Fourier and Wavelet Analysis, 6 credits

Fourier- och waveletanalys, 6 hp

TATA66

Course starting semester

All instances mentioned below are cancelled.

Main field of study

Mathematics Applied Mathematics

Course level

Second cycle

Course type

Programme course

Examiner

Bengt Ove Turesson

Director of studies or equivalent

Göran Forsling

Education components

Preliminary scheduled hours: 36 h
Recommended self-study hours: 124 h

Possible deviation from course syllabus/programme syllabus

ECV = Elective / Compulsory / Voluntary

Course offered for		Semester	Period	Timetable module	Language	Campus	ECV	N.B.
6CYYI	Applied Physics and Electrical Engineering - International, M Sc in Engineering	8 (Spring 2017)	1, 2	4, 2	Swedish	Linköping, Valla	E	CANCELLED
6CYYI	Applied Physics and Electrical Engineering - International, M Sc in Engineering	8 (Spring 2017)	1, 2	4, 2	Swedish	Linköping, Valla	E	CANCELLED
6CYYI	Applied Physics and Electrical Engineering - International, M Sc in Engineering	8 (Spring 2017)	1, 2	4, 2	Swedish	Linköping, Valla	E	CANCELLED
6CYYI	Applied Physics and Electrical Engineering - International, M Sc in Engineering	8 (Spring 2017)	1, 2	4, 2	Swedish	Linköping, Valla	E	CANCELLED
6CYYI	Applied Physics and Electrical Engineering - International, M Sc in Engineering	8 (Spring 2017)	1, 2	4, 2	Swedish	Linköping, Valla	E	CANCELLED
6CYYI	Applied Physics and Electrical Engineering - International, M Sc in Engineering (Applied Mathematics)	8 (Spring 2017)	1, 2	4, 2	Swedish	Linköping, Valla	E	CANCELLED
6CYYI	Applied Physics and Electrical Engineering - International, M Sc in Engineering (Applied Mathematics)	8 (Spring 2017)	1, 2	4, 2	Swedish	Linköping, Valla	E	CANCELLED
6CYYI	Applied Physics and Electrical Engineering - International, M Sc in Engineering (Applied Mathematics)	8 (Spring 2017)	1, 2	4, 2	Swedish	Linköping, Valla	E	CANCELLED
6CYYI	Applied Physics and Electrical Engineering - International, M Sc in Engineering (Applied Mathematics)	8 (Spring 2017)	1, 2	4, 2	Swedish	Linköping, Valla	E	CANCELLED
6CYYI	Applied Physics and Electrical Engineering - International, M Sc in Engineering (Applied Mathematics)	8 (Spring 2017)	1, 2	4, 2	Swedish	Linköping, Valla	E	CANCELLED
6CYYY	Applied Physics and Electrical Engineering, M Sc in Engineering	8 (Spring 2017)	1, 2	4, 2	Swedish	Linköping, Valla	E	CANCELLED
6CYYY	Applied Physics and Electrical Engineering, M Sc in Engineering (Applied Mathematics)	8 (Spring 2017)	1, 2	4, 2	Swedish	Linköping, Valla	E	CANCELLED
6MMAT	Mathematics, Master's programme	2 (Spring 2017)	1, 2	4, 2	Swedish	Linköping, Valla	E	CANCELLED
6MMAT	Mathematics, Master's programme (Applied and Computational Mathematics)	2 (Spring 2017)	1, 2	4, 2	Swedish	Linköping, Valla	E	CANCELLED
6MMAT	Mathematics, Master's programme (Mathematics)	2 (Spring 2017)	1, 2	4, 2	Swedish	Linköping, Valla	E	CANCELLED

Main field of study

Mathematics, Applied Mathematics

Course level

Second cycle

Advancement level

A1X

Course offered for

Applied Physics and Electrical Engineering, M Sc in Engineering
Mathematics, Master's programme
Applied Physics and Electrical Engineering - International, M Sc in Engineering

Specific information

The course is only offered every second year. It will not be offered during 2017.

Entry requirements

Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshold requirements for progression within the programme, or corresponding.

Prerequisites

Linear Algebra, Caculus in one and severel variables, Fourier Analysis or Transform Theory

Intended learning outcomes

To give the student a deeper understanding of Fourier analysis and to give a theoretical background to wavelets and applications in signal processing. The student should reach good comprehension in the following fields.

Basic Hilbert space theory
Fourier series and the Fourier transform
The theory of distributions
Multiresolution analysis (MRA)
Some commonly used wavelet systems

Course content

Introduction to the Lebesgue integral. Hilbert spaces: Inner products, orthogonal projection, convergence, completeness, orthonormal systems, orthonormal bases. Fourier series: Convergence theorems, Parseval’s identity. The Fourier transform: Basic properties, inversion, Plancherel’s identity, the Schwartz class. Distributions: Operations on distributions, tempered distributions, the Fourier transform, convolutions, periodic distributions, the Poisson summation formula, the sampling theorem. Wavelets: The Haar system, MRA (multiresolutional analysis), the Shannon wavelet, Meyer's wavelets, and wavelets with compact support, e.g., Daubechies' wavelets. Applications to differential equations and filter theory.

Teaching and working methods

Lectures.
The course runs over the entire spring semester.

Examination

UPG1

Written assignments

6 credits

U, 3, 4, 5

Grades

Four-grade scale, LiU, U, 3, 4, 5

Department

Matematiska institutionen

Director of Studies or equivalent

Göran Forsling

Examiner

Bengt Ove Turesson

Course website and other links

http://www.mai.liu.se/und/kurser/index-amne-tm.html

Education components

Preliminary scheduled hours: 36 h
Recommended self-study hours: 124 h

Course literature

C. Gasquet, P. Witomski: Fourier Analysis and Application. Filtering, Numerical Computation, Wavelets, Springer-Verlag, 1998.
Utdelat material

Code	Name	Scope	Grading scale
UPG1	Written assignments	6 credits	U, 3, 4, 5

Regulations (apply to LiU in its entirety)

The university is a government agency whose operations are regulated by legislation and ordinances, which include the Higher Education Act and the Higher Education Ordinance. In addition to legislation and ordinances, operations are subject to several policy documents. The Linköping University rule book collects currently valid decisions of a regulatory nature taken by the university board, the vice-chancellor and faculty/department boards.

LiU’s rule book for education at first-cycle and second-cycle levels is available at http://styrdokument.liu.se/Regelsamling/Innehall/Utbildning_pa_grund-_och_avancerad_niva.

Student rights and obligations (login required)
Linköping University common rules and regulations (in Swedish)

Course literature is preliminary.

C. Gasquet, P. Witomski: Fourier Analysis and Application. Filtering, Numerical Computation, Wavelets, Springer-Verlag, 1998. <br>Utdelat material

Download

Download course literature list as PDF

Note: The course matrix might contain more information in Swedish.

I = Introduce, U = Teach, A = Utilize

		I	U	A	Modules
1. DISCIPLINARY KNOWLEDGE AND REASONING
	1.1 Knowledge of underlying mathematics and science (G1X level)		X	X	UPG1
	1.2 Fundamental engineering knowledge (G1X level)	X
	1.3 Further knowledge, methods, and tools in one or several subjects in engineering or natural science (G2X level)
	1.4 Advanced knowledge, methods, and tools in one or several subjects in engineering or natural sciences (A1X level)
	1.5 Insight into current research and development work
2. PERSONAL AND PROFESSIONAL SKILLS AND ATTRIBUTES
	2.1 Analytical reasoning and problem solving		X	X	UPG1
	2.2 Experimentation, investigation, and knowledge discovery		X	X
	2.3 System thinking
	2.4 Attitudes, thought, and learning		X	X
	2.5 Ethics, equity, and other responsibilities		X
3. INTERPERSONAL SKILLS: TEAMWORK AND COMMUNICATION
	3.1 Teamwork
	3.2 Communications		X	X	UPG1
	3.3 Communication in foreign languages			X
4. CONCEIVING, DESIGNING, IMPLEMENTING AND OPERATING SYSTEMS IN THE ENTERPRISE, SOCIETAL AND ENVIRONMENTAL CONTEXT
	4.1 External, societal, and environmental context
	4.2 Enterprise and business context
	4.3 Conceiving, system engineering and management
	4.4 Designing
	4.5 Implementing
	4.6 Operating
5. PLANNING, EXECUTION AND PRESENTATION OF RESEARCH DEVELOPMENT PROJECTS WITH RESPECT TO SCIENTIFIC AND SOCIETAL NEEDS AND REQUIREMENTS
	5.1 Societal conditions, including economic, social, and ecological aspects of sustainable development for knowledge development
	5.2 Economic conditions for knowledge development
	5.3 Identification of needs, structuring and planning of research or development projects
	5.4 Execution of research or development projects
	5.5 Presentation and evaluation of research or development projects

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