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Time-frequency analysis and discrete representation of functions and distributions




Time-frequency analysis and discrete representation of functions and distributions


Gianluca Giacchi - University of Bologna - University of Lausanne


The Fourier transform, a fundamental tool of several modern branches of mathematics, fails in capturing the local frequency information carried by signals. On top of that, a reasonable definition of “local” frequencies through the Fourier transform is prevented by uncertainty principles, which manifest under different forms. To overcome these issues and get an overall time-frequency perspective, new integral transforms must be defined, such as the short-time Fourier transform and the (cross) Wigner distribution. Parallelly, Gabor analysis stems from the need of discretizing signals, focusing the attention on “atomic” decompositions by means of time-frequency operators, such as time-frequency shifts. We present a unifying approach to time-frequency representations and Gabor analysis that leverages metaplectic operators, which offer the advantage of tracing operator calculus back to ordinary matrix calculus.


Gianluca Giacchi was born in Milan, where he pursued his Bachelor’s and Master’s degrees in Mathematics at the University of Milan (Unimi). He acquired a general background during his bachelor’s period, covering both theoretical and applied aspects of mathematics. His master’s degree focused on modern mathematical analysis, with particular emphasis on Fourier analysis and its multiple branches. He is currently a Ph.D. student at the University of Bologna (Unibo) and the University of Lausanne - CHUV (Unil), with his research focused on convex optimization problems with applications to Magnetic Resonance Imaging (MRI). Specifically, he is studying the problem of determining the optimal tuning parameter for the LASSO problem mathematically. He is also working on time-frequency analysis, with particular emphasis on studying original joint time-frequency representations defined through metaplectic operators.


Monday March 11th, 4pm


Room 322 @ DIBRIS/DIMA, Via Dodecaneso 35, Genoa