Transform Methods, 7.5 Credits

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 explain basic concepts in the theory of continuous and discrete signals and explain the sampling theorem and its consequences,
• thoroughly explain the basic theory of discrete linear time-invariant systems (LTI-systems),
• describe in detail the basic properties of the Fourier transform of a continuous signal and the basic properties of the Fourier transform and z-transform of a discrete signal,
• define the wavelets based on a multi-resolution analysis and explain how wavelet coefficients  are calculated using filter banks.

Skills

•  apply transform methods to analyze signals and systems,
• use transformers for frequency analysis.

Required Knowledge

The course requires 90 ECTS including 22,5 ECTS in Calculus, a basic course in mathematical statistics minimum 6 ECTS and a course in Differential Equations. Proficiency in English equivalent to the level required for basic eligibility for higher studies. 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.