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227-0445-10L 6 Credits MSC D-ITET , D-MATH , D-INFK , D-PHYS

Mathematical Methods of Signal Processing

Lecturers & Examiners: Prof. Hans Georg Feichtinger
VVZ CR n/a

Last Updated: 2026-02-05 15:35:16

Abstract

This course offers a mathematical correct but still non-technical description of key objects relevant for signal processing, such as Diracmeasures, Dirac combs, various function spaces (like L^2), impulse response, transfer function, Gabor expansion, and so on. The approach is based on properties of "Feichtinger's algebra". MATLAB routines will serve as illustration.

Objective

The aim of the class to familiarize the participants with the idea of generalized functions (usual called distributions), and to provide a (novel approach) to a theory of mild distributions, which cannot be found in books so far (the course will contribute to the development of such a book). From the physical point of view, such an object is something, which can be measured or captured by (linear) measurements, such as an audio signal. The Harmonic Analysis perspective is, that the Fourier transform and time-frequency transforms are possible over any locally compact group. Engineers talk about discrete or continuous, periodic and non-periodic signals. Hence, a unified approach to these settings and a discussion of their interconnection (e.g. approximately computing the Fourier transform of a function using the DFT) is at the heart of this course.

Content

Mathematical Foundations of Signal Processing: 0. Recalling (on and off) concepts from linear algebra (e.g. linear mappings, etc.) and introducing concepts from basic linear functional analysis (Hilbert spaces, Banach spaces) 1. Translation invariant systems and convolution, elementary functional analytic approach; 2. Pure frequencies and the Fourier transform, convolution theorem 3. The subalgebra L1(Rd) of integrable functions (without Lebesgue integration), Riemann Lebesgue Lemma 4. Plancherels Theorem, L2(Rd) and basic Hilbert space theory, unitary mappings 5. Short-time Fourier transform, the Feichtinger algebra S0(Rd) as algebra of test functions 6. The dual space of mild distributions, relationship to tempered distributions (for this familiar); various characterization 7. Gabor expansions of signals, characterization of smoothness and decay, Gabor frames and Riesz bases; 8. Transition from continuous to discrete variables, from periodic to the non-periodic case; 9. The kernel theorem, as the continuous analogue of matrix representations; 10. Sobolev spaces (describing smoothness) and weighted spaces; 11. Spreading representation and Kohn-Nirenberg representation of operators; 12. Gabor multipliers and approximation of slowly varying systems; 13. As time permits: the idea of generalized stochastic processes 14. Further subjects as demanded by the audience can be covered on demand. Detailed lecture notes will be provided. This material will become part of an on-going book-project, which has many facets.

Resources

Lecture Notes

This material will be regularly updated and posted at the lecturer's homepage, athttps://www.univie.ac.at/nuhag-php/home/skripten.phpThere will be also a dedicated WEB page atwww.nuhag.eu/ETH20(to be installed in the near future).

Learning Materials (Links)

General Information

Language
English
Levels
MSC

Examination

Type
end-of-semester examination
Mode
oral 30 minutes

Course Components

Type Title Time & Place Hours
lecture with exercise Mathematical Methods of Signal Processing
Lecture start is on Tuesday, 6 October 2020. Remote lecture on Tusdays 8-10h, per Zoom: Meeting-ID: 994 133 7347 Kenncode: 530642 Remote exercise on Thursdays 14-16h, per Zoom. Meeting-ID: 863 7709 0433 Kenncode: 170548 The lecturer will communicate the exact lesson times of ONLINE courses.
  • Tue 08:00-10:00 (ON LI NE)
  • Thu 14:00-16:00 (ON LI NE)
4 h weekly

Offered In