drug-resistant Temporal Lobe Epilepsy Patients
09:32

Network Analysis of Foramen Ovale Electrode Recordings in Drug-resistant Temporal Lobe Epilepsy Patients

Published on: December 18, 2016

12.7K

Area of Science:

  • Harmonic analysis
  • Quantum mechanics
  • Signal processing

Background:

  • Harmonic analysis on the real line (R) and half-line (R+) is explored.
  • Symmetry groups of Hermite and Laguerre functions are investigated.
  • The relationship between these functions and Fourier analysis is a key focus.

Purpose of the Study:

  • To present recent findings in harmonic analysis on R and R+.
  • To establish a closed relation between Hermite/Laguerre functions, their symmetry groups, and Fourier analysis.
  • To introduce a unified framework using rigged Hilbert spaces.

Main Methods:

  • Utilizing rigged Hilbert spaces for a unified analytical framework.
  • Investigating the universal enveloping algebra of symmetry groups.
  • Developing a discretized Fourier transform on the circle.

Main Results:

  • A closed relation between Hermite/Laguerre functions, their symmetry groups, and Fourier analysis is demonstrated.
  • A connection is found between the universal enveloping algebra of symmetry groups and the fractional Fourier transform.
  • New functions on the circle, derived from Hermite functions, exhibit unique Fourier transform properties.

Conclusions:

  • The presented results offer a unified framework for harmonic analysis on R and R+.
  • The findings have significant implications for quantum mechanics and signal processing.
  • New insights into discretized Fourier transforms and circle functions are provided.