### 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
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

Control Engineering and Automation -> Electrical Engineering and Information Technology (Profile)

Electrical Engineering Systems and Technologies -> Electrical Engineering and Information Technology (Profile)

Electrical Power Engineering -> Electrical Engineering and Information Technology (Profile)

Electronic and Computer Engineering -> Electrical Engineering and Information Technology (Profile)

Electronics -> Electrical Engineering and Information Technology (Profile)

Information Processing -> Information and Communication Technology (Profile)

Telecommunication and Informatics -> Information and Communication Technology (Profile)

Wireless Technologies -> Information and Communication Technology (Profile)

Software Engineering and Information Systems -> Computing (Profile)

Computer Engineering -> Computing (Profile)

Computer Science -> Computing (Profile)

#### 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,

#### Lecturers in Charge

4 ECTS
L0 English Level
L1 e-Learning
45 Lecturers
0 Exercises
0 Laboratory exercises