# 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...