Math 4355: Mathematics of Signal Representations

Math 4355 -- Mathematics of Signal Representations
SPRING 2015

	Instructor: Demetrio Labate OFFICE: PGH 694 EMAIL ADDRESS: dlabate@math.uh.edu HOMEPAGE: http://www.math.uh.edu/~dlabate When and Where MEETING TIME: Mon, Wed, Fri 10-11, MEETING PLACE: SEC 203 OFFICE HOURS: Mon, Wed 11-12 (or by appointment) Course Description: This course is a self-contained introduction to Fourier analysis and wavelets, and their applications to problems of image and signal processing. The motivation for this course comes from fundamental questions about the analysis and processing of signals and images such as: what is the best way to store and transmit signals? how can we remove unwanted noise from data? how can we automatically identify features of interests in a signal? Fourier analysis and wavelets offer a very powerful conceptual framework to deal with these problems. The ideas covered in this course are at the core of a variety of technologies used in apoplications including image and video compression, electronic surveillance, remote sensing and data transmission. Textbook: A first course in wavelets with Fourier analysis by A. Boggess and F. Narcowich, Wiley, 2nd edition 2009. HOMEWORK: Homework 1 - Due 2/1 - Solution Homework 2 - Due 2/13 - Solution Homework 3 - Due 2/20 - Solution Homework 4 - Due 3/2 - Solution Homework 5: Ex 4,18,20,23(b),(c),(d),26 p.85-87 - Due 3/9 - Solution Homework 6: Ex 2,4,6, p.128-129 - Due 3/25. Solution Suggested review problems for Quiz #2: 1-11, 20-25 p.83-86 Homework 7: Ex 5,10,12 (use Matlab to produce the graphs), p.128-129 - Due 4/6. Solution Homework 8: Ex 7,8 (explore the use of Matlab commands fft and fftshift), p.156 - Due April 15. Solution Suggested review problems for Quiz #3: 2,4,6, p.128-129; 1,4,5, p.186-187. Homework 9: Ex 1,2,9,10 p.186-188 - Due April 27. Solution Suggested review problems for final exam: 1,3,5,7, p. 83-84; 3,5,7, p.186-187.	Fourier series approximation of square wave

Instructor: Demetrio Labate

dlabate@math.uh.edu

http://www.math.uh.edu/~dlabate

When and Where

Course Description:

This course is a self-contained introduction to Fourier analysis and wavelets, and their applications to problems of image and signal processing. The motivation for this course comes from fundamental questions about the analysis and processing of signals and images such as: what is the best way to store and transmit signals? how can we remove unwanted noise from data? how can we automatically identify features of interests in a signal? Fourier analysis and wavelets offer a very powerful conceptual framework to deal with these problems. The ideas covered in this course are at the core of a variety of technologies used in apoplications including image and video compression, electronic surveillance, remote sensing and data transmission.

Textbook:

A first course in wavelets with Fourier analysis by A. Boggess and F. Narcowich, Wiley, 2nd edition 2009.

HOMEWORK:

Suggested review problems for Quiz #2: 1-11, 20-25 p.83-86

Solution

Suggested review problems for Quiz #3: 2,4,6, p.128-129; 1,4,5, p.186-187.

Solution

Suggested review problems for final exam: 1,3,5,7, p. 83-84; 3,5,7, p.186-187.

Fourier series approximation of square wave

Useful Material:

Very brief linear algebra review [from http://alumni.media.mit.edu]
Another linear algebra review (with Matlab examples) [by G. Recktenwald Portland State University]

Test 1 with Solution - Test 2 with Solution - Test 3 with Solution

A list of old Tests #1,2,3 and their solutions: Tests+Solutions

Prerequisites:

MATH 2331 and one of the following: MATH 3333, MATH 3334, MATH 3330, MATH 3363. Students who wish to enroll without having one of the above junior-level courses are encouraged to discuss it with the instructor. While a prior knowledge of Matlab is not required, be aware that Matlab will be used for some homework. The use of the basic Matlab functions is very simple and it will be easy to acquire this basic-level knowledge during the course.

Course outline:

Inner product spaces [Ch.0, Sec.0.1-0.5]

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 [Ch.1, Sec 1.1-1.3; Ch.2, Sec. 2.1-2.4]

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 [Ch.3,Sec.3.1].

Wavelets [Ch.4, Sec 4.1-4.3; Ch.5, Sec. 5.1-5.2]

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.

Tests and Exam Dates:

The dates for the midterm exams are Fri Feb 20, Mon Mar 30 and Fri April 24. The final exam is scheduled on FRI MAY 8, 11am-2pm.

Grading:

Grades will be based on homework assignments counting 30% towards the final grade, on three midterm exams counting 40% towards the final grade (16% + 16% +8% for the worst one) and one final counting 30% towards the final grade.

The grade will be determined according to a set point scale: 90%-100%: A, 80%-89%: B, 70%-79%: C, 60-69% D; F is less than 60% (+ and - will also be used)..