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.
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.
Corresponding educational goals are:
(1) Specialist skills Fundamental specialist skills
(6) Firm fundamental specialist skills on electrical and electronic engineering, including areas such as electromagnetism, circuits, linear systems, and applied mathematics
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
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
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|
|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|
Mizumoto, Tetsuya. Mathematics for Electronics and Informatics. Tokyo: Baifukan; ISBN-13: 978-4563069957.(Japanese)
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)
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%.
Students must have successfully completed both Calculus I (LAS.M101) and Calculus II (LAS.M105) or have equivalentknowledge.