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Related Concept Videos

Definition of z-Transform01:26

Definition of z-Transform

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The z-transform is a powerful mathematical tool used in the analysis of discrete-time signals and systems. It is an essential analytical tool, analogous to the Laplace transform used in continuous-time systems. It plays a crucial role in the analysis of signals and systems, complementing the discrete-time Fourier transform. Both the z-transform and the Laplace transform convert differential or difference equations into algebraic equations, simplifying the process of solving complex problems.
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Relation of DFT to z-Transform01:20

Relation of DFT to z-Transform

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The Discrete Fourier Transform (DFT) is a crucial tool for analyzing the frequency content of discrete-time signals. It converts a sequence of N samples from the time domain into its corresponding sequence in the frequency domain, where each sample represents a specific frequency component.
To understand how the DFT works, it's helpful to consider the z-transform, which is a method for representing discrete sequences in the complex frequency domain. The z-transform involves summing the...
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Properties of the z-Transform II01:16

Properties of the z-Transform II

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The property of Accumulation in signal processing is derived by analyzing the accumulated sum of a discrete-time signal and using the time-shifting property to determine its z-transform. This principle reveals that the z-transform of the summed signal is related to the z-transform of the original signal by a multiplicative factor.
Moreover, the convolution property indicates that the convolution of two signals in the time domain corresponds to the product of their z-transforms in the frequency...
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Block Diagram Reduction01:22

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The process of deriving the transfer function of a control system often involves reducing its block diagram to a single block. This simplification can be achieved through a series of strategic operations, including relocating branch points and comparators. These operations preserve the overall function of the system while allowing for easier manipulation and combination of blocks.
The first step in this process is the identification and relocation of a branch point. A branch point, where a...
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Bewley Lattice Diagram01:12

Bewley Lattice Diagram

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The Bewley lattice diagram, developed by L. V. Bewley, effectively organizes the reflections occurring during transmission-line transients. It visually represents how voltage waves propagate and reflect within a transmission line, making it easier to understand the complex interactions that occur.
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Ziegler–Natta polymerization is another form of addition or chain‐growth polymerization used for synthesizing linear polymers over branched polymers. The catalyst used for polymerization is the Ziegler–Natta catalyst, named after Karl Ziegler and Giulio Natta, who developed it in 1953. This catalyst is an organometallic complex of titanium tetrachloride and triethyl aluminum, with the active form of the catalyst being an alkyl titanium compound. Using the Ziegler–Natta...
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Design and Synthesis of a Reconfigurable DNA Accordion Rack
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Trefftz co-chain calculus.

Daniele Casati1, Lorenzo Codecasa2, Ralf Hiptmair1

  • 1Seminar for Applied Mathematics, ETH Zurich, Zurich, Switzerland.

Zeitschrift Fuer Angewandte Mathematik Und Physik
|February 7, 2022
PubMed
Summary
This summary is machine-generated.

This study unifies discretizations for transmission problems using domain decomposition. It couples mesh-based and meshless methods for accurate numerical solutions in electromagnetics.

Keywords:
Co-chain calculusDiscrete exterior calculusFinite element exterior calculusTrefftz method

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Area of Science:

  • Computational mathematics
  • Numerical analysis
  • Electromagnetics

Background:

  • Linear transmission problems are fundamental in various scientific fields.
  • Discretization methods are crucial for solving these problems numerically.
  • Domain decomposition offers a powerful strategy for handling complex geometries and different physical domains.

Purpose of the Study:

  • To develop a unified framework for discretizing linear transmission problems.
  • To couple distinct numerical methods across different spatial domains.
  • To provide a robust approach for solving problems involving bounded and unbounded regions.

Main Methods:

  • Utilizing a mesh-based discrete co-chain model (e.g., finite element exterior calculus) in bounded domains.
  • Employing a meshless Trefftz-Galerkin approach with special solutions in unbounded domains.
  • Developing a unified coupling strategy based on discrete Hodge operators.

Main Results:

  • A novel method for coupling diverse discretizations across domain interfaces.
  • Derivation of the resulting linear system of equations from discrete Hodge operators.
  • Demonstration of the approach on a frequency-domain eddy-current problem.

Conclusions:

  • The proposed unified framework effectively couples different discretization techniques.
  • The method provides accurate numerical results for complex electromagnetic problems.
  • This approach offers a flexible and powerful tool for solving linear transmission problems.