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

Lecture

Fourier Transform and Convolution Product in Signal Processing

In course

MATH-207(a): Analysis IV (for SV, MT)

Ce cours est une introduction à la théorie des séries de Fourier, transformées de Fourier (y compris pour les distributions tempérées), à la transformée de Laplace, et à leurs utilisations pour résoudre des équations différentielles ordinaires et aux dériv ...

Description

This lecture covers the Fourier transform, its properties, and the convolution product, which is essential in signal processing. The convolution product is defined for two functions and has properties similar to regular multiplication. It is crucial in various contexts, especially in the study of linear time-invariant filters. These filters transform input signals into output signals, with the output signal obtained by convolving the input signal with a specific function. The lecture also explores practical applications of the convolution product in designing filters, such as low-pass and band-pass filters, and discusses the importance of the transfer function in signal processing.

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.

Instructors (3)

Official source

Ontological neighbourhood

Information engineering

Signal processing: Fourier analysis

Related concepts (28)

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.

Filter (signal processing)

In signal processing, a filter is a device or process that removes some unwanted components or features from a signal. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal. Most often, this means removing some frequencies or frequency bands. However, filters do not exclusively act in the frequency domain; especially in the field of many other targets for filtering exist.

Linear filter

Linear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant (or shift invariant) in which case they can be analyzed exactly using LTI ("linear time-invariant") system theory revealing their transfer functions in the frequency domain and their impulse responses in the time domain. Real-time implementations of such linear signal processing filters in the time domain are inevitably causal, an additional constraint on their transfer functions.

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).

Electronic filter

Electronic filters are a type of signal processing filter in the form of electrical circuits. This article covers those filters consisting of lumped electronic components, as opposed to distributed-element filters. That is, using components and interconnections that, in analysis, can be considered to exist at a single point. These components can be in discrete packages or part of an integrated circuit. Electronic filters remove unwanted frequency components from the applied signal, enhance wanted ones, or both.

Related lectures (50)

The convolution theorem

Explores the convolution theorem, DTFT reconstruction, frequency response effects, and signal building.

Properties of Fourier Transform

Explores the properties and applications of the Fourier transform in signal processing and mathematics.

Discrete Signals & Fourier Transform

Explores discrete signals, Fourier transform, modulation, convolution, DFT properties, and signal periodicity.

Signal Analysis and Filter Design

Explores signal analysis, FFT, filters, and power spectral density in signal processing.

Signal Processing: Image Restoration and Linear Prediction Log in to Mediaspace to watch this video