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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 3
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