Fourier Analysis

Course Description

Introduction to functional analysis: normed and unitary spaces, function spaces, orthogonality, basis, linear functionals, linear operators. General Fourier series, orthogonal systems and approximations. Fourier transform. Properties of Fourier transform. Haar system. Introduction to wavelets analysis. Examples and applications.

General Competencies

Learning of the vocabulary of functional analysis, using methods of Fourier series and Fourier transform analysis, understanding modern concepts of harmonic analysis.

Learning Outcomes

1. apply the properties of inner product and orthogonality in the inner product spaces
2. compute the Fourier series of periodic functions
3. distinguish various types of convergence of the Fourier series
4. operate with the properties of the Fourier transform
5. apply the discrete Fourier transform in signal analysis
6. understand the basic concepts in Haar wavelet analysis
7. operate with Haar decomposition and reconstruction algorithms

Forms of Teaching

Lectures

lectures are performed in croatian language

Exams

two exams during semester (each 120 minutes)

Consultations

once a week

Other

seminar during semester

Continuous Assessment Exam
Type Threshold Percent of Grade Threshold Percent of Grade
Seminar/Project 0 % 10 % 0 % 10 %
Mid Term Exam: Written 0 % 45 % 0 %
Final Exam: Written 0 % 45 %
Exam: Written 0 % 90 %

Week by Week Schedule

1. Inner product and inner product space. The spaces L^2 and l^2
2. Schwarz and triangle inequalities. Orthogonality
3. Gram-Schmidt ortogonalization. Linear operators and their adjoints
4. Introduction to Fourier analysis. Historical overview
5. Computation of Fourier series
6. The Riemann-Lebesgue lemma. Convergence at a point of continuity. Convergence at a point of discontinuity
7. Uniform convergence. Convergence in the mean
8. Exam
9. The Fourier transform. The Fourier inversion theorem
10. Properties of the Fourier transform
11. Linear filters. The sampling theorem. The uncertainty principle
12. The discrete Fourier transform and applications to signal analysis
13. Introduction to Haar wavelet analysis. Haar wavelets
14. Haar decomposition and reconstruction algorithms. Applications
15. Exam

Study Programmes

Computer Engineering (profile)
Mathematics and Science (2. semester)
Computer Science (profile)
Mathematics and Science (2. semester)
Control Engineering and Automation (profile)
Mathematics and Science (2. semester)
Electrical Engineering Systems and Technologies (profile)
Mathematics and Science (2. semester)
Electrical Power Engineering (profile)
Mathematics and Science (2. semester)
Electronic and Computer Engineering (profile)
Mathematics and Science (2. semester)
Electronics (profile)
Mathematics and Science (2. semester)
Information Processing (profile)
Mathematics and Science (2. semester)
Software Engineering and Information Systems (profile)
Mathematics and Science (2. semester)
Telecommunication and Informatics (profile)
Mathematics and Science (2. semester)
Wireless Technologies (profile)
Mathematics and Science (2. semester)

Literature

(.), A First Course in Wavelets with Fourier Analysis A. Boggess, F. J. Narcowich Prentice Hall 2001,
(.), An introduction to Wavelet Analysis D. F. Walnut Birkhauser 2004,
(.), Ten Lectures on Wavelets I. Daubechies SIAM 1992,

General

ID 34557
Summer semester
4 ECTS
L0 English Level
L1 e-Learning
45 Lectures
0 Exercises
0 Laboratory exercises
0 Project laboratory