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Related Concept Videos

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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Discrete Fourier Transform01:15

Discrete Fourier Transform

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The Discrete Fourier Transform (DFT) is a fundamental tool in signal processing, extending the discrete-time Fourier transform by evaluating discrete signals at uniformly spaced frequency intervals. This transformation converts a finite sequence of time-domain samples into frequency components, each representing complex sinusoids ordered by frequency. The DFT translates these sequences into the frequency domain, effectively indicating the magnitude and phase of each frequency component present...
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Discrete-Time Fourier Series01:20

Discrete-Time Fourier Series

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The Discrete-Time Fourier Series (DTFS) is a fundamental concept in signal processing, serving as the discrete-time counterpart to the continuous-time Fourier series. It allows for the representation and analysis of discrete-time periodic signals in terms of their frequency components. Unlike its continuous counterpart, which utilizes integrals, the calculation of DTFS expansion coefficients involves summations due to the discrete nature of the signal.
For a discrete-time periodic signal x[n]...
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Discrete-time Fourier transform01:26

Discrete-time Fourier transform

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The Discrete-Time Fourier Transform (DTFT) is an essential mathematical tool for analyzing discrete-time signals, converting them from the time domain to the frequency domain. This transformation allows for examining the frequency components of discrete signals, providing insights into their spectral characteristics. In the DTFT, the continuous integral used in the continuous-time Fourier transform is replaced by a summation to accommodate the discrete nature of the signal.
One of the notable...
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Properties of DTFT I01:24

Properties of DTFT I

705
In signal processing, Discrete-Time Fourier Transforms (DTFTs) play a critical role in analyzing discrete-time signals in the frequency domain. Various properties of the DTFTs such as linearity, time-shifting, frequency-shifting, time reversal, conjugation, and time scaling help understand and manipulate these signals for different applications.
The linearity property of DTFTs is fundamental. If two discrete-time signals are multiplied by constants a and b respectively, and then combined to...
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Properties of DTFT II01:24

Properties of DTFT II

490
In the study of discrete-time signal processing, understanding the properties of the Discrete-Time Fourier Transform (DTFT) is crucial for analyzing and manipulating signals in the frequency domain. Several properties, including frequency differentiation, convolution, accumulation, and Parseval's relation, offer powerful tools for signal analysis.
The frequency differentiation property is illustrated by considering a DTFT pair and differentiating both sides with respect to ω.
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Information encoding in spherical DFT.

Sol Samuels1, Chance M Baxter2, Susan R Atlas3

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Spherical density functional theory (DFT) now uniquely determines atomic positions and potentials using only sphericalized densities. This advance simplifies quantum mechanical calculations for Coulombic systems.

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Area of Science:

  • Quantum Chemistry
  • Computational Physics
  • Materials Science

Background:

  • Density Functional Theory (DFT) is a quantum mechanical modeling method.
  • Spherical DFT reformulates DFT using sphericalized electron densities.
  • Current spherical DFT proofs require explicit atomic location data.

Purpose of the Study:

  • To demonstrate that sphericalized densities alone determine atomic positions and potentials.
  • To establish a theoretical foundation for spherical DFT in Coulombic systems.
  • To provide a rationale for using spherical basis densities in simulations.

Main Methods:

  • Utilized geometric algebra and distance geometry.
  • Developed a mathematical proof for spherical DFT.
  • Applied the theory to LiF and glycine systems.

Main Results:

  • Spatial information is encoded within sphericalized densities.
  • Sphericalized densities uniquely determine the external potential.
  • Numerical examples validated the theoretical findings.

Conclusions:

  • Spherical DFT is theoretically sound for Coulombic systems.
  • Atomic locations are implicitly determined by spherical densities.
  • Supports the use of spherical basis densities for atomistic simulations.