Lectures: 13 h - Laboratory sessions: 11 h

Objectives

The goals of this lectures are as follows :

• Provide technical bases of signal filtering
  
• Evidence the many uses of Fourier Transform for linear systems
  
• Introduce to non-linear problems and some of their characteristics.

Syllabus

1. Non-linear systems and introduction to chaos
   
2. Classical concepts on signal processing
   
   
   • Laws of probability and applications to noisy signals
     
   • Central limit theorem
     
   • Its direct application to an experimental signal does not work!
     
   • Correlation time of an experimental signal
     
   • Averaging and lock-in detection
   
   
3. 1D Fourier transform
   
   
   • Signal decomposition on an orthogonal basis, example
     
   • orthogonal polynomials
     
   • Harmonics, Dirac signal, importance of phase
     
   • Fourier, an ideal basis for linear equations
     
   • Discrete transform and periodic signals. Principle of 2N FFT algorithm
     
   • Artefacts in FFT
     
   • Filtering, correlation, convolution, applications
   
   
4. Digitising and Shannon's theorem
   
   
   • Filtering before digital conversion, aliasing
     
   • Case of a camera, consequence of the lack of filtering in the time domain
   
   
5. 2D Fourier transform
   
   
   • Convolution and deconvolution, sharpening a blurred photograph
     
   • Reconstructing an image in Fourier space
     
   • X-rays - Principle of tomography
     
   • New microscopy techniques with a greater optical resolution than that given by the Rayleigh criterion
   
   
6. Physics of noise
   
   
   • Different types of noise and their physical origins
     
   • Shot noise and measurement of the elementary charge
     
   • Noise of a resistor, analogy with Brownian motion. Fluctuation-dissipation theorem
     
   • Spectral characteristics of physical noise. Spectral density of noise. 1/f noise
     
   • Noise variation with temperature
     
   • Adaptation of an amplifier in a measurement system

Laboratory sessions

Three half-day sessions are devoted to:

• Image and signal processing : rotation of images either simulated or recorded in tiff and jpeg formats. Filtering applied to signals, simulated images (fractals from Julia and Mandelbrot) and real images.
  
• Particle Image Velocimetry algorithm (PIV): this method enables to measure the velocity field of small particles advected by a fluid flow using video recording.
  
• Tomography reconstruction: reconstructing a 2D image using a set of 1D projections performed at different angles around the same axis.

Evaluation mechanism : a 2h written examination, and a report + matlab program illustrating one of the themes of the laboratory sessions.

Last Modification : Wednesday 8 March 2017