Transform Theory, 4 credits

Transformteori, 4 hp

TATA57

Main field of study

Mathematics Applied Mathematics

Course level

First cycle

Course type

Programme course

Examiner

Peter Basarab-Horwath

Director of studies or equivalent

Jesper Thorén

Education components

Preliminary scheduled hours: 46 h
Recommended self-study hours: 61 h

Available for exchange students

Yes
Course offered for Semester Period Timetable module Language Campus ECV
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 4 (Spring 2017) 2 1 Swedish Linköping C
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 4 (Spring 2017) 2 1 Swedish Linköping C
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 4 (Spring 2017) 2 1 Swedish Linköping C
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 4 (Spring 2017) 2 1 Swedish Linköping C
6CYYI Applied Physics and Electrical Engineering - International, M Sc in Engineering 4 (Spring 2017) 2 1 Swedish Linköping C
6CMED Biomedical Engineering, M Sc in Engineering 4 (Spring 2017) 2 1 Swedish Linköping C
6CIEI Industrial Engineering and Management - International, M Sc in Engineering - Chinese 4 (Spring 2017) 2 1 Swedish Linköping E
6CIEI Industrial Engineering and Management - International, M Sc in Engineering - Chinese (Specialization Electrical Engineering) 4 (Spring 2017) 2 1 Swedish Linköping C
6CIEI Industrial Engineering and Management - International, M Sc in Engineering - French 4 (Spring 2017) 2 1 Swedish Linköping E
6CIEI Industrial Engineering and Management - International, M Sc in Engineering - French (Specialization Electrical Engineering) 4 (Spring 2017) 2 1 Swedish Linköping C
6CIEI Industrial Engineering and Management - International, M Sc in Engineering - German 4 (Spring 2017) 2 1 Swedish Linköping E
6CIEI Industrial Engineering and Management - International, M Sc in Engineering - German (Specialization Electrical Engineering) 4 (Spring 2017) 2 1 Swedish Linköping C
6CIEI Industrial Engineering and Management - International, M Sc in Engineering - Japanese 4 (Spring 2017) 2 1 Swedish Linköping E
6CIEI Industrial Engineering and Management - International, M Sc in Engineering - Japanese (Specialization Electrical Engineering) 4 (Spring 2017) 2 1 Swedish Linköping C
6CIEI Industrial Engineering and Management - International, M Sc in Engineering - Spanish 4 (Spring 2017) 2 1 Swedish Linköping E
6CIEI Industrial Engineering and Management - International, M Sc in Engineering - Spanish (Specialization Electrical Engineering) 4 (Spring 2017) 2 1 Swedish Linköping C
6KFYN Physics, Bachelor´s Programme 2 (Spring 2017) 2 1 Swedish Linköping C
ECV = Elective / Compulsory / Voluntary

Main field of study

Mathematics, Applied Mathematics

Course level

First cycle

Advancement level

G1X

Course offered for

  • Industrial Engineering and Management - International, M Sc in Engineering
  • Biomedical Engineering, M Sc in Engineering
  • Applied Physics and Electrical Engineering - International, M Sc in Engineering
  • Physics, Bachelor´s Programme

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

Calculus, linear algebra

Intended learning outcomes

The course aims to give students a deeper knowledge of Fourier analysis and Transform Theory , which have many applications in both technology and mathematics. After successfully completing the course the student is expected to

  • be acquainted with necessary conditions for the existence of the transforms,
  • know and be able to derive simple properties of the transforms (e.g. behaviour at infinity, scaling and translation rules, rules for differentiation and integration as well as rules for multiplication by the time variable).
  • be able to derive the transforms of the elementary funktions,
  • know the invversion theorems, uniqueness theorems, the convolution formulas and the Parseval and Plancherel theorems,
  • be able to use the transforms to solve problems such as differential equations, Difference equations and convolution equations
  • be acquainted with and be able to use results about uniform convergence (continuity, differentiability and integrability of limit functions, Weierstrass’ Majorant Theorem).

Course content

In this course we study some important linear transformations which allow us to translate linear problems (differential, integral and difference equations) into more tractable algebraic problems, whose solutions can then be translated back to solutions of the original problem.
We study: Fourier series, which translate periodic functions into function series. These series are used to analyze periodic behaviour. The problem of convergence of the function series is important and we look att uniform and pointwise convergence as well as convergence in the mean for Fourier series. Bessel’s and Parseval’s Theorems are key results. Fourier transforms: these transforms are used to analyze non-periodic behaviour. The inversion formula for Fourier transforms is of central importance, and other tools at our disposal include the rules of calculation, the convolution formula and Plancherel’s Theorem. The Laplace transform: this transforms functions of a real variable into functions defined in the complex plane and it is used amongst other things for solving initial value problems. The tools at our disposal include rules of calculation, the convolution formula as well as initial and final value theorems. The Z-transform: transforms functions of the natural numbers into power series, and it is used to solve difference equations. The tools at our disposal include rules of calculation and the convolution formula.

Teaching and working methods

Teaching is done through lectures and problem classes.

Examination

TEN1Written examination4 creditsU, 3, 4, 5

Grades

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

Department

Matematiska institutionen

Director of Studies or equivalent

Jesper Thorén

Examiner

Peter Basarab-Horwath

Education components

Preliminary scheduled hours: 46 h
Recommended self-study hours: 61 h

Course literature

Pinkus, A., Zafrany, S.: Fourier Series and Integral Transforms. Kompletterande material (exempelsamling) utgivet av MAI.
Code Name Scope Grading scale
TEN1 Written examination 4 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. 

Pinkus, A., Zafrany, S.: Fourier Series and Integral Transforms. Kompletterande material (exempelsamling) utgivet av MAI.

Note: The course matrix is not fully translated to English.

I U A Modules Comment
1. ÄMNESKUNSKAPER
1.1 Kunskaper i grundläggande matematiska och naturvetenskapliga ämnen
X
X
X
I och U: Fourierserier, Fourier-, Laplace- och z-transform. Tillämpningar på differentialekvationer, differensekvationer och faltningsekvationer. A: Integraler, differentialekvationer och serier från Envariabelanalys. Linjära ekvationssystem, inre produkt och ortogonalitet från Linjär algebra.
1.2 Kunskaper i grundläggande (motsvarande G1X) teknikvetenskapliga ämnen
1.3 Fördjupade kunskaper (motsvarande G2X), metoder och verktyg inom något/några teknik- och naturvetenskapliga ämnen
1.4 Väsentligt fördjupade kunskaper (motsvarande A1X), metoder och verktyg inom något/några teknik- och naturvetenskapliga ämnen
1.5 Insikt i aktuellt forsknings- och utvecklingsarbete
2. INDIVIDUELLA OCH YRKESMÄSSIGA FÄRDIGHETER OCH FÖRHÅLLNINGSSÄTT
2.1 Analytiskt tänkande och problemlösning
X
X
Matematisk teori och problemlösning
2.2 Experimenterande och undersökande arbetssätt samt kunskapsbildning
X
2.3 Systemtänkande
2.4 Förhållningssätt, tänkande och lärande
X
X
Matematisk problemlösning, förmåga att översätta teorins villkor i problemets sammanhang. Att kunna verifiera eller kontrollera lösningens riktighet.
2.5 Etik, likabehandling och ansvarstagande
3. FÖRMÅGA ATT ARBETA I GRUPP OCH ATT KOMMUNICERA
3.1 Arbete i grupp
3.2 Kommunikation
X
Förmåga att tydligt presentera matematiska resonemang
3.3 Kommunikation på främmande språk
X
Engelskspråkig kurslitteratur. Kursen ges vid behov på engelska.
4. PLANERING, UTVECKLING, REALISERING OCH DRIFT AV TEKNISKA PRODUKTER OCH SYSTEM MED HÄNSYN TILL AFFÄRSMÄSSIGA OCH SAMHÄLLELIGA BEHOV OCH KRAV
4.1 Samhälleliga villkor, inklusive ekonomiskt, socialt och ekologiskt hållbar utveckling för kunskapsutveckling
4.2 Företags- och affärsmässiga villkor
4.3 Att identifiera behov samt strukturera och planera utveckling av produkter och system
4.4 Att konstruera produkter och system
4.5 Att realisera produkter och system
4.6 Att ta i drift och använda produkter och system
5. PLANERING, GENOMFÖRANDE OCH PRESENTATION AV FORSKNINGS- ELLER UTVECKLINGSPROJEKT MED HÄNSYN TILL VETENSKAPLIGA OCH SAMHÄLLELIGA BEHOV OCH KRAV
5.1 Samhälleliga villkor, inklusive ekonomiskt, socialt och ekologiskt hållbar utveckling
5.2 Ekonomiska villkor för kunskapsutveckling
5.3 Att identifiera behov samt strukturera och planera forsknings- eller utvecklingsprojekt
5.4 Att genomföra forsknings- eller utvecklingsprojekt
5.5 Att redovisa och utvärdera forsknings- eller utvecklingsprojekt

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