Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Fourier Transformation and Dirac Pulse Train

In course

MICRO-310(a): Signals and systems I (for MT)

Présentation des concepts et des outils de base pour la caractérisation des signaux ainsi que pour l'analyse et la synthèse des systèmes linéaires (filtres ou canaux de transmission). Application de ces techniques au traitement du signal et aux télécommuni ...

Description

This lecture covers the Fourier transformation, including the sampling property, generalized Fourier transformation, and its relation with the sign function. It also delves into the Dirac pulse train, its definition, Fourier transformation, and formal calculations. The development in series of Fourier, convergence of the Fourier series, and partial sums are discussed. Additionally, the lecture explores the interpretation of Fourier series, convolutional shift, periodization, frequency sampling, and the Fourier series of a periodic signal. The synthesis of periodic signals, simple generators, and the efficiency of periodization and frequency sampling are also addressed.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructor

Michaël Unser

Official source

Related concepts (25)

Fourier analysis

In mathematics, Fourier analysis (ˈfʊrieɪ,_-iər) is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics.

Fourier transform

In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function.

Fourier series

A Fourier series (ˈfʊrieɪ,_-iər) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation.

Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies.

Fractional Fourier transform

In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power, where n need not be an integer — thus, it can transform a function to any intermediate domain between time and frequency. Its applications range from filter design and signal analysis to phase retrieval and pattern recognition.

Related lectures (35)

Sampling and Reconstruction Theory

Covers the concepts of analog, discrete, and digital signals, sampling times, frequencies, and pulses.

The Fourier Transform MOOC: The Radio Sky II: Observational Radio Astronomy

Introduces the basics of the Fourier Transform, covering concepts like resolution, periodicity, 2D transforms, amplitude, phase, and convolution.

Fourier Transforms: Dirac Delta, Fourier Integral, and Transform

Explores the Dirac delta, Fourier integral, and Fourier transform applications in solving PDE problems.

Fourier Series and Analysis

Covers Fourier series, analysis, the heat equation, Gibbs phenomenon, and Fourier transform properties.

Fourier Analysis: Fourier Series and Transform