MATH 4355 - Mathematics of Signal Representations - University of Houston

# MATH 4355 - Mathematics of Signal Representations

***This is a course guideline.  Students should contact instructor for the updated information on current course syllabus, textbooks, and course content***

Prerequisites: MATH 2415 and six additional hours of 3000-4000 level Mathematics.

Course Description: Fourier series of real-valued functions, the integral Fourier transform, time-invariant linear systems, band-limited and time-limited signals, filtering and its connection with Fourier inversion, Shannon’s sampling theorem, discrete and fast Fourier transforms, relationship with signal processing.

Textbook: “A first course in wavelets with Fourier Analysis” by A. Boggess and F. Narcowich, Prentice Hall 2nd Edition, ISBN: 978-0470431177

Topics Covered

Inner product spaces

• The linear algebra of inner product spaces: Linear Subspaces, linear independence, linear bases. Linear mappings and the matrix representation of a linear mapping (operators). (These will be cover from "Linear Algebra" of the Shaum's Outline series). This will take us 2 weeks.
• Inner product spaces (At this ponit we switch to the textbook).
• The spaces L 2 and l 2 .
• Schwarz and triangle inequalities
• Linear operators and their adjoints.
• Best fit line for data.

Fourier series

• Introduction
• Compuation of Fourier series
• Convergence theorems for Fourier series.

The Integral Fourier transform

• The definition of the Integral Fourier transform
• Properties  Integral Fourier transform
• Convolutions
• Linear filters
• The sampling theorem: Analog to Digital and digital to Analog conversions
• Uncertainty principle (we will simply review this section).
• A brief overview of Computerized tomography (Radon transform) and the back-projection algorithm (this is material is not contained in the textbook. Instead we will use Epstein's “Introduction to Mathematics of Medical Imaging” for this part of the course).

The Discrete Fourier transform

• Definition, properties, FFT, FFT used for the approximate computation of integral Fourier transforms.
• Discrete signals, time-invariance, convolution and linear filters