Course syllabus
Course PM
This page contains the program of the course: lectures, exercise sessions and computer labs. Other information, such as learning outcomes, teachers, literature and examination, are in a separate course PM.
News:
The Class meets13-15 Tuesday, Jan 07, and Friday , Jan 10, 10-12 both in Pascal. We go through previous exams.
There will an exercise section, Wednesday, Dec 11, at MVH12, 13:15-15. Welcome!
The class will not meat on Tuesday December 3. I shall compensate this in coming weeks.
Course requirements
A good knowledge on Fourier Analysis, distribution theory and pdf is helpful but not required.
Computer assignments require some programming skills.
Course Literature
I. Lecture Notes (primary)
II. External course material
-Begh, Notes on Generalized Functions and Fourier Transforms, Chalmers and Göteborg University: comp.pdf
-Problem from the book Wavelets by Bergh, Ekstedt, Lindberg, Studentlitteratur. The Book is not available to purchase but can be downloaded from the authors website: Bergh et al, Wavelets.pdf
-Bracewell, The Fourier Transform and its Applications. Selected parts of Chapters 1-14 are included in the course. Notes will be available and the book is not essential for the course.
Program
Office hours: Jimmy Aronsson Thursdays: 13:15-14, Room 3065.
The schedule of the course is in TimeEdit.
Lectures
| Day | Sections | Content |
|---|---|---|
| 04 Nov | Fourier transforms | Basic properties, The Fourier Inversion formula, Fixed point. |
| 05 Nov | Fundamentals | Henkel-, Radon-, discrete- transforms, Hausdorff-Young, convolution, |
| 08 Nov | The Classes S, S' | Properties of S, Fourier Inversion formula, generalised functions, tempered distribution |
| 11 Nov | F-transform of distributions | F-inversion for tempered distributions, canonical examples of generalised functions. Convergence properties. |
| 12 Nov | Discrete F-trans. for distributions | Discrete fixed point for F-transform, periodic functions |
| 15 Nov | Poisson sum, DFT, FFT | Poisson summation formula, convergence of F-transforms, |
| 18 Nov | Computational rules | Interpolation, cyclic convolution,. discrete Parseval. |
| 19 Nov | Similarity, decimations | Similarity theorem, decimation rules (down-, up-sampling), stretch and repeat. |
| 22 Nov | DFT, FFT | DFT in matrix form. Diagram for fast Fourier transform, time-decimation, decimation in frequency. |
| 25 Nov | Fundamental relations | Auto-correlation, uncertainty, Heisenberg. |
| 26 Nov | Filters, Signals | Linearity,. time-invariance, impulse response, LTI-Filter. |
| 29 Nov | Infinite sequences | Convolution, Frequency response, all pass Filters, Filter banks |
| 02 Dec | Complete filters | Perfect res resolution, product filter, orthogonal filter, bi-orthogonal filter. |
| 03 Dec | Multi-resolution Analysis. | Hilbert space, closed subspaces, Riesz basis, bi-orthogonal basis. |
| 06 Dec | Wavelets | Haar sampling, Scaling, Wavelet and detail spaces, MRA and wavelet decomposition, Orthogonal system. |
| 09 Dec | Wavelets | ON-condition, conditions on scaling and wavelet functions, The continuous wavelet transform. |
| 10 Dec | Wavelets | doubly indexed wavelets, Frame, tight frame, Bi-orthogonal system, |
| 13 Dec | Wavelets | discrete wavelets, Approximation, 2D signal processing, shift wavelets. |
| 20 Dec | Wavelets |
2D Separable scaling functions, Multidimensional F-transform, Distribution in R^n, F-transform of tensor product. |
| 07 Jan | Wavelets | Henkel transform, homogenisation funktions and distributions, and their F-transform, Abel transform, Radon transform, Hilbert transform. |
| 10 Jan | Wavelets | Analytical signal, Fourier analysis and probability, the central limit theorem, there law of large number, Bochner's theorem. |
Recommended exercises
| Day | Exercises |
|---|---|
| 15 Nov | Exercises_1.pdf Bracewell: 2.10, 2.13, 2.15, 2.17, (3.10), 3.123, 3.20, 3.22 |
| 22 Nov | Exercises_1-2.pdf Bracewell: 5.29, 6.1, 6.3, 6.5, 6.9, 6.15 |
| 30 Nov | Exercises_1-1.pdf Bracewell: 6.32, 8.1, 8.3, 8.7, 8.11, 8.23, 8.29 |
| 06 Dec | Bergh etal: 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 3.1, 3.2, 3.4, 3.6, 3.7 |
| 13 Dec | Bergh etal: 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 4.10, 4.11, 4.12 |
| 20 Dec | Exercises from old Exams (2012, 2014, 2016) |
| 10 Jan | Exercises from old Exams (2012, 2014, 2016) |
Computer labs
There will be 3 computer assignments on FFT, Introduction to Wavelets and Image compression.
You may warm up and then work through the assignments.
- Lab 1: lab1.pdf Try filtering different signals, e.g., linear combination of few a sines and cosines. See FFT manual.
Carification for Lab1, by Jimmy that would be very much helpful to work: Lab1_clarifications-1.pdf
For the high- and low-pass filters you may use H and G of the of the first example on Lecture Notes, Part34page 43. Here is an example of how to work with Assignment 1: assignment1_supplement.html
-Lab 2: lab2.pdf Introduction to wavelets
To carry out this lab you will need to download the file: signals.mat
-Lab 3: lab3.pdf Imager Compressions;
Image compression manual: TestbildTV12gray16-1.tif
Additional lab 3 files: BangaloreGray16.tif, TestbildTV12-1024gray16.tif, compress.m, dctcompress.m, wcshow.m
Notes: You need to hand in solutions for assignments 1 and 3. You do not need to hand in solution of assignment 2 (however, to prepare for assignment 3, it is useful that you are familiar with assignment 2).
You may work on a group of 2 (or individually), but the hand ins should be composed for the group or individual. (However, working with the assignments, cooperation among the groups is encouraged.) The hand ins should be delivered in pdf format sent to the lab-assistant not later than the study weak 4 for assign 1 and before the date for the written exam for assign 3. The reports (pdf files) should be sufficiently complete to be read independently.
Reference literature:
- Learning MATLAB, Tobin A. Driscoll. Provides a brief introduction to Matlab to the one who already knows computer programming. Available as e-book from Chalmers library.
-
Physical Modeling in MATLAB 3/E, Allen B. Downey
The book is free to download from the web. The book gives an introduction for those who have not programmed before. It covers basic MATLAB programming with a focus on modeling and simulation of physical systems.
Some Previous Exams with solutions:
Course summary:
| Date | Details | Due |
|---|---|---|