University of Houston

Department of Mathematics

Math 4355: Mathematics of Signal Representations. SPRING 2009

**Prerequisites: **MATH 2431 and one of the following: MATH 3333, MATH 3334, MATH 3330, MATH 3363. MATH 3321 can be used instead of MATH 2431. Students may attempt the course without having one of the above junior-level courses but they first have to obtain the consent of the instructor.

**Textbook: **

*“A first course in wavelets with Fourier analysis” * by A. Boggess and F. Narcowich, Wiley, 2009. Note: New edition in preparation, text available to course members by special arrangement with publisher.

**Description of MATH 4355: **

**Inner product spaces **

- Inner product spaces.
- The spaces of square integrable functions and square summable series.
- Schwarz and triangle inequalities.
- Orthogonal projections and the least squares fit.

**Fourier series and transform**

- Computation of Fourier series.
- Convergence of Fourier series.
- The Fourier transform.
- Convolutions.
- Linear filters.
- The sampling theorem: Analog to digital and digital to analog conversions.
- From analog to digital filters.
- The Discrete Fourier transform (DFT), FFT, its use for the approximate computation of integral Fourier transforms.

**Wavelets**

- The Haar wavelet.
- Multiresolution analysis.
- The scaling relation.
- Properties of the scaling function.
- Decomposition and reconstruction.
- Wavelet design in the frequency domain.
- The Daubechies wavelet.

**Grading: **

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.