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Discrete-Time Fourier Series

Discrete-Time Fourier Series

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]...

Discrete-time Fourier transform

Discrete-time Fourier transform

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...

Properties of DTFT I

Properties of DTFT I

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...

Properties of DTFT II

Properties of DTFT II

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. Multiplying by j...

Discrete Fourier Transform

Discrete Fourier Transform

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...

Mesh Analysis for AC Circuits

Mesh Analysis for AC Circuits

In the domain of radio communication, the significance of impedance matching must be considered. It is crucial to ensure the efficient transmission of signals between radio transmitters and receivers. Achieving this balance involves using impedance-matching circuits, with one fundamental configuration comprising a resistor, capacitor, and inductor.
The process of harmonizing these impedances begins with a clear understanding of the input and output signals. Once these signals are known, the...

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Related Experiment Video

Updated: Jun 10, 2026

Transport Properties of Ibuprofen Encapsulated in Cyclodextrin Nanosponge Hydrogels: A Proton HR-MAS NMR Spectroscopy Study

Transport Properties of Ibuprofen Encapsulated in Cyclodextrin Nanosponge Hydrogels: A Proton HR-MAS NMR Spectroscopy Study

Published on: August 15, 2016

A Mathematical Analysis of IPT-DMFT.

Eric Cancès^1,2, Alfred Kirsch^1,2, Solal Perrin-Roussel^1,2

¹CERMICS, Ecole des Ponts, 6-8 Avenue Blaise Pascal, 77455 Marne-la-Vallée, France.

Communications in Mathematical Physics

|June 9, 2026

Summary

This study mathematically analyzes Dynamical Mean-Field Theory (DMFT), proving its equations have solutions under specific conditions. The research also details properties of these solutions and the Iterated Perturbation Theory solver.

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Last Updated: Jun 10, 2026

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Published on: August 6, 2018

Area of Science:

Quantum Mechanics
Condensed Matter Physics

Background:

Dynamical Mean-Field Theory (DMFT) is a key approximation method in quantum mechanics.
DMFT is a type of embedding method used for strongly correlated systems.

Purpose of the Study:

To provide a rigorous mathematical analysis of Dynamical Mean-Field Theory (DMFT).
To prove the existence and properties of solutions for DMFT equations under specific assumptions.
To offer a pedagogical formulation of DMFT for the finite Hubbard model.

Main Methods:

Mathematical analysis of DMFT equations.
Formulation of DMFT for the finite Hubbard model.
Utilizing one-body time-ordered Green's functions and self-energies.
Employing the Iterated Perturbation Theory (IPT) impurity solver with Matsubara Green's functions.

Main Results:

Proof that DMFT equations admit solutions for any physical parameters under certain assumptions.
Establishment of properties of the obtained DMFT solutions.
A self-contained mathematical formulation of DMFT for the finite Hubbard model is provided.

Conclusions:

The mathematical framework supports the applicability of DMFT.
The study clarifies the behavior of solutions within the DMFT approximation.
The provided formulation and analysis aid in understanding and applying DMFT.