Discrete Fourier transform (DFT) is used for processing sounds and graphics in digital computers. This lecture aims at being
able to do Fourier series expansion, which forms the basis for DFT. As an introduction to Fourier series expansion we illustrate
the least-square method and the orthogonal function expansion. Fourier series expansion is an instance of the orthogonal function
expansion. Understanding Fourier series expansion forms the basis for understanding Fourier transform and DFT, which are also
explained.

By learning the least-square method, the orthogonal function expansion, Fourier series expansion, Fourier transform, and discrete
Fourier transform, we acquire the basis for processing signals like sounds and images.

- Understanding the least-square method and being able to approximate given sequences of data or functions by linear functions or quadratic functions
- Understanding orthogonal functions and being able to do the orthogonal function expansion for given functions by some given set of orthogonal functions
- 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.
- Being able to do Fourier series expansion, which is an important instance of the orthogonal function expansion.
- Being able to do Fourier transform and discrete Fourier transform for simple examples.

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% | - |

Mid-term exam is evaluated on a 40-point scale, final exam a 50-point, and reports a 10-point. When the mid-term exam is M
point, the final exam F point, and the repots R point, the overall score is R+M+F*(100-(R+M))/50.

If you can solve basic problems about least square method, Fourier series expansion, and Gram-Schmidt orthogonalization, you are in the level of obtaining 60-point.

If you can solve basic problems about least square method, Fourier series expansion, and Gram-Schmidt orthogonalization, you are in the level of obtaining 60-point.

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 |

15. | |||

Total. | - | - | 2650minutes |

ways of feedback | specific contents about "Other" |
---|---|

Feedback in the class |

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.

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.

- Tuesday 15:00-15:10 or any time agreed on by email via zoom

- Course that cultivates a basic problem-solving skills

Work experience | Work experience and relevance to the course content if applicable |
---|---|

N/A | N/A |

Last modified : Sat Jun 29 04:20:09 JST 2024