ECE 513: Vector Space Signal Processing

Spring 2012

Lectures: Tuesdays, Wednesdays, 11-12:25 p.m.; 163 Everitt Lab.

Instructor: Prof. Yoram Bresler (ybresler at illinois.edu, 112 Coordinated Science Lab., 244-9660)

Office Hours: appointments by email.

TA: Kiryung Lee (klee81 at illinois.edu, 341 Coordinated Science Lab.)

TA Office Hours: Mondays, 4:30 ~ 6 p.m.; 361 Everitt Lab.

Overview:
Rigorous presentation of key mathematical tools in a vector space framework, and their applications in signal processing, including: finite and infinite dimensional vector spaces, Hilbert spaces, linear operators, inverse problems (e.g. deconvolution, tomography, Fourier imaging), least-squares methods, conditioning and regularization, matrix decompositions, subspace methods, bases and frames for signal representation (e.g. generalized Fourier series, wavelets, splines), Hilbert space of random variables, random processes, signal and spectral estimation, compressed sensing.

Topics:

Inverse problems and matrix theory (10 hours): linear inverse problems; orthogonal projections; minimum-norm least squares solutions; Moore-Penrose pseudoinverse; singular value decomposition; matrix decomposition and approximation; conditioning and regularization.

General linear vector spaces (17 hours): finite and infinite dimensional vector spaces; Hilbert spaces; projection theorem; inverse problems in infinite dimensional vector spaces; approximation and Fourier series; pseudoinverse operators; iterative methods for optimization and inverse problems; bases and frames for signal representation;

Hilbert space of random variables (6 hours): random processes; least-squares estimation; Wiener filtering; Wold decomposition; discrete-time Kalman filter.

Applications in signal processing (12 hours, during the course): deconvolution, optimal filter design, temporal and spatial spectrum estimation, tomography, harmonic retrieval, subspace methods, sensor array processing, extrapolation of band-limited sequences, generalized sampling, wavelets, splines, subset selection, sparse approximation and compressed sensing.

Handouts:

Course Syllabus

Lecture Notes: (access restricted to only people registered in the course)

Chapter 1 Updated 1/28/2012