EESC 6350: Signal Theory

Fall 2021

Instructor Aria Nosratinia,
ECSN 4.504 Tel: 972-883-2894
Time Mon-Wed. 2:30-3:45pm
Place ECSN 2.112
Office Hours By appointment
Textbook Moon and Stirling: Mathematical Methods and Algorithms for Signal Processing, Prentice-Hall
TA TBA
Course Notes Posted on eLearning
Grading Midterm 1, (25%), Midterm 2, (25%), Final (40% – time and place to be announced by Registrar), Homework and Quizzes (10%). Optional mini-project for 10%, reducing the final exam to 30%.
Useful Dates Classes begin Aug. 23. Last day of class December 6

This course is designed to prepare the students for advanced studies and research in (digital) signal processing and communications. It explores the fundamentals of signal representation and approximation, with an emphasis on minimum mean square theory and Hilbert space approximation. The course starts with a basic overview of matrix and vector analysis, followed by a coverage of least squares (LS) solutions, where the power and breadth of the orthogonality principle and its applications in signal processing are emphasized. Minimum-norm and MNLS solutions, psuedo-inverses, eigen-value and singluar-value decompositions are treated in detail. Time permitting, additional advanced topics will also be visited.

Contents:

  • Signal Spaces
    • Vector spaces
    • Norms
    • Inner products and orthogonality
    • Linear transformations
    • Projections

  • Representation and Approximation in Vector Spaces
    • The orthogonality principle
    • Matrix representation of LS problems
    • Applications

  • Linear Operators and Matrix Inverses
    • Linear Operators
    • Operator Norms
    • Adjoint Operators
    • Four fundamental subspaces
    • Matrix inverses, min. norm and least squares solutions
    • Psuedo-inverses

  • Matrix Factorizations
    • LU factorization
    • Cholesky factorization
    • QR factorization

  • Eigen-Decompositions
    • Eigenvalues and eigenvectors
    • Diagonalization
    • Invariant subspaces
    • Quadratic forms
    • Applications

  • Singular Value Decomposition
    • The Eckart-Young Theorem
    • SVD approximation
    • Relation with psuedo-inverses

  • Advanced Topics
    • Extension to infinite dimensions
    • Hilbert spaces
    • The Hilbert space of zero-mean random variables
    • MMSE approximation of random variables
    • Generalized Fourier decompositions
    • Energy preserving transformations, the generalized Parseval equality