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Related Concept Videos

Definition of Laplace Transform01:22

Definition of Laplace Transform

The Laplace transform is an indispensable mathematical technique for simplifying the resolution of differential equations by converting them into more manageable algebraic expressions. The Laplace transform of a function is denoted by L[x(t)], where x(t) is the time-domain function. The laplace transform is mathematically expressed as
Properties of Laplace Transform-I01:15

Properties of Laplace Transform-I

The Laplace transform is a powerful mathematical tool used to convert functions from the time domain into the frequency domain, greatly simplifying the analysis and solution of linear time-invariant systems. This transformation is facilitated by several universal properties: Linearity, Time-Scaling, Time-Shifting, and Frequency Shifting.
The Linearity property is foundational to the Laplace transform. It states that the transform of a linear combination of functions is equivalent to the same...
Properties of Laplace Transform-II01:16

Properties of Laplace Transform-II

Time differentiation, convolution, integration, and periodicity are fundamental concepts in analyzing functions and signals over time. Each concept provides a unique perspective on how functions evolve, interact, and repeat, offering essential tools for various scientific and engineering applications.
Time differentiation involves analyzing the rate of change of a function over time. Mathematically, it is the derivative of a function with respect to time. This concept can be likened to tracking...
Second Derivatives and Laplace Operator01:22

Second Derivatives and Laplace Operator

The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
Consider a scalar function. The curl of its...
Region of Convergence of Laplace Tarnsform01:20

Region of Convergence of Laplace Tarnsform

The Region of Convergence (ROC) is a fundamental concept in signal processing and system analysis, particularly associated with the Laplace transform. The ROC represents an area in the complex plane where the Laplace transform of a given signal converges, determining the transform's applicability and utility.
Consider a decaying exponential signal that begins at a specific time. When deriving its Laplace transform, the time-domain variable is replaced with a complex variable. This substitution...
Basic signals of Fourier Transform01:07

Basic signals of Fourier Transform

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 zero. It...

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Related Experiment Video

Updated: May 27, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Wigner functions defined with Laplace transform kernels.

Se Baek Oh1, Jonathan C Petruccelli, Lei Tian

  • 1Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA.

Optics Express
|November 24, 2011
PubMed
Summary
This summary is machine-generated.

We introduce a novel Laplace kernel Wigner function, extending phase-space analysis to complex variables. This new method enhances signal representation and analyzes evanescent waves, offering broader applications in physics.

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Last Updated: May 27, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Published on: June 8, 2018

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Published on: August 12, 2013

Area of Science:

  • Quantum mechanics
  • Signal processing
  • Optics

Background:

  • The traditional Wigner function is a cornerstone for phase-space representation in quantum mechanics.
  • Its limitation lies in real momentum variables, restricting the analysis of certain signal types.
  • Evanescent waves, crucial in phenomena like surface plasmon polaritons, pose challenges for conventional phase-space methods.

Purpose of the Study:

  • To introduce a generalized Wigner-type function utilizing Laplace transform kernels.
  • To explore the representation capabilities of this new function with complex momentum variables.
  • To demonstrate its utility in analyzing complex wave phenomena, specifically evanescent waves.

Main Methods:

  • Development of the Laplace kernel Wigner function by incorporating Laplace transform kernels.
  • Mathematical analysis of the function's properties, particularly its marginal distributions.
  • Application of the function to model and analyze evanescent waves in surface plasmon polariton systems.

Main Results:

  • The Laplace kernel Wigner function allows for complex momentum variables, expanding signal representation in phase-space.
  • It retains key properties of the traditional Wigner function, such as marginal distribution characteristics.
  • Successful analysis of evanescent waves using the proposed Laplace kernel Wigner function was achieved.

Conclusions:

  • The Laplace kernel Wigner function offers a powerful extension to phase-space analysis, accommodating a wider range of signals.
  • Its ability to handle complex variables and analyze evanescent waves makes it valuable for advanced physics applications.
  • This generalized approach provides new insights into wave phenomena and signal processing.