# Mathematical foundations of signal processing

COM-514 / **6 credits**

**Teacher(s): ** Fageot Julien René Pierre, Simeoni Matthieu Martin Jean-André, Bejar Haro Benjamin

**Language:** English

## Summary

Signal processing tools are presented from an intuitive geometric point of view which is at the heart of all modern signal processing techniques. The student will develop the mathematical depth and rigor needed for the study of advanced topics in signal processing and approximation theory.

## Content

**Sequences, Discrete-Time Systems, Functions and Continuous-Time Systems** (review of discrete-time Fourier transform; DFT; Fourier transform and Fourier series).

**From Euclid to Hilbert: Linear Algebra Fundamentals for Representation Theory** (vector spaces; Hilbert spaces; approximations, projections and decompositions; bases and frames; linear operators; adjoint; generalized inverses; matrix representations; computational aspects)

**Sampling and Interpolation** (sampling and interpolation with normal and non orthogonal vectors, sequences and functions; sampling and interpolation of bandlimited sequences and functions)

**Polynomial and Spline Approximation** (Legendre and Chebyshev polynomials; Lagrange interpolation; minimax approximation; Taylor expansions; B-splines)

**Regularized Inverse Problems** (regularized convex optimisation; Tikhonov regularisation; penalised basis pursuit; proximal algorithms; pseudo-differential operators and L-splines; representer theorems for continuous inverse problems with Tikhonov penalties)

## Learning Prerequisites

## Required courses

Signal processing for communications (or Digital signal processing on Coursera)

Linear Algebra I and II (or equivalent).

## Recommended courses

Signals and Systems

## Important concepts to start the course

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

## Learning Outcomes

By the end of the course, the student must be able to:

- Master the right tools to tackle advanced signal and data processing problems
- Develop an intuitive understanding of signal processing through a geometrical approach
- Get to know the applications that are of interest today
- Learn about topics that are at the forefront of signal processing research
- Identify and implement the algorithm best suited to solve a given convex optimisation problem
- Assess the computational cost and numerical stability of a numerical solver

## Transversal skills

- Collect data.
- Write a scientific or technical report.
- Use a work methodology appropriate to the task.
- Demonstrate the capacity for critical thinking
- Use both general and domain specific IT resources and tools

## Teaching methods

Ex cathedra with exercises, homeworks and practicals.

## Expected student activities

Attending lectures, completing exercises.

## Assessment methods

homeworks and project assignement 50%, final exam (written) 50%

## Supervision

Office hours | Yes |

Assistants | Yes |

Forum | Yes |

## Resources

## Virtual desktop infrastructure (VDI)

No

## Bibliography

M. Vetterli, J. Kovacevic and V. Goyal, *"Signal Processing: Foundations"*, Cambridge U. Press, 2014.

Available in open access at http://www.fourierandwavelets.org

## Ressources en bibliothèque

## Moodle Link

## 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

**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