Teaching plan for the course unit

 

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General information

 

Course unit name: Applied Harmonic Analysis

Course unit code: 568183

Academic year: 2018-2019

Coordinator: F. Javier Soria de Diego

Department: Department of Mathematics and Computer Science

Credits: 6

Single program: S

 

 

Estimated learning time

Total number of hours 150

 

Face-to-face learning activities

60

 

-  Lecture

 

30

 

-  Lecture with practical component

 

15

 

-  Laboratory session

 

15

Supervised project

20

Independent learning

70

 

 

Recommendations

 


Students should have completed introductory courses in Calculus, Functional Analysis and Measure Theory.


Further recommendations


Students should have some knowledge of computer programming.

 

 

Competences to be gained during study

 


— Capacity to understand the concepts and rigorous proofs of fundamental theorems of certain specific areas of mathematics.

— Capacity to apply the results and acquired techniques to solve complex problems in certain areas of mathematics in academic or professional contexts.

— Ability to prepare and develop logical mathematical reasoning and to identify errors in incorrect reasoning.

— Capacity to construct, interpret, analyze and validate mathematical models developed to simulate real situations.

— Ability to enunciate and verify statements, and to communicate the acquired mathematical knowledge orally or in writing.

— Capacity to select and use software tools to address problems related to mathematics.
 

 

 

 

 

Learning objectives

 

Referring to knowledge


— To know the classical results of Fourier series and the Fourier transform, and the main techniques for almost everywhere convergence of the series.

— To learn the applications to information theory.

— To acquire a broad knowledge of the applications of Fourier theory in image processing and digital signal analysis, learning to compress an image or filter a noisy signal.

— To acquire basic research skills in the field of harmonic analysis.
 

 

 

Teaching blocks

 

1. Fourier Series; Orthonormal basis and L2 theory; Convergence of the series

2. Fourier transform in Lp; Sampling theorem

3. Maximal operators; Almost everywhere convergence

4. Marcinkiewicz Theorem; Interpolation Theory

5. Convolution operators; Fourier multipliers; Hilbert transform

6. MATLAB

6.1. Introduction to MATLAB

6.2. Programming

6.3. Averaging filter

6.4. Convolution

6.5. Fourier transform: DFT and FFT

6.6. Spatial filters

6.7. Frequency domain filters

6.8. Compression

6.9. JPEG and DCT

6.10. Audio

6.11. Wavelets

 

 

Teaching methods and general organization

 


Different topics will be presented in the theory lectures; For each topic, a collection of problems will be handed in; Students will prepare these exercises at home, and solve them in class; Students will also be assigned a series of programming exercises with MATLAB, to apply digital processing to images and sounds
 

 

 

Official assessment of learning outcomes

 


The continuous assessment has two sections:

Activity 1: Resolution of exercises and MATLAB assignments given in class.

Activity 2: Theoretical examination.

The final grade will be calculated as the average between 50% of Activity 1 and 50% of Activity 2 (a minimum grade of 4/10 is required on each activity).

 

Examination-based assessment


The single assessment consists of a final examination with theoretical and practical questions (exercises and MATLAB programs).

In order to be eligible for reassessment, students must have obtained a grade of at least 3.5/10 as a final grade in the continuous or single assessment.