Lecture

Fourier Series and Transforms: Introduction and Properties

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

In course

ENG-366: Signals, instruments and systems

The goal of this course is to transmit knowledge in sensing, computing, communicating, and actuating for programmable field instruments and, more generally, embedded systems. The student will be able to put in practice the knowledge acquired using concrete ...

Description

This lecture introduces the concepts of Fourier series and Fourier transforms, explaining how any periodic function can be decomposed into sines and cosines. It covers the properties of these transforms, including linearity, translation, modulation, and scaling. The lecture also discusses the convolution operation in both time and frequency domains, highlighting its commutativity, associativity, and distributivity properties. The Discrete Fourier Transform (DFT) is presented as a discretized version of the Fourier transform, suitable for computer implementation. Examples and demonstrations using Matlab are provided to illustrate these concepts.

Instructor

Alcherio Martinoli

I received my Diploma in Electrical Engineering from the Swiss Federal Institute of Technology in Zurich (ETHZ), and a Ph.D. in Computer Science from the Swiss Federal Institute of Technology in Lausanne (EPFL). I am currently an Associate Professor at the School of Architecture, Civil, and Environmental Engineering and the head of the Distributed Intelligent Systems and Algorithms Laboratory. Before joining EPFL I carried out research activities at the Institute of Biomedical Engineering of the ETHZ, at the Institute of Industrial Automation of the Spanish Research Council in Madrid, Spain, and at the California Institute of Technology, Pasadena, U.S.A. Additional information can be found on my full CV.

Official source

Ontological neighbourhood

Information engineering

Signal processing: Fourier analysis

Related concepts (32)

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.

Non-uniform discrete Fourier transform

In applied mathematics, the nonuniform discrete Fourier transform (NUDFT or NDFT) of a signal is a type of Fourier transform, related to a discrete Fourier transform or discrete-time Fourier transform, but in which the input signal is not sampled at equally spaced points or frequencies (or both). It is a generalization of the shifted DFT. It has important applications in signal processing, magnetic resonance imaging, and the numerical solution of partial differential equations.

Discrete Fourier transform over a ring

In mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex numbers, over an arbitrary ring. Let R be any ring, let be an integer, and let be a principal nth root of unity, defined by: The discrete Fourier transform maps an n-tuple of elements of R to another n-tuple of elements of R according to the following formula: By convention, the tuple is said to be in the time domain and the index j is called time.

Digital signal processing

Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a sequence of numbers that represent samples of a continuous variable in a domain such as time, space, or frequency. In digital electronics, a digital signal is represented as a pulse train, which is typically generated by the switching of a transistor.

Signal processing

Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing signals, such as sound, , potential fields, seismic signals, altimetry processing, and scientific measurements. Signal processing techniques are used to optimize transmissions, digital storage efficiency, correcting distorted signals, subjective video quality and to also detect or pinpoint components of interest in a measured signal. According to Alan V. Oppenheim and Ronald W.

Related lectures (64)

Fourier Series and Analysis

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

Frequency Estimation (Theory)

Covers the theory of numerical methods for frequency estimation on deterministic signals, including Fourier series and transform, Discrete Fourier transform, and the Sampling theorem.

Introduction to the Laplace Transform

Covers the Laplace Transform, LTI systems, Fourier Transforms, and properties.

Fourier Analysis: Fourier Series and Transform

Explores Fourier series, DFT, FFT, and their applications in signal processing and spectroscopy.

Sampling and Reconstruction Theory