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 topics
covered in lectures of signal processing.

By learning the least-square method, the orthogonal function expansion, and Fourier series expansion, we acquire the basics
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.

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 |
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 |
Confer Problem 16 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 |
Read Section 5.7 and 5.8 of the reference book. Confer Example 2 in Section 5.8 for Legendre polynomials. Confer Section 7.1 for column vectors. |
190minutes |

6. | The orthogonal function expansion (2) - An orthogonal set of functions --- Trigonometric functions - The orthogonal function expansion |
Read Section 5.8 of the reference book. Confer Appendix 3 for formulae about 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 last one. Confer Problem 20 in Section 5.7 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 for Fourier series expansion. | 190minutes |

9. | The orghogonal function expansion (4) - Examples --- Hermite polynomials and Laguerre polynomials |
Confer Problem 18 in Section 5.8 for Hermite polynomials. Confer Example 2 in Section 5.8 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 5.7 and 5.8 of the reference book for the orthogonal function expansion. Read Section 7.9 for the inner product spaces. Confer Example 3 in Section 7.9 for the n-dimensional Euclidean space. |
190minutes |

11. | The orthogonal function expansion (6) - Cauchy-Schwarz inequality - Triangle inequality - Orthonormal basis |
Read Section 7.9 for Cauchy-Schwarz inequality and Triangle inequality. Read Section 5.7 for the definition of orthonormality Confer Section 7.4 for basis. |
190minutes |

12. | The orthogonal function expansion (7) - Orthogonal projection - Orthogonal basis - Gram-Schmidt orthogonalisation |
Read Section 9.2 for projections. Gram-Schmidt orthogonalisation is not treated in the reference book. Consult some linear algebra textbook. |
190minutes |

13. | The orthogonal function expansion (8) - Obtaining Legendre polynomials by Gram-Schmidt orthogonalisation |
Gram-Schmidt orthogonalisation is not treated in the referece book. Consult some linear algebra textbook. | 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 fourteenth |
Review the contents of all lectures | 190minutes |

Total. | - | - | 2650minutes |

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% | 20% | 25% |

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 and Fourier series expansion, you are in the level of obtaining 60-point.

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

A reference book is:

Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons Inc; 9th International edition, 2006.

The tenth edition was published in 2011 and it may be equally fine.

Note that I do not use this book as a textbook. Note also that this book is thick and covers topics much more than this lecture covers.

Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons Inc; 9th International edition, 2006.

The tenth edition was published in 2011 and it may be equally fine.

Note that I do not use this book as a textbook. Note also that this book is thick and covers topics much more than 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 : Tue Apr 26 04:03:46 JST 2022