Wavelets and signal processing 2017/2018

Ondículas y tratamiento de señales (máster)



*More challenges posted in Exercises.
*Problem set 3 is posted in Exercises. It is due by 18/May/2018.



Course details

The contents are included in the guía docente.

The lectures are scheduled on Monday and Wednesday 5:30pm - 19:00pm in room 320 (Building 17, Science Faculty, Mathematics).

For general reference, please check the master program website in the Department of Mathematics.




I shall post periodically notes on the topics of the course. If you print them, please, note the possible page overlapping between the end of a section and the beginning of the next one.

File Version
Page title 25/Jan/2018
About this course
Simple waves
1.1 Physical principles
Harmonic oscillators
Electromagnetic waves and simple circuits
Sound waves
1.2 Mathematical methods and results
Basic Fourier series and integrals
Some properties
Gibbs phenomenon
More flavors of harmonic analysis
Introduction to digital signals
2.1 Sampling and A/D, D/A conversion
Shannon sampling theorem
Basic quantization
Data approximation
2.2 Discrete Fourier analysis
Some discrete transforms
An example: JPEG
Uncertainty revisited [not covered this course]

2.3 Digital filters
Convolutions from analog to digital [not covered this course]
Basics on filter design [not covered this course]
Simple filters in image processing [not covered this course]
Wavelets: theory and practice
3.1 Multiresolution analysis
The limits of Fourier analysis
The theoretical framework
Construction of wavelets
3.2 Wavelets in practice
The D4 transform (see pp.44-47 of this)

Mallat's algorithm

Some examples and applications

Some computational aspects
4.1 The Fast Fourier Transform
The basic algorithm

Some variants and applications

4.2 Coding and compression
Entropy and Shannon theorems

Huffman coding

Dictionary methods

The bibliography will be updated from time to time. Some of the topics in the previous list might not be covered in the course if time does not permit. I intend to write the corresponding notes anyway.

Corrections and comments are welcome.

- p.2, l.-9. y_p=Ge^{i\omega_e t}
- p.11, l.3. N\to\infty
- p.7, (1.23). Missing L^{-1} in the RHS.
- p.49, (2.23). Extra 2 exponent.
- p.49, (2.24). The denominator in the formula for a_j is without x.
- p.50, formula fo L_k after (2.25) the factors of the product should be (x-x_j)/(x_k-x_j).
- p.55, (2.44). In the ranges t and t should be x and y
- p.57, (2.47). Adding a number to a matrix means adding it to each entry.
- p.57, (2.48). The second matrix is M_2.
- p.77. Change the title of 3.1.1 to "Windowed transforms".
- p.78, (3.7). Extra hat in w.
- p.82, (3.21). Missing da.
- p.83. Missing "Suggested readings".
- p.85, after (3.40) is = \phi(x)-\phi(x-1)
- p.88, p.93. Include references to [GG12] and [Her17].
- p.89 Th 3.1.9. "be such that {\phi ...  }_{k\in\Z} is orthonormal" (missing end of the sentence).
- p.89, 2^{j/2} in (3.65) and 2^{j} in (3.66) (the sign of the exponent is positive).
- p.90, (3.72) second parenthesis should be big.

Material and results of some of the in class experiments:




The following sheets are home assignments that contribute to the grading.

Home assignments
File Status Deadline
Problem set 1
Available 7/March/2018
Problem set 2 Available 18/Apr/2018
Problem set 3 Available 18/May/2018

Note: In the firstly posted version of Problem set 1, in the third line of 5) there was a missing absolute value (this is forced because the functions are even). It should read |x|\not\in [...,...]. The integral of f_\delta is 2 instead of 1. I have posted a corrected version. I am very sorry for the inconvenience.

Note:In the first problem of Problem set 2, with "smooth" I assume that it decays as quick as you want, otherwise its Fourier transform could not exist or not being regular.

The grades for officially enrolled students can be checked in the Moodle server after the home assignments are corrected.

As mentioned below, a 25% of the grade can be got with (non mandatory) extra activities. From time to time some items will be added to the following list. The plan is to propose around 12 extra activities or so. A student completing 3 of them, not all theoretical, will get the full 25%. More than 3 activities increase the global grade.
The number preceding each activity indicates approximately the chapter in the lecture notes it is related to.

Experimental challenges
Name and link
1. The strange beat (interference.mp3)
7/Mar/2018 (end)
1. A not so free fall
7/Mar/2018 (end)
1. Grooving on the waves
7/Mar/2018 (end)
2. Moiré, qu'est-ce que c'est? (sectors, video)
18/Apr/2018 (end)
2. Echo echo echo End of the course
2. Unhuman voice End of the course
2. Checkerboard oddity End of the course
3. The new wave (The Meyer wavelet) End of the course
3. Step by step (sound quantization) End of the course
Theoretical challenges
Name and link Deadline
1. Breaking up a sawtooth 7/Mar/2018 (end)
3. Periodically explicit
End of the course




The grading is as follows:

1) Home assignments 75%.
2) Final exam* or extra activities 25%.

For extra activities, see Exercises.

*According with the information "Decanato" sent me, the final exam is scheduled at 01.17.SS.320 in 18/May/2018. Please check this information if you are willing to attend.



For this subject: