Geophysical Data Processing Theory

Code:

MEMG0006

Acronym:

TMG

Keywords
Classification	Keyword
OFICIAL	Mathematics, Physics, Earth Sciences

Instance: 2015/2016 - 2S

Active?	Yes
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
MEMG	19	Plano de estudos oficial a partir de 2008/09	1	-	6	56	162

Teaching language

Suitable for English-speaking students

Objectives

To familiarize students with Fourier Analysis, based on the Theory of Tempered Distributions. Illustration of the theory with practical applications in Matlab environment. This knowledge should be well consolidated, enabling students to deepen their knowledge, if demanded during their professional practice.

Learning outcomes and competences

In completing this course, basically the student should be able to:

- Conveniently apply the sampling theorem in A/D operations;

- Conveniently apply the direct and inverse Fourier transform;

- Understand the application of filters in signal processing in time/space and frequency domains;

- Develop signal processing algorithms in Matlab environment.

Working method

Presencial

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

Mathematical Analysis I and II

Program

Concept of distribution. Examples of application of the concept. Properties of distributions. Distribution as a generalized function. Properties of tempered distributions. Convolution. Identification of a distribution with an interval function. Identity of two distributions in an interval. Limited support distribution. Derivative of a distribution. Odd and even distributions. Integral of a distribution. Properties of Dirac distribution. Distribution limits. Distributions as function limits. Example using pedestal and Gaussian function. Physical quantities as distributions – impulse response Convolution: definition, commutative, associative property, etc Translation as convolution. Derivation as convolution. Integration as convolution. Z transform. Deconvolution.

Fourier transform. Definition. Euler relation. Cissoid properties. Presentation of Fourier transform. Fourier theorem (inversion formula). Sine and cosine transform. Transform representation in module and phase. Demonstration of Fourier theorem. Fourier integral of a real function. Fourier integral of a pure imaginary. Fourier integral of a real and even function. Fourier integral of a real and odd function. Fourier integral of any real function. Fourier integral of a hermitian function. Fourier integral of a causal function – interdependence of a real part and imaginary spectrum. Physical meaning of Fourier transform: spectrum. Amplitude spectrum, phase spectrum, spectral density, properties of Fourier operator. Alternative definitions. Spectrum elementary properties: Linearity, symmetry, translation, modulation, scale, derivation, integration. Spectrum of some functions and interesting distributions: Dirac distribution spectrum, sinusoid, Heaviside step, pedestal, sinus cardinal, triangle, sampling function. Convolution theorem. Parseval’s theorem or energy theorem. Rayleigh’s theorem. Relationship between signal compression and spectral expansion. Heisenberg theorem or uncertainty principle. Functions and spectrum of limited support. Flatness. Gibbs phenomenon. Periodic functions and Fourier series. Shannon and Kotielnikov theorem. Gabor theorem. Current and instantaneous spectrum, spectral density. Linear time-invariant systems. Impulse response. Transfer function. Introduction to filter design.

Mandatory literature

Papoulis, Athanasios; The Fourier integral and its applications. ISBN: 07-048447-3
Novais Madureira - Abilio Cavalheiro; Teoria dos Métodos Geofísicos, 2008
Karl, John H.; An introduction to digital signal processing. ISBN: 0-12-398420-3

Complementary Bibliography

Bracewell, Ronald N.; The Fourier transform and its applications. ISBN: 0-07-007013-X

Teaching methods and learning activities

This course unit is theoretical-practical. The theoretical-practical examples aim to consolidate students’ knowledge. Furthermore the theoretical-practical examples show the constrictions imposed by discrete sampling in a small interval (non-infinitesimal) of numerical examples, in contrast with the continuous "sampling" from minus infinity to plus infinity with infinitesimal step which corresponds to the analytical formulation. At a more advanced stage, all theory converges to the understanding of sampling methodologies and signal processing, as well as synthesis skills based on transfer functions describing linear systems.

Software

Matlab

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Designation	Weight (%)
Participação presencial	25,00
Teste	50,00
Trabalho escrito	25,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Elaboração de relatório/dissertação/tese	20,00
Estudo autónomo	90,00
Frequência das aulas	56,00
Total:	166,00

Eligibility for exams

Attended classes in number equal or exceeding the minimum allowed by FEUP General Evaluation Rules and a minimum final grade of 6,5 out of 20, in the students continuous assessment. The assessment comprises: 2 tests, 50%, assignments and students’ performance, 50%. Students with special status have to perform the same practical assignments and attend the same tests as regular students.

 

Calculation formula of final grade

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

Non-approved students having obtained frequency/ “frequência”, may access a written/oral resit exam. Final classification weighted average: 75% for resit exam plus 25% for assignments and performance component of “Avaliação Distribuída”.

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

Examinations or Special Assignments

Not applicable

Special assessment (TE, DA, ...)

According to General Evaluation Rules of FEUP

Classification improvement

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