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.
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
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
|✔ ・Fundamental specialist skills on EEE|
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||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.|
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.
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 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%).
Students must have successfully completed both Calculus I (LAS.M101) and Calculus II (LAS.M105) or have equivalentknowledge.