ECE 608 - Signal Theory

 

INSTRUCTOR

Prof. Marco F. Duarte, Marcus 215I, mduarte@ecs.umass.edu
Office Hours: Tuesdays 10:00am-11:00am, Wednesdays 3:00pm-4:00pm, or by appointment.

LECTURES

325 Engineering Lab 2:30pm-3:45pm Tuesday and Thursday.

DESCRIPTION

Unified treatment of techniques for representation of signals and signal processing operations. Emphasis on physical interpretation of vector spaces, linear operators, transform theory, and digital signal processing with wavelet filter banks. This course is an introduction to the mathematical foundations of signal processing.  Students completing this course will know:

  1. •The properties of linear and Hilbert spaces and their uses for signal representations.

  2. •The principles of continuity, convergence, and orthogonal projection, and their applications in signal processing operations.

  3. •Properties, representations, and some applications of linear functionals and operators.

  4. •The fundamental techniques for constrained and unconstrained optimization.

PREREQUISITES

The course is open to graduate students only. Prerequisites include undergraduate-level linear algebra and signals and systems. As in many courses in signals and systems, a reasonable degree of mathematical sophistication will be very helpful.

TEXTBOOK

Todd K. Moon and Wynn C. Stirling, “Mathematical Methods and Algorithms for Signal Processing,” Prentice Hall, 1999.

The following books will also be useful:


  1. •David G. Luenberger, “Optimization by Vector Space Methods,” Wiley, New York, 1968.

  2. •L. E. Franks, “Signal Theory”, Prentice Hall, 1968.

  3. •A. Mertins, “Signal Analysis”, Wiley, 1999.


LECTURE SCHEDULE (TENTATIVE)


Part 1: Linear Spaces

Week 1: Introduction. Signal spaces. Vector spaces.

Week 2: Norms and normed vector spaces. Induced norms. Cauchy-Schwarz inequality. Orthogonality.

Part 2: Hilbert Spaces

Week 3: Inner products. Orthogonal subspaces. Linear transformations.

Week 4: Projections and orthogonal projections. Projection Theorem.

Part 3: Approximation in Hilbert Spaces

Week 5: Orthogonality principle. Least squares.

Week 6: Complete orthonormal sequences. Fourier series.

Part 4: Linear Functionals and Operators

Week 7: Linear operators. Operator norms. Adjoint operators.

Week 8: Eigenvalues and eigenvectors. Singular value decomposition. Karhunen-Loève Transform.

Part 5: Local, Global, and Constrained Optimization

Week 9: Derivatives and Differentials. Euler-Lagrange Equations.

Week 10: Convexity. Equality constraints.

Week 11: Lagrange multipliers. Inequality constraints.

Week 12: Kuhn-Tucker conditions.

Part 6: Applications

Week 13: Transform coding.