Fourier Series and Transforms: Introduction and Properties

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.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related lectures (469)

Related concepts (169)

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

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 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.

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.

Fast Fourier transform

A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical.

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 Transform: Concepts and Applications

Covers the Fourier transform, its properties, applications in signal processing, and differential equations, emphasizing the concept of derivatives becoming multiplications in the frequency domain.

Frequency Estimation (Theory)ME-301: Measurement techniques

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

Fourier Transform: Concepts and Applications MATH-202(c): Analysis III

Covers the Fourier transform, its properties, and applications in signal processing and differential equations, demonstrating its importance in mathematical analysis.

Sampling Theorem and Control Systems ME-326: Control systems and discrete-time control

Explores the Sampling Theorem, digital control, signal reconstruction, and anti-aliasing filters.

Spectrogram Analysis: Psychoacoustic Foundations & Signal Processing

Explores spectrogram analysis, including DFT, STFT, and CQT, for tasks like music transcription and onset detection.