Syllabus:

# Transform Methods, 7.5 Credits

Swedish name: Transformmetoder

This syllabus is valid: 2017-08-21 and until further notice

Course code: 5MA181

Credit points: 7.5

Education level: Second cycle

Main Field of Study and progress level: Mathematics: Second cycle, has only first-cycle course/s as entry requirements

Responsible department: Department of Mathematics and Mathematical Statistics

Established by: Faculty Board of Science and Technology, 2017-10-01

## Contents

Element 1 (6.5 hp): Theory
In the course, continuous, discrete and stochastic signals, sampling and reconstruction, and discrete linear time-invariant systems (LTI systems) and their connection to convolution, are treated. Further, LTI systems given by difference equations, Fourier series and Fourier transform, discrete Fourier transform, z transform, frequency analysis of signals and discrete LTI systems, and digital filters, are included. Finally, multiresolution analysis and wavelets, and calculation of wavelet coefficients using filter banks are covered.

Element 2 (1 hp) Computer labs
Frequency analysis, effects of sampling, noise reduction and image compression using wavelets.

## Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding
• independently account for basic notions in the theory of continuous and discrete signals, and The Sampling Theorem
• thoroughly account for the basic theory of discrete linear time-invariant systems (LTI systems)
• thoroughly describe the basic properties of Fourier series and Fourier transforms of a continuous signal, and the basic properties of Fourier series, fourier transforms and z transforms of a discrete signal
• define wavelets starting with a multiresolution analysis, and explain how wavelet coefficients are calculated using filter banks
Skills
• decide systen answers for discrete LTI systems given by a differens equation
• apply transforms for frequency analysis

## Required Knowledge

The course requires 90 ECTS including 22,5 ECTS in Calculus and a course in Differential Equations.
Proficiency in English equivalent to Swedish upper secondary course English 5/A. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.

## Form of instruction

The teaching in Element 1 takes the form of lectures and lessons. The teaching in Element 2 takes the form of supervised lab work.

## Examination modes

Element 1 is assessed through a written examination. Element 2 is assessed through written lab reports. For Element 1, one of the following judgements is awarded: Fail (U), Pass (3), Pass with merit (4), Pass with distinction (5). For Element 2, one of the following judgements is awarded: Fail (U), or Pass (G). For the course as whole, one of the following grades is awarded: Fail (U), Pass (3), Pass with merit (4), Pass with distinction (5). The grade for the whole course is determined by the judgement given for Element 1. To pass the whole course, all elements must have been passed. The grade is only set once all compulsory elements have been assessed.

A student who has been awarded a passing grade for the course cannot be reassessed for a higher grade. Students who do not pass a test or examination on the original date are given another date to retake the examination. A student who has sat two examinations for a course or a part of a course, without passing either examination, has the right to have another examiner appointed, provided there are no specific reasons for not doing so (Chapter 6, Section 22, HEO). The request for a new examiner is made to the Head of the Department of Mathematics and Mathematical Statistics. Examinations based on this course syllabus are guaranteed to be offered for two years after the date of the student's first registration for the course.

Credit transfer
All students have the right to have their previous education or equivalent, and  their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be adressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications

## Other regulations

This course can not be included in a degree together with another course with similar contents. When in doubt, the student should consult the director of study at the department of mathematics and mathematical statistics. The course can also be included in the subject area of computational science and engineering.