


Politecnico di Torino  
Academic Year 2017/18  
01RMQNG Timefrequency and multiscale analysis 

Master of sciencelevel of the Bologna process in Mathematical Engineering  Torino 





Esclusioni: 01RLM 
Subject fundamentals
The course is mostly addressed to the students of Mathematical Engineering, but it is suggested to all students interested in the mathematical foundations of digital signal and image processing, with a solid mathematical background. It is essentially a course in applied harmonic analysis, devoted to the principles of signal analysis, starting from discrete Fourier analysis, up to the timefrequency and timescale (wavelets) analysis.
From a modern point of view, signals are modelled by vectors in Hilbert spaces (of sequences or functions), and the problems of approximation, compression, etc., reduce to the representation of the signal with respect to a suitable basis. One is then interested in looking for optimal and structured bases, often constructed by operations such as translations, dilations and modulations, starting from a suitable window. The classical results from signal processing (sampling, interpolation, filters, filter banks, etc.) can be reinterpreted by this geometricanalytic language from Functional Analysis, which is illuminating in its own right and is also the language currently used in timefrequency analysis and mathematical signal processing. 
Expected learning outcomes
Understanding of the subjects of the course and computational skill. Familiarity with the mathematical content of engineering disciplines.

Prerequisites / Assumed knowledge
The topics contained in the courses of Mathematical Analysis I, II, Geometry and Functional Analysis.

Contents
Review on Hilbert spaces, orthonormal bases, Riesz bases.
Discrete Fourier Analysis (DTFT, z, DFT transforms). Sampling and interpolations. Localization (STFT transform) and uncertainty principle. Filter banks. Wavelets bases of sequences. Wevelets bases of functions. Introduction to variational methods. 
Delivery modes
Theoretical lessons: 40 hours. Exercise hours: 10 hours in computer lab. Theoretical lessons are devoted to the presentation of the topics, with definitions, properties and the proofs which are believed to facilitate the learning process. Every theoretical aspect is associated with introductory examples. The exercise hours are devoted to the analysis of small projects in computer lab.

Texts, readings, handouts and other learning resources
 Lecture notes (in italian), provided by the instructor
 M. Vetterli, J. Kovacevic, V. Goyal, Foundations of signal processing, Cambridge University Press, 2014  J. Kovacevic, V. Goyal, M. Vetterli, Fourier and wavelets signal processing, in preparation (available on authors’ web page) Other material will be available on the Portale della Didattica. 
Assessment and grading criteria
The goal of the exam is to test the knowledge of the candidate on the topics included in the official program of the course and to verify the computational and theoretical skills in solving exercises and developing small projects. Marks range from 0 to 30 and the exam is successful if the mark is at least 18.
The exam is oral and includes a discussion of an exercise or project assigned during the course. 
