# MAT-02457 Fourier Methods, 5 cr

#### Additional information

The course will be lectured in English every second year.

The implementation will not be executed during the academic year 2019-2020.

#### Person responsible

Antti Kuusisto

#### Lessons

Implementation | Period | Person responsible | Requirements |

MAT-02457 2019-01 | - |
Antti Kuusisto |
Exam and weekly course exercises |

#### Learning Outcomes

After completing the course, a student knows the definition and basic properties of Fourier series and Fourier transform. The student can represent a periodic function as a Fourier series (with real or complex coefficients) and can form new ones from known series. In addition, the student can calculate the Fourier transform of non-periodic functions for some simple example cases by using the definition and the basic properties. The student knows what is Gibbs phenomenon and can predict when it occurs in some example cases. The student knows the Dirac delta function and how to use it. A student that masters the course contents have prerequisite for understanding the practical value of the above concepts. The Fourier series of a periodic function or the Fourier transform of a non-periodic function decomposes the function into its frequency components. In practice however, the function of interest may not be available, but one only can calculate some samples of it. Then one should be able estimate the frequency decomposition by using the samples. This is done by using the discrete Fourier transform. The discovery of fast Fourier transform in 1960's was groundbreaking for signal processing and is cited as one of the most important algoritms in mathematics. The importance of the frequency decomposition stems from the ability to filter unwanted frequencies.

#### Content

Content |
Core content |
Complementary knowledge |
Specialist knowledge |

1. |
Real Fourier series for periodic function and determining its coefficients, even and odd functions as special cases, Gibbs phenomenon. | Dirichlet conditions for fourier series convergence, expanding a function defined on a bounded interval to a periodic function. | |

2. |
Complex Fourier series, Parseval's theroem, series as frequency decomposition. | ||

3. |
Discrete Fourier transform and its properties. | Significance of the fast Fourier transform | |

4. |
Fourier transform of non-periodic functions: definition and basic properties, Fourier transform as frequency decomposition. | Dirac delta function, Convolution, Parseval's theorem |

#### Instructions for students on how to achieve the learning outcomes

For grade 5, the student masters the core course content as well as completementary knowledge. For 3-4, the student is understands the core content and has a working knowledge of the complementary knowledge issues. For 1-2, the student has a working knowledge of the core content.

#### Assessment scale:

Numerical evaluation scale (0-5)

#### Prerequisites

Course | Mandatory/Advisable | Description |

MAT-01166 Mathematics 1 | Mandatory | |

MAT-01366 Mathematics 3 | Mandatory |

**Additional information about prerequisites**

or courses of Engineering Mathematics / Mathematics in Finnish.

#### Correspondence of content

Course | Corresponds course | Description |

MAT-02457 Fourier Methods, 5 cr | MAT-02456 Fourier Methods, 4 cr |