# Mathematical foundations of signal processing

COM-514 / **6 credits**

**Teacher: **

**Language:** English

**Remark:** cours pas donné en 2023-24

## Summary

A theoretical and computational framework for signal sampling and approximation is presented from an intuitive geometric point of view. This lecture covers both mathematical and practical aspects of modern signal processing, with hands-on projects, applications and algorithmic aspects.

## Content

**From Euclid to Hilbert (1/2):** Hilbert Spaces and Linear Operators (Vector spaces, Hilbert/Banach spaces; adjoint and inverse operators; projection operators)

**From Euclid to Hilbert (2/2):** Hilbert Representation Theory (Riesz bases; Gramian; basis expansions; approximations & projections; matrix representations)

**Application (1/2):** Sampling and Interpolation (Fourier transforms and Fourier series; sampling & interpolation of sequences and functions; Shannon sampling theorem revisited; bandlimited approximation)

**Application (2/2):** Computerized Tomography (line integrals and projections, Radon transform, Fourier projection/slice theorem, filtered backprojection algorithm).

**Regularized Inverse Problems (1/2):** Theory (Discrete and functional inverse problems; Tikhonov regularisation; sparse recovery; convex optimisation; representer theorems; Bayesian interpretation)

**Regularized Inverse Problems (2/2):** Algorithms (Proximal algorithms; gradient de- scent; primal-dual splitting; computational aspects; numerical experiments and examples)

## Learning Prerequisites

## Important concepts to start the course

Good knowledge of linear algebra concepts. Basics of Fourier analysis and signal processing. Basic knowledge of Python and its scientific packages (Numpy, Scipy).

## Supervision

Office hours | No |

Assistants | Yes |

## In the programs

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Mathematical foundations of signal processing**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

## Reference week

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