Signal Processing Theory

Code:

M.EMG0029

Acronym:

TTS

Keywords
Classification	Keyword
OFICIAL	Technology and Applied Sciences
OFICIAL	Mathematics

Instance: 2022/2023 - 2S

Active?	Yes
Web Page:	http://moodle.fe.pt
Responsible unit:	Mining Engineering Department
Course/CS Responsible:	Master in Mining and Geo-Environmental Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
M.EMG	9	Plano de estudos oficial a partir de 2008/09	1	-	6	45,5	162

Teaching language

Portuguese and english
Obs.: Inglês, se necessário

Objectives

Familiarise the student with Fourier Analysis and its applications in signal processing, based on the formalism of the Theory of Temperate Distributions.
It is intended that this core of knowledge is well assimilated, allowing future deepening if professional practice so requires.

Learning outcomes and competences

On completion of this course unit the student should be able to

- Understand the formalism, meaning and applications of tempered distributions, the Fourier transform and the convolution operation. In particular:
- Apply conveniently the sampling theorem in the digitisation of analogue signals (A/D conversion);
- Critically use the Fourier transform, direct and inverse, as an efficient converter between domains;
- Understand the application of filters in signal processing in time/space and frequency domains;
- Develop algorithms in Matlab environment for signal processing/filtering.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Mathematical Analysis I and II;

Program

Concept of distribution. The distribution as a generalized function. Properties of tempered distributions. Application examples. Properties of distributions. Properties of the Dirac distribution. Distributions as limits of functions. Examples using the pedestal and Gauss function. Physical quantities as distributions - impulse response. Convolution: definition, properties; translation, derivation and integration as convolution. Translation of a convolution. Z-transform. Deconvolution.

Fourier transform. Definition. Euler relation. Fourier's theorem (inversion formula). Sine and cosine transforms. Modulus and phase representation of the transform. Existence issues. Fourier integral of a real function, of a pure imaginary function, of a real and even function, of a real and odd function, of any real function, of a Hermitian function and of a causal function - interdependence of the real and imaginary part of the spectrum. Physical meaning of Fourier transform: spectrum. Amplitude spectrum, phase spectrum, spectral density, properties of the Fourier operator. Elementary properties: linearity, symmetry, translation, modulation, scaling, derivation, integration. Spectra of some interesting functions and distributions: spectrum of Dirac distribution, sinusoid, Heaviside step, pedestal, cardinal sine, triangle, sampling comb. Convolution theorem. Parseval's theorem or energy theorem. Relations between signal compression and spectrum expansion. Heisenberg uncertainty theorem. Bounded support functions and spectra. Gibbs phenomenon. Periodic functions and Fourier series. Shannon and Kotielnikov theorem. Current and instantaneous spectra, spectral density. Time invariant linear system. Impulse response and pass function. Introduction to the use and design of filters.

Mandatory literature

Francisco Vaz; Análise de Fourier para engenharia eletrotécnica. ISBN: 978-989-746-135-4
Novais Madureira, Abilio Cavalheiro; Teoria dos Métodos Geofísicos, 2008
Athanasios Papoulis; The Fourier integral and its applications. ISBN: 07-048447-3
John H. Karl; An introduction to digital signal processing. ISBN: 0-12-398420-3

Teaching methods and learning activities

The course is of a theoretical and practical nature. The practical examples are intended to illustrate and consolidate the theoretical knowledge acquired. The theoretical-practical examples will highlight the constraints imposed by discrete and small-interval (not infinitesimal) sampling - numerical examples, as opposed to analytical formulation. In the most advanced phase of the course, the whole body of theory converges in the understanding of sampling and signal processing methodologies as well as in the synthesis capacity evidenced by the transfer/impulse function as a descriptor of the behaviour of linear systems.

Development of algorithms in Matlab environment for signal processing/filtering.

Software

Excel spreadsheet
Matlab

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Designation	Weight (%)
Participação presencial	10,00
Teste	50,00
Trabalho escrito	40,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	71,00
Frequência das aulas	45,00
Trabalho escrito	46,00
Total:	162,00

Eligibility for exams

Not exceed the number of absences established in the General Evaluation Rules and have a mark in the Distributed Evaluation equal to or higher than 6.0 points.

Calculation formula of final grade

The final mark will be the mark of the distributed evaluation/“Avaliação Distribuída”.

Assessment during the semester includes: 2 tests, assignments and performance.
The marks for the Distributed Assessment are obtained by a weighted average using the following weights:
- 55% for two assessment tests: 20% 1st test and 35% 2nd test;
- 45% for assignments and performance.

Non-approved students having obtained frequency/ “Obtenção de frequência”, may access a written resit exam. In these cases final classification will be a weighted average: 70% for resit exam and 30% for assignments and performance components of “Avaliação Distribuída”.

Accreditation of marks exceeding 18 points shall be validated through an oral exam.

Examinations or Special Assignments

Not foreseen.

Special assessment (TE, DA, ...)

According with FEUP general evaluation rules.

Classification improvement

Written and/or oral exam, possibly coincident with the resit/appeal exam.