Fourier Methods (5cr)

Objectives

After completing the course, a student knows the definition and basic properties of Fourier series and Fourier transform. The student can represent a periodic function as a Fourier series (with real or complex coefficients) and can form new ones from known series. In addition, the student can calculate the Fourier transform of non-periodic functions for some simple example cases by using the definition and the basic properties. The student knows what is Gibbs phenomenon and can predict when it occurs in some example cases. The student knows the Dirac delta function and how to use it. A student that masters the course contents have prerequisite for understanding the practical value of the above concepts. The Fourier series of a periodic function or the Fourier transform of a non-periodic function decomposes the function into its frequency components. In practice however, the function of interest may not be available, but one only can calculate some samples of it. Then one should be able estimate the frequency decomposition by using the samples. This is done by using the discrete Fourier transform. The discovery of fast Fourier transform in 1960's was groundbreaking for signal processing and is cited as one of the most important algoritms in mathematics. The importance of the frequency decomposition stems from the ability to filter unwanted frequencies.

Content

Core contentReal Fourier series for periodic function and determining its coefficients, even and odd functions as special cases, Gibbs phenomenon. Complex Fourier series, Parseval's theroem, series as frequency decomposition.Discrete Fourier transform and its properties.Fourier transform of non-periodic functions: definition and basic properties, Fourier transform as frequency decomposition. Complementary knowledgeDirichlet conditions for fourier series convergence, expanding a function defined on a bounded interval to a periodic function.Significance of the fast Fourier transformDirac delta function, Convolution, Parseval's theorem

Prerequisites

PrerequisiteCode: MAT-01166Name: Mathematics 1ECTS credits: 5Mandatory: MandatoryPrerequisiteCode: MAT-01366Name: Mathematics 3ECTS credits: 5Mandatory: Mandatoryor courses of Engineering Mathematics / Mathematics in Finnish. The courses are not compulsory, but corresponding knowledge (basics of real differential and integral calculus, and  complex numbers) is required.

Further information

The course will be lectured in English every second year.This course belongs to the SEFI 2 level of engineering mathematics.