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Lecture

Discrete Signals & Fourier Transform

In course

MICRO-311(a): Signals and systems II (for MT)

Ce cours aborde la théorie des systèmes linéaires discrets invariants par décalage (LID). Leurs propriétés et caractéristiques fondamentales y sont discutées, ainsi que les outils fondamentaux permettant de les étudier (transformée de Fourier et transformé ...

Description

This lecture covers the concept of discrete signals and the Fourier transform, including topics such as discreet ideal low-pass filters, DTFT, signal modulation, convolution, DFT properties, and the link between DFT and continuous-time Fourier transform. The instructor explains the matrix representation of signals, the n-periodicity of signals, and the Dirichlet core function. Additionally, the lecture delves into DTFT resampling, causal averaging windows, and the importance of calculating DTFT for non-periodic discrete signals. Through examples and explanations, students will gain a comprehensive understanding of signal processing and Fourier analysis.

Instructors (2)

Pierre Vandergheynst

Pierre Vandergheynst received the M.S. degree in physics and the Ph.D. degree in mathematical physics from the Université catholique de Louvain, Louvain-la-Neuve, Belgium, in 1995 and 1998, respectively. From 1998 to 2001, he was a Postdoctoral Researcher with the Signal Processing Laboratory, Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland. He was Assistant Professor at EPFL (2002-2007), where he is now a Full Professor of Electrical Engineering and, by courtesy, of Computer and Communication Sciences. As of 2015, Prof. Vandergheynst serves as EPFL’s Vice-Provost for Education. His research focuses on harmonic analysis, sparse approximations and mathematical data processing in general with applications covering signal, image and high dimensional data processing, computer vision, machine learning, data science and graph-based data processing. He was co-Editor-in-Chief of Signal Processing (2002-2006), Associate Editor of the IEEE Transactions on Signal Processing (2007-2011), the flagship journal of the signal processing community and currently serves as Associate Editor of Computer Vision and Image Understanding and SIAM Imaging Sciences. He has been on the Technical Committee of various conferences, serves on the steering committee of the SPARS workshop and was co-General Chairman of the EUSIPCO 2008 conference. Pierre Vandergheynst is the author or co-author of more than 70 journal papers, one monograph and several book chapters. He has received two IEEE best paper awards. Professor Vandergheynst is a laureate of the Apple 2007 ARTS award and of the 2009-2010 De Boelpaepe prize of the Royal Academy of Sciences of Belgium.

Michaël Unser

Official source

Ontological neighbourhood

Information engineering

Signal processing: Fourier analysis

Related concepts (32)

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.

Discrete-time Fourier transform

In mathematics, the discrete-time Fourier transform (DTFT), also called the finite Fourier transform, is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function.

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.

Convolution

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function () that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity).

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