### 2020　Fourier Transform and Laplace Transform

Font size  SML

Undergraduate major in Electrical and Electronic Engineering
Instructor(s)
Kakushima Kuniyuki  Ito Hiroyuki
Class Format
Lecture    (ZOOM)
Media-enhanced courses
Day/Period(Room No.)
Tue3-4(S221)  Fri3-4(S221)
Group
-
Course number
EEE.M211
Credits
2
2020
Offered quarter
1Q
Syllabus updated
2020/9/18
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

This course focuses on the Fourier transform and Laplace transform used in the analysis of linear systems. Topics include linear systems, Fourier series of periodic functions, Fourier transforms of aperiodic functions, the properties of Fourier transforms, convolution, discrete Fourier transforms, fast Fourier transforms, Laplace transforms, the properties of Laplace transforms, and solving differential equations with the Laplace transform. By combining lectures and exercises, the course enables students to understand and acquire the fundamentals of mathematical tools widely applicable to linear systems.
Analysis in the frequency domain is vital in the field of electrical engineering and information communication engineering. For example, when the response of a linear electronic circuit to an input voltage is given as a function of time, the problem is dealt with in the time domain. By using a mathematical approach such as the Fourier transform,however, the response can be discussed in the frequency domain, and this can provide us with very useful results.Mathematical approaches taught in this course are not only useful in analyzing electronic circuits, but are applicable to various other types of linear systems, and are highly effective in the field of engineering. Students will experience the satisfaction of solving practical problems by using their mathematical knowledge acquired through this course.

### Student learning outcomes

By the end of this course, students will be able to:
1) Understand linear systems and mathematically transform signals between the time and frequency domains.
2) Expand periodic functions in Fourier series and represent the response of a linear electronic circuit using a transfer function.
3) Compute the frequency spectra of aperiodic functions by using the Fourier transform.
4) Explain the principles and properties of the discrete Fourier transform and fast Fourier transform together with their applications.
5) Acquire the fundamentals of the Laplace transform, and based on this knowledge, compute the transient response of a linear electronic circuit.
6) Apply Fourier and Laplace transforms to solve problems.

### Keywords

Fourier series, Fourier transform, discrete Fourier transform, Laplace transform, time domain, frequency domain,transient response analysis, sampling theorem, linear system, electronic circuit, transfer function, system stability

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills ✔ ・Fundamental specialist skills on EEE

### Class flow

At the beginning of each class, solutions to exercise problems that were assigned during the previous class are reviewed. Towards the end of class, students are given exercise problems related to the lecture given that day to solve.To prepare for class, students should read the course schedule section and check what topics will be covered.Required learning should be completed outside of the classroom for preparation and review purposes.

### Course schedule/Required learning

Course schedule Required learning
Class 1 Linear systems and Fourier series of periodic functions Understand the definition of a linear system. Compute the Fourier series of periodic functions.
Class 2 Properties of Fourier series Compute Fourier coefficients considering the even/odd properties of the function. Discuss term-by-term differentiation of Fourier series.
Class 3 Steady state response of linear electronic circuits to a periodic input Discriminate the linearity of electronic circuits. Compute the steady state response of linear electronic circuits to an arbitrary periodic input.
Class 4 Aperiodic functions, Fourier integral, and Fourier transform Derivation of the Fourier transform and the Fourier inverse transform. The Fourier cosine transformation and the Fourier sine transformation
Class 5 Properties of the Fourier transform Understand conditions to exist the Fourier transform. Properties of the Fourier transform
Class 6 Relation between time domain and frequency domain, Temporal and frequency responses of linear electronic circuits Explain duality of time and frequency domains. Explain and illustrate the relation between the temporal and frequency responses of a linear electronic circuit.
Class 7 Exercise and evaluation of the first part of the course Exercise problems and evaluate achievement for classes 1–6.
Class 8 Shannon sampling theorem Explanation of Shannon sampling theorem
Class 9 Basics of discrete Fourier transform Basics of discrete Fourier transform
Class 10 Application of discrete Fourier transform Discrete Fourier transform for discrete periodic signal
Class 11 Laplace transform and partial fraction expansion Nature of Laplace transform
Class 12 Inverse Laplace transform and transient response of a linear circuit Inverse Laplace transform and application to differential equation. Linear circuit transient response.
Class 13 Stability of a system and applications of Laplace transform in electrical engineering. z transform. Stability of a system assessed by Laplace transform.
Class 14 Exercise and evaluation of the 2nd part of the course Exercise problems and evaluate achievement for classes 8–13.

### Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

none

### Reference books, course materials, etc.

Matsushita, Yasuo. Fourier analysis: Fundamentals and Applications. Tokyo: Baifukan; ISBN-13: 978-4563011093.(Japanese)
Terada, Fumiyuki. Fourier Transform and Laplace Transform. Tokyo: Science-sha; ISBN-13: 978-4781908939.(Japanese)

### Assessment criteria and methods

Students' knowledge of Fourier series, Fourier transform, and Laplace transform, and their ability to apply them to problems will be assessed.
Reports of classes 1-6 (10%), exercise in class 7 (40%), reports of classes 8-13 (10%), and exercise in class 14 (40%).

### Related courses

• EEE.M201 ： Analysis for Electrical and Electronic Engineers
• IEE.C431 ： Applied Statistical Analysis
• EEE.M241 ： Discrete-Time System
• EEE.C201 ： Electric Circuits I
• EEE.C202 ： Electric Circuits II
• EEE.C261 ： Control theory

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students must have successfully completed both Calculus I (LAS.M101) and Calculus II (LAS.M105) or have equivalentknowledge.