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)

LectureNotes_Part1.pdf

LectureNotes_Part2.pdf

LectureNotes_Part3A.pdf

LectureNotes_Part4.pdf

LectureNotes_Part5.pdf

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. 

packing_Bracewell.pdf

PoissonSummationFormula-1.pdf

 

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.

 

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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)

 

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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:

1. 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.
2. 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: 

tenta+sol_170109.pdf

tenta+sol_150117.pdf

tenta_130820.pdf

tenta_121220.pdf

tenta_110428.pdf

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