Course Information
SemesterCourse Unit CodeCourse Unit TitleL+PCreditNumber of ECTS Credits
6MATH334INTRODUCTION TO WAVELETS AND APPLICATIONS3+036

Course Details
Language of Instruction English
Level of Course Unit First Cycle
Department / Program MATHEMATICS
Mode of Delivery Face to Face
Type of Course Unit Elective
Objectives of the Course This is an introductory course on wavelet analysis, with an emphasis on the fundamental mathematical principles and basic algorithms, including continuous and discrete wavelet transform, orthogonal and biorthogonal wavelets of compact support, wavelet base and wavelet packages. Applications related, for example to signal analysis, image processing, numerical analysis will be also discussed. It is designed as a broad introduction to wavelets for engineers, mathematicians, and physicists.
Course Content Mathematical preliminaries for continuous and discrete Fourier transform. Short Time Fourier Transform. Continuous wavelet transform. Discrete wavelet transform. Daubechies wavelets. Designing orthogonal wavelet systems. Orthonormality of translates of scaling functions. Orthonormality of scaling and wavelet functions. Approximation conditions (Smoothness conditions). Discrete wavelet transform and relation to filter banks. Signal decomposition (Analysis), Relation with filter banks. Down-sampling, up-sampling and filtering. Multi-Resolution Analysis (MRA). Biorthogonal Wavelets. B-Splines. Applications of wavelets: Applications of wavelets in signal and image processing and other related engineering fields.
Course Methods and Techniques
Prerequisites and co-requisities None
Course Coordinator Associate Prof.Dr. Nasser AGHAZADEH nasseraghazadeh@iyte.edu.tr
Name of Lecturers Associate Prof.Dr. Nasser AGHAZADEH nasseraghazadeh@iyte.edu.tr
Assistants None
Work Placement(s) No

Recommended or Required Reading
Resources P. Soman, K.I. Ramachandran, N.G. Resmi, Insight into Wavelets, From Theory to Practice, 3rd edition, PHI Learning Private Limited, New Delhi, 2010
Martin J. Mohlenkamp MarĂ­a Cristina Pereyra, Wavelets, Their Friends, and What They Can Do for You, European Mathematical Society, 2008
David F. Walnut, An Introduction to Wavelet Analysis, Springer Science+Business Media New York, 2004
Edward Aboufadel and Steven Schlicker, DISCOVERING WAVELETS, JOHN WILEY & SONS, INC, 1999
Jaideva C. Goswami, Andrew K. Chan, Fundamentals of Wavelets Theory, Algorithms, and Applications, Second Edition, John Wiley & Sons, 2011

Course Category

Planned Learning Activities and Teaching Methods
Activities are given in detail in the section of "Assessment Methods and Criteria" and "Workload Calculation"

Assessment Methods and Criteria
In-Term Studies Quantity Percentage
Midterm exams 1 % 30
Quizzes 0 % 0
Homeworks 2 % 10
Other activities 0 % 0
Laboratory works 0 % 0
Projects 2 % 20
Final examination 1 % 40
Total
6
% 100

ECTS Allocated Based on Student Workload
Activities Quantity Duration Total Work Load
Weekly Course Time 1 42 42
Outside Activities About Course (Attendance, Presentation, Midterm exam,Final exam, Quiz etc.) 5 14 70
Laboratory 1 6 6
Exams and Exam Preparations 6 12 72
Total Work Load   Number of ECTS Credits 6 190

Course Learning Outcomes: Upon the successful completion of this course, students will be able to:
NoLearning Outcomes
1 Understanding wavelet transform and different wavelet families.
2 Understanding orthogonal wavelet systems, and their properties.
3 Understanding the relation of discrete wavelet transform to filter banks.
4 Ability to decompose and reconstruct a given signal.
5 Understanding biorthogonal wavelet systems.
6 Ability to apply wavelets in practical problems, e.g., in signal and image processing.


Weekly Detailed Course Contents
WeekTopicsStudy MaterialsMaterials
1 Fundamentals of Linear Algebra: Vector spaces, Bases, Orthogonality, Orthonormality, Projection, Functions and function spaces, Orthogonal functions, Orthonormal functions, Orthogonal basis functions, Signal Representation in Fourier domain, Fourier series, Orthogonality, Orthonormality and the method of finding the Fourier coefficients, Complex Fourier series, Orthogonality of complex exponential bases Insight into wavelets, Chapter 2
2 Mathematical preliminaries for continuous and discrete Fourier transform, Limitations of Fourier domain signal processing Insight into wavelets, Chapter 3
3 Short Time Fourier Transform (STFT): Signal representation with continuous and discrete STFT, Concept of time-frequency resolution, Resolution problem associated with STFT, Heisenberg's uncertainty principle and time frequency tiling Insight into wavelets, Chapter 3
4 Why wavelet transform? Introduction to wavelet transform: The origins of wavelets, Wavelets and other wavelet like transforms, Different families of wavelets, Continuous wavelet transform: Continuous time-frequency representation of signals, Properties of wavelets used in continuous wavelet transform, Continuous versus discrete wavelet transform Insight into wavelets, Chapter 3
5 Discrete wavelet transform: Haar scaling functions and function spaces, Translation and scaling of phi(t), Orthogonality of translates of phi(t), Function space V0 , Finer Haar scaling functions, Concepts of nested vector spaces, Haar wavelet function, Scaled and translated Haar wavelet functions Insight into wavelets, Chapter 4
6 Orthogonality of phi(t) and psi(t), Normalization of Haar bases at different scales, Refinement relation with respect to normalized bases, Support of a wavelet system, Daubechies wavelets, Plotting the Daubechies wavelets Insight into wavelets, Chapter 4
7 Other wavelet Families
8 Designing orthogonal wavelet systems (A direct approach): Refinement relation for orthogonal wavelet systems, Restrictions on filter coefficients, Unit area under scaling function, Orthonormality of translates of scaling functions, Orthonormality of scaling and wavelet functions Insight into wavelets, Chapter 5
9 Approximation conditions (Smoothness conditions), Designing Daubechies orthogonal wavelet system coefficients, Constraints for Daubechies' 6 tap scaling function. Insight into wavelets, Chapter 5
10 Discrete wavelet transform and relation to filter banks: Signal decomposition (Analysis), Relation with filter banks Insight into wavelets, Chapter 6
11 Frequency response, Signal reconstruction: Synthesis from coarse scale to fine scale, Downsampling, upsampling and filtering, Perfect reconstruction filters, QMF conditions, Computing initial sj+1 coefficients, Concepts of Multi-Resolution Analysis (MRA) Insight into wavelets, Chapter 6
12 Biorthogonal Wavelets: Biorthogonality in vector space, Introduction to biorthogonal Wavelet Systems, Signal representation using biorthogonal wavelet system Insight into wavelets, Chapter 8
13 Biorthogonal synthesis, Construction of biorthogonal wavelet system, B-Splines Insight into wavelets, Chapter 8
14 Applications of wavelets: Applications of wavelets in signal and image processing and other related engineering fields


Contribution of Learning Outcomes to Programme Outcomes
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14
C1 4
C2 4
C3 3 3 3
C4 3 3 3
C5 4
C6 3 4 4

Contribution: 0: Null 1:Slight 2:Moderate 3:Significant 4:Very Significant


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