Article | REF: AF1440 V1

Fourier transform and its applications (Part 1)

Author: Joël LE ROUX

Publication date: April 10, 2007, Review date: November 19, 2019

You do not have access to this resource.
Click here to request your free trial access!

Already subscribed? Log in!


Overview

Français

ABSTRACT

Fourier transform is a tool enabling the understanding and implementation of a large number of numerical methods for signal and image processing. This tool has many applications in domains such as vocal recognition, image quality improvement, digital transmission, the biomedical sector and astronomy. The aim of this article is to describe the single-dimensional Fourier transform (Fourier series, frequency analysis and extensions of the transform) and present its various applications. Sampling, the z transform and numerical filters are also dealt with.

Read this article from a comprehensive knowledge base, updated and supplemented with articles reviewed by scientific committees.

Read the article

AUTHOR

  • Joël LE ROUX: École polytechnique universitaire (EPU) - University of Nice Sophia-Antipolis

 INTRODUCTION

The Fourier transform, or more generally frequency or spectral analysis, is a fundamental tool for understanding and implementing many digital signal and image processing techniques. It can be found in direct applications such as the harmonic analysis of vibrations and musical signals, but also in a wide variety of other fields. These include all applications where signals measured by sensors need to be shaped by filtering. It is used in low bit-rate coding of music and speech, speech recognition, image quality enhancement and compression, digital transmissions, new radio and TV broadcasting systems, biomedical applications (scanner, nuclear magnetic resonance imaging), astronomy (interferometric image synthesis), wave propagation modeling, spectral analysis for the study of molecular structures and crystallography. Its extension (finite field calculations) is used in error correction methods for digital transmission. It is also used in quantum computing to factor numbers.

The aim of this presentation is to provide the reader with both the theoretical and practical knowledge required to apply frequency analysis tools, and to offer an overview of how they are used in different fields. It does not claim to be mathematically rigorous, and places greater emphasis on operational aspects.

This presentation has been divided into three parts.

The first part (this folder [AF 1440]) gives fundamental results on the transform of one-dimensional signals as continuous and then sampled functions of time, with particular emphasis on its use in digital filtering.

We start with the simplest case, the analysis of periodic functions by Fourier series, then continue with the analysis of continuous time functions, mentioning distribution theory. We'll look at the main properties, such as the convolution transform. Then we'll see how the Fourier transform can be used to tackle the problems posed by sampling and the formulation of digital filtering.

In the second part , we will look at Fourier transform expressions in the case of digital processing, describing the fast Fourier transform algorithm in particular. We will give the main results concerning the spectral analysis of random signals, then turn to the case of two-dimensional signals and images.

The third part begins with the study of filtering and spectral analysis of two-dimensional signals, and ends with a presentation of some multi-dimensional signal processing involving the Fourier transform, such as medical imaging.

You do not have access to this resource.

Exclusive to subscribers. 97% yet to be discovered!

You do not have access to this resource.
Click here to request your free trial access!

Already subscribed? Log in!


The Ultimate Scientific and Technical Reference

A Comprehensive Knowledge Base, with over 1,200 authors and 100 scientific advisors
+ More than 10,000 articles and 1,000 how-to sheets, over 800 new or updated articles every year
From design to prototyping, right through to industrialization, the reference for securing the development of your industrial projects

This article is included in

Mathematics

This offer includes:

Knowledge Base

Updated and enriched with articles validated by our scientific committees

Services

A set of exclusive tools to complement the resources

Practical Path

Operational and didactic, to guarantee the acquisition of transversal skills

Doc & Quiz

Interactive articles with quizzes, for constructive reading

Subscribe now!

Ongoing reading
The Fourier transform and its applications (part 1)