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 2433 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

Grades will be based on homework assignments and on two exams (midterm and final). You can improve the grade of your midterm exam with the final, but this implies that for those who choose to do so the final is comprehensible. The midterm gives you 80 pts and the final 110pts. Each homework assignment gives a different number of points (5 for each problem). All grades are summed and divided by the total number of pts you can collect in the course. This gives you your final grade. A quotient of  0.43  or more is  D- , of  0.46  or more is D, of 0.54 or more is C, of 0.64 is  B-, of 0.80 or more is A- ,  of 0.85 or more  is  A.

CSD Accommodations:

Accommodation Forms: Students seeking academic adjustments/auxiliary aids must, in a timely manner (usually at the beginning of the semester), provide their instructor with a current Student Accommodation Form (SAF) (paper copy or online version, as appropriate) from the CSD office before an approved accommodation can be implemented.

Details of this policy, and the corresponding responsibilities of the student are outlined in The Student Academic Adjustments/Auxiliary Aids Policy (01.D.09) document under [STEP 4: Student Submission (5.4.1 & 5.4.2), Page 6]. For more information please visit the Center for Students with Disabilities Student Resources page.

Additionally, if a student is requesting a (CSD approved) testing accommodation, then the student will also complete a Request for Individualized Testing Accommodations (RITA) paper form to arrange for tests to be administered at the CSD office. CSD suggests that the student meet with their instructor during office hours and/or make an appointment to complete the RITA form to ensure confidentiality.

*Note: RITA forms must be completed at least 48 hours in advance of the original test date. Please consult your counselor ahead of time to ensure that your tests are scheduled in a timely manner. Please keep in mind that if you run over the agreed upon time limit for your exam, you will be penalized in proportion to the amount of extra time taken.

UH CAPS

Counseling and Psychological Services (CAPS) can help students who are having difficulties managing stress, adjusting to college, or feeling sad and hopeless. You can reach (CAPS) by calling 713-743-5454 during and after business hours for routine appointments or if you or someone you know is in crisis. No appointment is necessary for the "Let's Talk" program, a drop-in consultation service at convenient locations and hours around campus.