VAMK

Change language: Suomi

Front Page > Archived Education > Electrical Engineering (I-ST) > 2007 > Year 3 > Integral Transforms (IXS9107)

Integral Transforms

Structure Type: Study unit
Code: IXS9107
Type: Optional obligatory / Professional Studies
Curriculum: I-ST 2007
Level: Bachelor of Engineering
Year of Study: 3 (2009-2010)
Credits: 4 cr
Responsible Teacher: Mäkelä, Jarmo
Language of Instruction: Finnish

Learning Outcomes

The solution of various mathematical problems becomes considerably easier, if the functions under study are replaced by their integral transforms. When a function is replaced by its integral transform, the function transforms, by means of integration, to a new function of a new variable. The most imnportant integral transforms are the Fourier transform, which is applied, in particular, in the analysis of vtyhe oscillatory phenomena, and the Laplace transform, which is used in the solution of differential equations. Closely related to the differential equations are the so-called difference equations, which have sequences as their solutions. The difference equations may be solved by means of the so-called z-transforms. In this course, which is heavily based on the complex analysis learned in the Advanced Analysis course, the student learns the basics of all these transforms and their application.

Student's Workload

81 h, which contains 42 h of contact studies at VAMK and 24 h at University of Vaasa.

Prerequisites / Recommended Optional Courses

Integral Calculus, Differential Equations and Series, Advanced Analysis.

Contents

1. A brief summary of complex analysis and the thorem of residues from the Advanced Analysis course.
2. Fourier series; the Dirichlet theorem.
3. Complex Fourier series.
4. A analysis of standing waves by means of the Fourier series.
5. Evaluation of selected series by means of the Fourier series.
6. Solution of differential equations by means of the Fourier series.
7. Fourier transforms and inverse transforms.
8. Examples of the determination of the Fourier transforms and inverse transforms by means of the therem of residues.
9. Solution of differential equations by means of the Fourier transforms.
10. Laplace transform.
11. Inverse Laplace transforms.
12. Bromwich integral. (Or Mellin’s inverse formula)
13. Determination of inverse Laplace transforms by means of residues.
14. Solution of linear differential equations by means of the Laplace transforms.
15. Solution of systems of differential equations using Laplace transforms.
16. The convolution theorem.
17. Shift of the origin in the Laplace transforms and inverse transforms.
18. Causal sequence.
19. z-transform of a causal sequence.
20. Inverse z-transform.
21. Determination of inverse z-transforms by means of residues.
22. Difference equations.
23. Solution of difference equations by means of z-trasforms.
24. Elements of Mellin transforms.

Recommended or Required Reading and Other Learning Resources/Tools

Kreyszig, E: Advanced Engineering Mathematics (John Wiley & Sons). Material prepared by the lecturer.

Mode of Delivery / Planned Learning Activities and Teaching Methods

Theory, examples and exercises during the lectures, homework exercises.

Assessment Criteria

Grade 5: The student is able to creatively solve problems related with almost all the contents of the course.
Grade 3: The student is able to solve problems related with the central contents of the course.
Grade 1: The student is able to solve basic problems related with the central contents of the course.

Assessment Methods

Homework exercises, practical works, an examination.


Back