# 227-0445-10L Mathematical Methods of Signal Processing

Semester | Autumn Semester 2020 |

Lecturers | H. G. Feichtinger |

Periodicity | non-recurring course |

Language of instruction | English |

Abstract | This course offers a mathematical correct but still non-technical description of key objects relevant for signal processing, such as Dirac measures, 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. |

Lecture notes | This material will be regularly updated and posted at the lecturer's homepage, at Link There will be also a dedicated WEB page at Link (to be installed in the near future). |

Prerequisites / Notice | We encourage students who are interested in mathematics, but also students of physics or mathematics who want to learn about application of modern methods from functional analysis to their sciences, especially those who are interested to understand what the connections between the continuous and the discrete world are (from continuous functions or images to samples or pixels, and back). Hans G. Feichtinger (Link) For any kind of questions concerning this course please contact the lecturer. He will be in Zurich most of the time, even if the course has to be held offline. It will start by October 1st 2020 only. |