Interference and Diffraction
             Fractional Fourier 
                 Transform
                       SÉBASTIEN COËTMELLEC
                                            
                                                                  Summary
                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                           Summary                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                3
                                           I - Lesson                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             5
                                                          A. Fractional Fourier Transformation........................................................................................................................................6
                                                                        1. Definition.........................................................................................................................................................................6
                                                                        2. The intrinsic properties of fractional operators...................................................................................................................8
                                                          B. Fractional Fourier transformation and Wigner Radon distributions..............................................................................10
                                                                        1. The Wigner Distribution Function.................................................................................................................................11
                                                                        2. Example of a Wigner distribution function.....................................................................................................................12
                                                                        3. The Radon-Wigner transformation.................................................................................................................................14
                                                                        4. Relation between FRFTs and Radon-Wigner distributions............................................................................................16
                                                          C. Differential equation solving.................................................................................................................................................18
                                                                        1. Illustration and Resolving...............................................................................................................................................18
                                                          D. Digital fractional Fourier Transforms.................................................................................................................................20
                                                                        1. From continuous to discrete............................................................................................................................................20
                                                                        2. Sampling optimization...................................................................................................................................................21
                                           Bibliography                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    25
                                            
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