SIgnal Processing

Objectifs

Acquire the basic concepts for :
Handle analog and digital signals
Model and represent signals
Perform simple processing operations
Signal Processing (TdS) brings a different perspective to concepts already seen in statistics and economics courses, such as correlation, convolution, stationarity, causality…
One of the fundamentals of TdS is time-frequency analysis (spectrogram) with the Fourier transform. We introduce the motivations behind the wavelet basis.
We also introduce the notion of parsimonious representation through signal sampling. Parsimonious representation is at the origin of penalized regression models (LASSO) and regularization. These concepts will appear in all 3rd-year engineering courses, as well as in the Smart Data and Public Statistics Masters.
In addition, signals come into play in many forms in most areas of technology. These include image processing, telecommunications, acoustics, astronomy, economics, biology, optics, mechanics, electricity and electronics. More and more companies are asking for skills in this discipline. Finally, this subject not only provides students with a complementary background for their 3rd year courses, but also gives them an insight into the basic concepts of signal processing, so they can better understand the needs of companies for an internship or a job.

Plan

Definition of a signal, energy, power, periodicity, correlation, convolution, noise (disturbance signal, signal-to-noise ratio, etc.). Notion of different signals (by dimension, phenomenon, energy, morphology)
Frequency representation: notion of frequency (fundamental + harmonic), Dirac distribution (neutral element of the convolution operator), Fourrier transform (properties, examples, spectrum), Gabor theorem (Heisenberg Uncertainty Principle: quantum mechanics), Parseval Plancherel equality (conservation of energy).
Filter theory: linearity, superposition principle, transfer function, impulse response, low-band/high-pass.
Discretion in time and frequency (related to digital), signal sampling, Shannon-Nyquist, time-frequency analysis (spectrogram)

Prérequis

A little bit of math: sequences and series of functions, integration, complex numbers, basic measurement theory et Knowledge of Python for practical exercises