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- Liberal arts and basic science courses
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- School of Engineering
- Instructor(s)
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- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
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- Group
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- Course number
- XEG.S606
- Credits
- 1
- Academic year
- 2019
- Offered quarter
- 2Q
- Syllabus updated
- 2019/5/23
- Lecture notes updated
- -
- Language used
- English
- Access Index
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Course description and aims
This course introduces the Fourier series and the Fourier integral transform and related transforms in a rigorous formulation. Reasoning and insight are emphasized over rote manipulation. Areas of application include antennas, signal and image formation and processing, communications, physics and statistics.
This course provides a first-principles introduction, and the goal is to enable the student to formulate and derive conditions for applying the theory in various contexts.
Student learning outcomes
Successful completion of this course will give the student a deeper knowledge of the essence of the basic principles in Fourier theory, series expansions and Fourier integrals. The student will be ready to apply the Fourier theory and acquire the tools to be able to formulate and solve problems not covered in the lectures, and to produce novel research outcomes.
Keywords
Harmonic Motion, Hilbert Space, Fourier Series, Convolution and Correlation, Sampling and Replication, Fourier Transform, Hankel Series and Hankel Transform
Competencies that will be developed
| ✔ Specialist skills | ✔ Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
1) At the beginning of each class, solutions to exercise problems assigned in the previous class are reviewed.
2) Attendance is taken in every class.
3) Students must familiarize the contents assigned in the previous class before coming to the class.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Signals, Signal Spaces and Operators | Students must make sure they understand what significance the course holds for them by checking their learning portfolio. Elementary operations acting on signals, concatenation of operations, groups of operations are introduced. Peruse lecture notes. |
| Class 2 | Harmonic Motion and Phasors Periodic Motion and Fourier Series (Dirichlet theory) | Phasor representation, Notions of resonance, Fourier Polynomials and their Infinite Limits. Notion of Infinite Dimension should be well understood. Brief review of limits, sequences and series. Peruse lecture notes. |
| Class 3 | Introduction to Hilbert Space and Fourier Series extensions | Notion of duality, square-summable sequences, Complete Ortho-Normal Sets (CONS), and Dirac’s bra-ket notation. Peruse lecture notes. |
| Class 4 | One-dimensional Fourier Transforms | Fejér Theory, Gibbs phenomenon, Properties of FT, and the duality in Fourier theory. Peruse relevant chapters of textbook and lecture notes. |
| Class 5 | Convolution and Correlation, Sampling and Replication | Convolution property and correlation property of FT, Power Signal Spaces, Application to Sampling and Replication. Peruse relevant chapters of the course textbook and lecture notes. |
| Class 6 | Multi-dimensional Fourier Transform | Multi-dimensional extensions of the Fourier transform. Peruse relevant chapter of the course textbook and lecture notes. |
| Class 7 | Hankel Series and Hankel Transform | Basics of Bessel functions with applications to FT in polar coordinates. Peruse relevant chapter of the course textbook and lecture notes. |
| Class 8 | Applications in Mathematical Physics | Solving the ODE’s of Mathematical Physics, Radon transform and connections to the Laplace transform. Peruse relevant chapter of the course textbook and lecture notes. |
Textbook(s)
[Bracewell: The Fourier Transform and Its Applications] is recommended for reading and a typed lecture notes of mine will be provided.
Reference books, course materials, etc.
Lecture notes
Assessment criteria and methods
Students will be assessed on their understanding of the concepts, theory, principles of Fourier analysis and their applications. The course scores are based on take-home exercise problems. Even though the series of lectures are the same as the course XEG.S506 which is for masters level, the grades in this doctoral level course are evaluated with different problem sets from XEG.S506.
Related courses
- SCE.A201 ： Mathematics for Systems and Control A
- SCE.I203 ： Digital Signal Processing
- SCE.I203 ： Digital Signal Processing
- EEE.M211 ： Fourier Transform and Laplace Transform
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students must have basic knowledge of Fourier analysis at the undergraduate level.
