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## Syllabus for

TMA462 - Fourier and wavelet analysis

Owner: TM
5,0 Credits (ECTS 7,5)
Grading: TH - Five, Four, Three, Not passed
Level: C
Department: 11 - MATHEMATICAL SCIENCES

Teaching language: English

 Course module Credit distribution Examination dates Sp1 Sp2 Sp3 Sp4 No Sp 0101 Examination 5,0 c Grading: TH 5,0 c Contact examiner, 25 Aug 2007 am V

#### In programs

CEMAS MSc PROGRAMME IN COMPUTATIONAL AND EXPERIMENTAL TURBULENCE, Year 1 (elective)
TTFYA ENGINEERING PHYSICS, Year 4 (elective)
TM Teknisk matematik, Year 2 (elective)
EMMAS MSc PROGR IN ENGINEERING MATHEMATICS, Year 1 (elective)

#### Examiner:

Bitr professor  Jöran Bergh

#### Eligibility:

For single subject courses within Chalmers programmes the same eligibility requirements apply, as to the programme(s) that the course is part of.

#### Course specific prerequisites

Basic Fourier analysis

#### Aim

Fourier analysis (frequency analysis) is an indis-pensable tool for deterministic and statistical signal analysis and processing (and in the theory of partial differential equations). Presently, the so-called wavelet transform is widely used as a complement to traditional Fourier transforms. The aim of the course is, in part, to describel how these trans-forms are used in practice, e.g., for 'sampling' of signals, in antenna theory, in geometrical topics, in computer homography, in probability theory, and also the 'fast' transforms which are now executed by computers in this context, for example in image processing.

#### Content

A basic mathematical tool for Fourier analysis is the concept of generalized functions (distributions). These provide a unified approach to Fourier series and Fourier integrals. We treat the Sampling Theorem, the Paley-Wiener theorem, the connec-tion autocorrelation - probability measure, the un-certainty relation, the Hilbert transform, antennas, optical lenses, discrete and fast Fourier and wavelet transforms. In higher dimensions, Fourier, Hankel, Abel, Radon, and wavelet transforms are treated, and applied to computer tomography and image processing.

#### Literature

R.N. Bracewell: The Fourier Transform and Its Applications (3 Ed.), McGraw-Hill 1999
J. Bergh, F. Ekstedt, M. Lindberg: Wavelets, Studentlitteratur 1999
Locally produced material.

#### Examination

Minor written examination and exercises handed in, partly based on computer calculations.

Page manager Published: Thu 03 Nov 2022.