### 2018　Fourier Transform and Laplace Transform

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Undergraduate major in Electrical and Electronic Engineering
Instructor(s)
Kakushima Kuniyuki  Ito Hiroyuki
Course component(s)
Lecture
Mode of instruction

Day/Period(Room No.)
Wed1-2(S221)  Fri3-4(S221)
Group
-
Course number
EEE.M211
Credits
2
2018
Offered quarter
2Q
Syllabus updated
2018/4/3
Lecture notes updated
2018/7/27
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

### 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 Practice and explanation of the Fourier series Review and practice of the Fourier series and related knowledge
Class 5 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 6 Properties of the Fourier transform Understand conditions to exist the Fourier transform. Properties of the Fourier transform
Class 7 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 8 Test level of understanding with exercise problems and summary of the first part of the course Test level of understanding and self-evaluate achievement for classes 1–7.
Class 9 Shannon sampling theorem Explanation of shannon sampling theorem
Class 10 Basics of discrete Fourier transform Basics of discrete Fourier transform
Class 11 Application of discrete Fourier transform Discrete Fourier transform for discrete periodic signal
Class 12 Laplace transform and partial fraction expansion Nature of Laplace transform
Class 13 Inverse Laplace transform and transient response of a linear circuit Inverse Laplace transform and application to differential equation. Linear circuit transient response.
Class 14 Stability of a system and applications of Laplace transform in electrical engineering. z transform. Stability of a system assessed by Laplace transform.
Class 15 Exam for the 2nd part of the lecture Exam for the 2nd part of the lecture

### Textbook(s)

Mizumoto, Tetsuya. Mathematics for Electronics and Informatics. Tokyo: Baifukan; ISBN-13: 978-4563069957.(Japanese)

### 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 toproblems will be assessed.
Midterm and final exams 80%, exercise problems 20%.

### 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. 