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Signal 1

Presentation

1) Convolution and Digital image: convolution of functions and sequences, sampling, windowing, noise, quantification.
2) Fourier Transform
    a) Fourier Transform of a function: definitions, properties, examples, links with regularity, link with convolution. Gibbs effect, Heisenberg Uncertainty Principle.
      b) Discrete Fourier Transform: FFT algorithm, convolution theorem, Shannon, Noise and Fourier transform.
3)  Hilbertian Analysis
     1) Linear forms, préhilbertiens spaces.
     2) Hilbert spaces, projection theorem on a convex
     3) Bases Hilbert, Examples (Fourier bases, Haar bases ). cosine base application to compression.

Objectives

At the end of this module, the student will have understood and be able to explain (main concepts):

 

1) Creation of a digital image via windowing operations, sampling
2)FFTalgorithm
3) the notions of Hilbert spaces and Hilbert basis.

 

The student will be able to:

1) Implement FFT numerically and understand  the result of a FFT.
2) Process a signal or an image via the FFT.



Needed prerequisite

- Analyse fonctionnelle.
- Langage C.

Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...