Mathematical Foundations of Signal Processing

Overview

The goal of this class is to present signal processing tools from an intuitive geometric point of view which is at the heart of all modern signal processing techniques from Fourier transforms and sampling theorems to time-frequency analysis and wavelets. The course is designed to provide the mathematical depth and rigor needed for the study of advanced topics in signal processing and also features introductions to current applications where such tools are crucial. In particular, several applications will be studied, including image compression with linear and non-linear approximation, audio coding, mobile sensing, and prediction of the stock market.

During this course, students will:
– Master the right tools to tackle advanced signal and data processing problems
– Have 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

Content:

1. From Euclid to Hilbert
  • Vector spaces
  • Hilbert spaces
  • Approximations
  • Projections and decompositions
  • Bases and frames

2. Sequences and Discrete-Time Systems

  • Sequences
  • Systems
  • Discrete-time Fourier transform
  • z-transform
  • DFT
  • Multirate sequences and systems
  • Stochastic processes and systems

3. Functions and Continuous-Time Systems

  • Functions
  • Systems
  • Fourier transform
  • Fourier series
  • Continuous stochastic processes and systems

4. Sampling and Interpolation

  • Sampling and interpolation with
  1. finite-dimensional vectors
  2. sequences
  3. functions
  4. periodic functions
  5. stochastic processes

5. Approximation and Compression

  • Approximation by
  1. polynomials
  2. splines
  3. series truncation
  • Compression and transform coding

6. Localization and Uncertainty

  • Localization for
  1. functions
  2. sequences
  3. bases
  • Local Fourier and wavelet bases
  • Time, frequency and resolution in the real world

7. Compressed Sensing

  • Sparse signals
  • Compressed acquisition
  • Reconstruction algorithms
  • Applications

Pre-requisite:

Circuits and Systems

Signal processing for communications

Type of teaching:

Ex cathedra with exercises

Form of examination:

Written

Course website:

All the material for this course can be found on this link.

Form of examination:

Homeworks 20%, midterm (written) 30%, final exam (written) 50%

Reading material:

M. Vetterli, J. Kovacevic and V. Goyal, “Foundations of Signal Processing”, Cambridge U. Press, 2014. ISBN-13: 978-1107038608

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

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