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Discrete-time Fourier transform01:26

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...
The Uncertainty Principle04:08

The Uncertainty Principle

Werner Heisenberg considered the limits of how accurately one can measure properties of an electron or other microscopic particles. He determined that there is a fundamental limit to how accurately one can measure both a particle’s position and its momentum simultaneously. The more accurate the measurement of the momentum of a particle is known, the less accurate the position at that time is known and vice versa. This is what is now called the Heisenberg uncertainty principle. He mathematically...
Convergence of Fourier Series01:21

Convergence of Fourier Series

The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
Discrete-Time Fourier Series01:20

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]...
Uncertainty: Overview00:59

Uncertainty: Overview

In analytical chemistry, we often perform repetitive measurements to detect and minimize inaccuracies caused by both determinate and indeterminate errors. Despite the cares we take, the presence of random errors means that repeated measurements almost never have exactly the same magnitude. The collective difference between these measurements - observed values - and the estimated or expected value is called uncertainty. Uncertainty is conventionally written after the estimated or expected value.
Properties of DTFT II01:24

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...

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Related Experiment Video

Updated: Jul 4, 2026

The Frequency Domain Thermoreflectance Technique for Thermal Property Measurements
09:10

The Frequency Domain Thermoreflectance Technique for Thermal Property Measurements

Published on: December 5, 2025

Uncertainty relation for the discrete Fourier transform.

Serge Massar1, Philippe Spindel

  • 1Laboratoire d'Information Quantique, CP 225, Université Libre de Bruxelles (ULB), Bruxelles, Belgium.

Physical Review Letters
|June 4, 2008
PubMed
Summary

This study introduces a new uncertainty relation for unitary operators, crucial for quantum state localization in mutually unbiased bases. It bridges discrete and continuous variable uncertainty principles, with applications in quantum information and signal processing.

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Area of Science:

  • Quantum Mechanics
  • Quantum Information Theory

Background:

  • Uncertainty relations are fundamental in quantum mechanics, limiting simultaneous precision of certain observables.
  • Existing relations, like Heisenberg's, apply to specific variable pairs (e.g., position-momentum).

Purpose of the Study:

  • To derive a generalized uncertainty relation for two unitary operators satisfying a specific commutation relation.
  • To explore its application in quantum state localization within mutually unbiased bases.
  • To connect discrete and continuous variable uncertainty principles.

Main Methods:

  • Derivation of a novel uncertainty relation for unitary operators (U, V) with UV = e^(iφ)VU.
  • Application of this relation to analyze simultaneous localization in discrete Fourier transform-related bases.
  • Identification of minimum uncertainty states as solutions to Harper's equation.

Main Results:

  • A new uncertainty relation is established for unitary operators.
  • This relation quantifies the trade-off in localizing quantum states in mutually unbiased bases.
  • It interpolates between Pauli operators and continuous variable uncertainty relations.
  • An uncertainty relation for modular variables is also presented.

Conclusions:

  • The derived uncertainty relation offers a unified framework for discrete and continuous variables.
  • It has potential applications in quantum state manipulation, quantum information processing, and signal processing.
  • Minimum uncertainty states are identified as discrete analogs of coherent and squeezed states.