Mathématiques et traitement du signal

Mathématiques et traitement du signal
Diplôme d'ingénieur de l'École et observatoire des sciences de la Terre (EOST)Parcours Diplôme d'ingénieur de l'EOST

ComposanteÉcole et observatoire des sciences de la Terre (EOST)

Description

Le cours est structuré en trois segments bien distincts :

  • Variables complexes (7h) :
    Fonctions analytiques, fonctions holomorphes, séries de Laurent, singularités, intégration dans le plan complexe, théorème de Cauchy, théorème des résidus, application au calcul d'intégrales réelles.
  • Théorie des distributions (7h) :
    Fonctions test, distributions régulières et singulières, exemples et propriétés élémentaires (dérivation, changement d’échelle, etc.), représentation des fonctions discontinues, convolution.
  • Analyse de Fourier (10h) :
    Séries de Fourier, transformées de Fourier (fonctions et distributions): définition, exemples, propriétés (changement d'échelle, dérivation, convolution, etc.). Applications au traitement du signal: phénomène de Gibbs, apodisation, échantillonnage, repliement spectral.

The course is structured in three distinct segments:

  • Complex variables (7h):

Analytic functions, holomorphic functions, Laurent's series, singularities, integration in the complex plane, Cauchy's theorem, residue theorem, application to the calculus of real integrals.

  • Theory of distributions (7h):

Test functions, regular and singular distributions, examples and elementary properties (derivation, change of scale), representation of discontinous functions, convolution.

  • Fourier analysis (10h):

Fourier series, Fourier transforms (fonctions and distributions): definition, examples, properties (change of scale, derivation, convolution, etc.). Applications to signal processing: Gibbs phenomenon, tapering, sampling, aliasing.

A la fin de ce cours, vous serez capable de :

- Calculer des intégrales de contour dans le plan complexe.
- Représenter et manipuler des signaux physiques singulières ou non
à l’aide de distributions.
- Calculer des séries et des transformées de Fourier de fonctions e t de distributions.

- Evaluate contour integrals in the complex plane.
- Represent and manipulate singular and non-singular physical signals using distributions. - Compute Fourier series and Fourier transforms of functions and distributions.
 

Compétences visées

L'objet de ce cours est de présenter les outils mathématiques de base pour le traitement des signaux numériques.

The purpose of this course is to present the basic mathematical tools for numerical signal analysis.
 

School regulations

The curriculum includes three years of study: admissions, the organisation of studies, assessments, placements and vivas, graduation and international exchanges are all explained in the current school rules (pdf).

First and second year courses

First and second year courses

  • General modules: mechanics, geology, mathematics, IT, digital analysis, signal processing, inverse methods.
  • Geophysical methods: physics of the Earth, seismology, seismic modelling and imaging, geodesy, gravimetry, potential methods, geomagnetism, electromagnetism, rock physics and fracture, hydrology.
  • Practical work: geophysical measurements in the field (photo) and in the laboratory, geology field placements in the Alps.
  • Languages and economic and social sciences: English, modern language 2, economics, industrial property, management, sustainable development, ethics, quality, company health and safety
  • IT and research projects, shared with the first year of the master’s degree
  • Summer placements at a laboratory or company, with numerous opportunities abroad (international placement contact: Mike Heap)

Third year course

Students have a choice of 3 specialisations in the third year:

  • Geophysics applied to the exploration and production of raw materials: seismic and hydrodynamic characterisation of reservoirs, seismic processing and interpretation, potential methods.
  • Geophysics applied to geotechnics: geotechnics and the resistance of materials applied in civil engineering, geomechanics, hydrogeophysics, electromagnetic methods, earthquake.
  • Hydrogeology, hydrogeochemistry, hydrogeophysics (HydroG3).

Additional teaching:

  • Languages and economic and social sciences: English, energy economy, company strategy and structure.
  • Geophysics field camp in Alsace (photo).  Here are images of a normal fault in the Rhine Graben taken by students.
  • 6-month industry placement culminating in the writing of a dissertation and a viva before a jury in order to obtain the engineering degree. The placements are carried out all over the world.