Introduction to the discrete Fourier series considering both mathematical and engineering aspects - A linear-algebra approach

Abstract

The discrete Fourier series is a valuable tool developed and used by mathematicians and engineers alike. One of the most prominent applications is signal processing. Usually, it is important that the signals be transmitted fast, for example, when transmitting images over large distances such as between the moon and the earth or when generating images in computer tomography. In order to achieve this, appropriate algorithms are necessary. In this context, the fast Fourier transform (FFT) plays a key role which is an algorithm for calculating the discrete Fourier transform (DFT); this, in turn, is tightly connected with the discrete Fourier series. The last one itself is the discrete analog of the common (continuous-time) Fourier series and is usually learned by mathematics students from a theoretical point of view. The aim of this expository/pedagogical paper is to give an introduction to the discrete Fourier series for both mathematics and engineering students. It is intended to expand the purely mathematical view; the engineering aspect is taken into account by applying the FFT to an example from signal processing that is small enough to be used in class-room teaching and elementary enough to be understood also by mathematics students. The MATLAB program is employed to do the computations.

Keywords

• discrete Fourier series
• mathematical and engineering aspects
• discrete Fourier transform (DFT)
• fast Fourier transform (FFT)