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Lecture

Trigonometric Fourier Series: Calculation and Interpretation

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 calculation and interpretation of trigonometric Fourier series, including the orthogonality of co-sinusoids, the spectrum of periodic signals, and the concept of discrete spectrum. It also delves into the Hilbert interpretation, the Hilbert space of finite duration and energy signals, and the associated dot product. The lecture further explores the generalized Fourier series, the complex case of Fourier series, and the polar representation of Fourier coefficients.

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Instructor

Michaël Unser

Official source

Ontological neighbourhood

Information engineering

Signal processing: Fourier analysis

Related concepts (32)

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.

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.

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.

Generalized Fourier series

In mathematical analysis, many generalizations of Fourier series have proved to be useful. They are all special cases of decompositions over an orthonormal basis of an inner product space. Here we consider that of square-integrable functions defined on an interval of the real line, which is important, among others, for interpolation theory. Consider a set of square-integrable functions with values in or , which are pairwise orthogonal for the inner product where is a weight function, and represents complex conjugation, i.

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