In the broad range of the information and communications engineering, signal analysis has been a powerful mathematical tool box for understanding the behaviors of signals and systems through alternative expressions, , as the sum / integral of well-understood elementary functions, e.g, monomial functions and trigonometric functions. Starting from the elementary complex analysis as a prerequisite of the signal analysis, this lecture surveys its central ideas found, e.g., in Taylor series expansion, Laurent series expansion, Fourier series expansion, Fourier transform, Sampling theorems, Discrete time Fourier transform and Laplace transform.
Through the lectures and seminars, the students will be able to:
1) understand mathematical treatments of complex functions.
2) understand the mathematical ideas and calculation methods of the major signal analyses, e.g., Fourier series, Fourier transform, Laplace transform.
3) understand the technical value of the frequency analysis and apply to the broad range of Information and Communications engineering.
Linear systems, Eigen functions, Complex Analysis, Euler's formula, Complex derivative, Cauchy-Riemann equation, Holomorphic function, Complex integral, Cauchy's integral theorem, Cauchy's integral formula, Cauchy's residue theorem, Laurent series expansion, Taylor series expansion, Fourier series expansion, Complex Fourier series expansion, Fourier Integral, Fourier transform, Sampling theorem, Discrete time Fourier transform, Laplace transform, Differential equation, Linear time invariant systems and frequency responses.
✔ Specialist skills | Intercultural skills | Communication skills | ✔ Critical thinking skills | ✔ Practical and/or problem-solving skills |
Two lectures and one seminar are given in every week.
Course schedule | Required learning | |
---|---|---|
Class 1 | Invitation to Signal and System Analysis: Eigen functions of Linear Systems | Explain about eigen functions of linear systems. |
Class 2 | Complex number system: four arithmetic operations, Euler's formula | Explain how real field can be extended to complex field. Explain about Euler's formula |
Class 3 | Complex functions: Continuity, Differentiability, Cauchy-Riemann's equation, Holomorphic function | Explain about the relation between differentiability of two variable real function and complex function. |
Class 4 | Holomorphic function and Complex Integral: Cauchy's integral theorem, Cauchy's integral formula | Explain Cauchy's integral theorem and Cauchy's integral formula. |
Class 5 | Taylor series expansion and Laurent series expansion | Explain about the relation between Taylor series expansion and Laurent series expansion. |
Class 6 | Cauchy's residue theorem and its application to definite integral | Explain about Cauchy's residue theorem and how this theorem can be applied to application to definite integral. |
Class 7 | Supplementary Lecture on Complex Analysis | Explain about the elements of the complex analysis. |
Class 8 | Fourier Analysis and Signal Processing | Explain about Fourier Series and Fourier Transform from Signal Processing view point. Explain about the relation between Sampling theorem and Discrete time Fourier transform. |
Class 9 | Fourier series 1: Trigonometric polynomial approximation, Fourier series expansion, Convergence theorems | Explain about (i) the relation between the trigonometric polynomial approximation and Fourier series, and (ii) convergence theorem of Fourier series expansion. |
Class 10 | Fourier series 2: Gibbs phenomenon, Parseval's equation，Orthogonal function expansion | Explain about Gibbs' phenomenon and Parseval's equation. |
Class 11 | Fourier integral: Derivation of Fourier and Inverse Fourier transforms, Parseval's equation, Convolution theorem | Explain about the relation between Fourier transform and Fourier series expansion. |
Class 12 | Laplace transform 1: Laplace transform and its inversion, Basic properties and calculation formulae | Explain about the relation between Laplace transform and Fourier transform. |
Class 13 | Laplace transform 2: Applications to differential and integral equations | Explain how to apply Laplace transform to differential equation and integral equation. |
Class 14 | Laplace transform 3: Applications to transient analysis of linear systems | Apply Laplace transform to transient analysis of linear systems. |
To enhance effective learning, students are encouraged to spend a certain length of time outside of class on preparation and review (including for assignments), as specified by the Tokyo Institute of Technology Rules on Undergraduate Learning (東京工業大学学修規程) and the Tokyo Institute of Technology Rules on Graduate Learning (東京工業大学大学院学修規程), for each class.
They should do so by referring to textbooks and other course material.
M. Sakawa, Ouyou kaisekigaku no kiso, Morikita Publishing, 2014 (in Japanese).
Lecture materials will be given if necessary.
Learning achievement is evaluated based on a terminal exam(9) and quizzes(1). The numbers indicate approximate weights for the evaluation.
As a general rule, we accept only applications from students in the department of Information and communications Engineering