Signal Processing

Course Summary:

This course provides geophysicists with the theoretical and practical foundations of signal processing applied to geophysical data. It covers essential concepts such as orthogonal decompositions, Fourier series and transforms, filtering techniques, and spectral analysis, emphasizing the differences and connections between continuous and discrete formulations.

The course also introduces time-frequency analysis methods, which are particularly effective for studying complex and non-stationary geophysical signals. Through practical examples and applications, students will develop the skills needed to analyze, process, and interpret seismic and other geophysical data efficiently.

Course Information

Course Objectives

The objective of this course is to provide a body of knowledge that enables students to engage with the extensive literature published in this field.

The aim is to develop practical skills in fundamental signal-processing techniques, including orthogonal decompositions, Fourier series decomposition, filtering, and spectral analysis. Particular attention will be given to comparing continuous and discrete formulations. In this context, suitable tools—especially time-frequency methods—will be studied, as their application is particularly effective for many geophysical signals.

Prerequisites

A solid foundation in mathematics and physics is required for effective understanding of the course material.

Course Content

I – SIGNAL REPRESENTATION
I-1 – Mathematical model of a signal
  I-1-1 – Deterministic signals
  I-1-2 – Random signals
I-2 – Energy and power of signals
I-3 – Representation using elementary signals
  I-3-1 – Principle
  I-3-2 – Application to signal approximation
  I-3-3 – Main developments using orthogonal functions
  I-3-4 – Useful duration and bandwidth of a signal
I-5 – Singular signals
I-6 – Discrete signals

II – FOURIER TRANSFORM
II-1 – Concept of signal representation
II-2 – Orthogonal functions
II-3 – Fourier transform
  II-3-1 – Theoretical foundations and definitions
  II-3-2 – Properties of the Fourier transform
II-4 – Fourier series
  II-4-1 – Theoretical foundations and definitions
  II-4-2 – Fourier series decomposition of a periodic signal
  II-4-3 – Properties of Fourier series
  II-4-4 – Characterization of periodic signals

III – CONVOLUTION
III-1 – Definition
III-2 – Properties of convolution
  III-2-1 – Commutativity, associativity, distributivity
  III-2-2 – Fourier transform of a convolution
  III-2-3 – Differentiation of a convolution

IV – CORRELATION
IV-1 – Definition
IV-2 – Fourier transform of a correlation
IV-3 – Physical significance of correlation

Tutorials (TD) Content

·       Mathematical model of a signal

·       Energy and power of signals

·       Fourier transform

·       Properties of convolution

·       Properties of correlation

Evaluation Method

Assessment is based on continuous evaluation and a final written examination.

References

- Delmas J.P. Eléments de Théorie du Signal : Signaux Déterministes. Collection

pédagogique de Télécommunication, Ellipses, 1991.

- Gasquet C. et P.Witomski, 2004. Analyse de Fourier et Application, Masson

- Bracewell, R,N, 1986. The Fourier Transform and its application, McGraw-H