2021 Theory of Linear System

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Academic unit or major
Undergraduate major in Transdisciplinary Science and Engineering
Ohashi Takumi 
Class Format
Lecture / Exercise     
Media-enhanced courses
Day/Period(Room No.)
Mon3-4(S513)  Thr3-4(S513)  
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
Language used
Access Index

Course description and aims

The purpose of this course is to learn how to handle complex numbers and their functions, the concept of frequency, and the theory of linear systems necessary for analyzing these systems, which are important in the study of engineering.

Student learning outcomes

To learn the basics of linear algebra, the function of complex numbers, Fourier transform, Laplace transform, z-transform, theory to model systems and to understand linear circuits and the basis of control theory.


Determinant, eigenvalue, eigenvector, function of complex numbers, Cauchy-Riemann equations, Taylor series, Laurant series, pole, Fourier expansion, Fourier transform, Laplace transform, discrete time Fourier transform, discrete Fourier transform, z-transform, continuous time system, discrete time system, controllability, observability, and stability.

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Lecture and Practice

Course schedule/Required learning

  Course schedule Required learning
Class 1 Review and definition of basic operations Understand the notations and basic terminologies used such as the fundamental operations for vectors and matrices.
Class 2 Matrix operations and Cramer's Rule Review matrix operations such as determinants, permutations, inversions, transpositions. Be able to solve for minor of matrices, and cofactors.
Class 3 Eigen equations, matrix of functions and their operations and characteristics. Understand the use of Eigen equations, eigenvectors, and eigenvalues. Refresh on additional matrix operations and types, and matrix of functions.
Class 4 Complex sine waves and Bode plots Understand complex sine waves and its application to RL circuits. Bode plots will be introduced as a tool to examin frequency reponses of magnitude and phase of a system.
Class 5 Fourier series Understand the definition, properties, and importance of Fourier Series. Various forms of the Fouriers Series such as complex Fourier series and its truncated form will be taught to solve various functions (e.g. step function).
Class 6 Fourier Transform and Impulse Response Understand the Fourier transform, inverse Fourier transform, and impulse response.
Class 7 Laplace (Inverse) Transform, Initial and Final Value Theorem Understand and be able to solve the Laplace transform, inverse Laplace transform, and its difference with the Fourier transform. Partial fraction decomposition to solve the inverse Laplace Transform will be introduced.
Class 8 Complex integrals and the Cauchy's Integral Theorem Understand the concept of complex integrals as a necessary tool to solving the inverse Laplace transform using the Residual Theorem. Understan the Cauchy's Integral Theorem.
Class 9 Laurent's Series, residual theorem, and state-space representation Understand the Laurent's Series expansion and be able to solve the inverse Laplace transform using the Residual Theorem. Understand State-space analyses and apply to a mechanical system.
Class 10 Block diagrams, transfer functions, and state-space solutions Learn how to construct and interpret a block diagram, solve state-space equations, and understand the concept of transfer functions.
Class 11 Control Theory Understand the basic concepts of classical and modern control theory.
Class 12 Minimal Realization, Similarity Transformations, and Stability Understand the concept of minimal realization, similarity transformations, and stability as tools to assess linear systems.
Class 13 Sampling theorem and discrete forms of Fourier and Laplace transforms Understand the concept of sampling theorem and the discrete forms of Fourier transform and Laplace transforms (Z-transform). Learn how to solve for the discrete (inverse) Fourier transform and (inverse) Z-transform.
Class 14 Sampling theorem and discrete forms of Fourier and Laplace transforms Understand the concept behind discrete state-space equations, its block diagram, and bilinear transformation.

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.


Hwei P. Hsu, "Signals and Systems"

Reference books, course materials, etc.

None in particular

Assessment criteria and methods

Evaluated based on the final examination and exercises.
(It might be changed depending on the situation of COVID-19.)

Related courses

  • TSE.M201 : Ordinary Differential Equations and Physical Phenomena

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

None in particular


In the current syllabus, the contents of the English version do not necessarily coincide with the Japanese version.
The syllabus will be changed as needed.

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