Course title
L00155002
Engineering Mathematics

 SASANO Isao
Middle-level Diploma Policy (mDP)
Program / Major mDP Goals
Computer Science and Engineering B-1 コンピュータサイエンスの数理的基礎と問題分析のスキルを身に付けることができる。
Purpose of class
By mastering the least-squares method, orthogonal function expansion, and Fourier series expansion, along with an introduction to Fourier transforms and the Discrete Fourier Transform (DFT), students will acquire a fundamental framework for digital signal processing. This provides the essential mathematical basis for analyzing and processing real-world data, such as audio and visual signals.
Course description
The Discrete Fourier Transform (DFT) is a fundamental tool for processing audio and visual data in digital computing. This course aims to provide a comprehensive understanding of Fourier series expansion, which serves as the theoretical foundation for the DFT. As an introduction, we will explore the least-squares method and orthogonal function expansion, positioning the Fourier series as a specific application of these broader concepts. Building on this, the lecture will also briefly cover the Fourier transform and the DFT, illustrating how Fourier series expansion transitions into these essential techniques for modern signal processing.
Goals and objectives
  1. Understanding the least-square method and being able to approximate given sequences of data or functions by linear functions or quadratic functions
  2. Understanding orthogonal functions and being able to do the orthogonal function expansion for given functions by some given set of orthogonal functions
  3. Understanding Gram-Schmidt orthogonalisation, which is a method (algorithm) for orthogonalising a set of vectors in an inner product space, and being able to construct an orthogonal set of functions from a given set of functions.
  4. Being able to do Fourier series expansion, which is an important instance of the orthogonal function expansion.
  5. Being able to do Fourier transform and discrete Fourier transform for simple examples.
Relationship between 'Goals and Objectives' and 'Course Outcomes'

 reports mid-term exam final exam Total.
1. 5% 30% 20% 55%
2. 0% 10% 5% 15%
3. 0% 0% 5% 5%
4. 5% 0% 15% 20%
5. 0% 0% 5% 5%
Total. 10% 40% 50% -
Evaluation method and criteria
Your final grade will be calculated based on a mid-term exam (40 points), a final exam (50 points), and reports (10 points). Let M be the mid-term score, F the final exam score, and R the report score. The overall score (S) is determined by the following formula: S = R+M+F*(100-(R+M))/50. A passing grade (60 points) reflects a fundamental mastery of the course material. Students who can successfully solve basic problems regarding the least-squares method, Fourier series expansion, and Gram-Schmidt orthogonalization will achieve this proficiency level.
Language
English
Class schedule

 Class schedule HW assignments (Including preparation and review of the class.) Amount of Time Required
1. Introduction and the least-square method (1)
- Approximation of sequences of data in linear functions 		Read Section 20.5 of the reference book. 180minutes
2. The least-square method (2)
- Approximation of sequences of data in quadratic functions 		See Example 2 in Section 20.5 of the reference book 190minutes
3. The least-square method (3)
- Approximation of sequences of data in linear combination of some fixed set of functions 		It is not treated in the reference book. 190minutes
4. The least-square method (4)
- Approximation of functions in linear combination of some fixed set of functions 		See Problem 14 in Section 20.5 of the reference book. 190minutes
5. The least-square method (5) and the orthogonal function expansion (1)
- Approximation of column vectors
- Approximation of functions in linear combination of some fixed set of orthogonal functions
- An orthogonal set of functions --- Legendre polynomials 		See Problem 14 (c) for approximation of functions in linear combination of some fixed set of orthogonal functions.
See Section 5.2 of the reference book for Legendre polynomials.
See Section 4.0 of the reference book for column vectors. 		190minutes
6. The orthogonal function expansion (2)
- An orthogonal set of functions --- Trigonometric functions
- The orthogonal function expansion 		Read Section 11.1 and Section A3.1 of the reference book for trigonometric functions. 190minutes
7. Mid-term examination and explanation of the answers
- Pencil-and-paper test for checking the understanding of the contents of the lectures from the first to the eighth
(We resume the lecture after the mid-term examination.) 		Review the contents of all the lectures until the sixth one.
See Problem 14 in Section 11.5 of the reference book for Chebyshev polynomials. 		190minutes
8. The orthogonal function expansion (3)
- An example of the orthogonal function expansion --- Fourier series expansion
- Orthogonal set of functions with a weight function
- An example --- Chebyshev polynomials 		Read Section 11.1 of the reference book for Fourier series expansion. 190minutes
9. The orghogonal function expansion (4)
- Examples --- Hermite polynomials and Laguerre polynomials 		See Problem 14 in Section 11.6 of the reference book for Hermite polynomials.
See Example 2 in Section 5.2 of the reference book for Legendre polynomials. 		190minutes
10. The orthogonal function expansion (5)
- The orthogonal function expansion in Chebyshev, Hermite, and Laguerre polynomials
- Inner product spaces
- An inner product space --- n-dimensional Euclidean space 		Read Section 11.6 of the reference book for the orthogonal function expansion.
Read Section 7.9 for the inner product spaces.
See Example 3 in Section 7.9 for the n-dimensional Euclidean space. 		190minutes
11. The orthogonal function expansion (6)
- Cauchy-Schwarz inequality
- Triangle inequality
- Orthogonal basis
- Orthonormal basis
- Orthogonal projection 		See Problem 24 in Section 20.4 of the reference book for Cauchy-Schwarz inequality, Section 13.2 of the reference book for the triangle inequality, Section 8.3 of the reference book for the definition of orthonormality, and Section 9.2 for orthogonal projections. 190minutes
12. The orthogonal function expansion (7)
- Gram-Schmidt orthogonalization
- Obtaining Legendre polynomials by Gram-Schmidt orthogonalisation 		Gram-Schmidt orthogonalization is not treated in the reference book. Consult some linear algebra textbook. 190minutes
13. Fourier transform and discrete Fourier transform See Section 11.9 of the reference book for Fourier transform and discrete Fourier transform 190minutes
14. Final examination and explanation of the answers
- Paper-and-pencil test for checking the understanding of the contents of the lectures from the first to the thirteenth 		Review the contents of all lectures 190minutes
Total. - - 2650minutes
Feedback on exams, assignments, etc.
ways of feedback specific contents about "Other"
Feedback in the class
Textbooks and reference materials
A reference book is:
Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley ＆ Sons Inc; 10th International edition, 2011.
Note that I do not use this book as a textbook. Note also that this book is thick and covers topics much more than ones this lecture covers.
Prerequisites
Basic knowledge of linear algebra and analysis
Office hours and How to contact professors for questions
  • Tuesday 15:00-15:10 or any time agreed on by email via zoom
Regionally-oriented
Non-regionally-oriented course
Development of social and professional independence
  • Course that cultivates a basic problem-solving skills
Active-learning course
More than one class is interactive
Course by professor with work experience
Work experience Work experience and relevance to the course content if applicable
N/A N/A
Education related SDGs:the Sustainable Development Goals
    Last modified : Sat Mar 14 14:27:08 JST 2026