Fourier transform and its applications (Part 1)

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.