Analisi di Fourier e ricostruzione di segnali a partire da dati campionati

Picardello, Massimo A.

La Matematica nella società e nella cultura, Série 1, Tome 5 (2012) no. 3, pp. 399-444

Résumé

We explain how to reconstruct exactly a continuous signal (apart from numerical round-offs) starting from a sequence of its values sampled on a grid ofuniform step. The mathematics of this reconstruction is based upon Fourier Analysis, and requires that the signal has bounded Fourier spectrum and the sampling step is sufficiently small: this fact has been widely known for a long time (the Shannon sampling theorem is dated 1948, but the mathematicians knew all this much before). Our account of these facts avoids all mathematical technicalities but gives most statements and ideas. The steps to process the discrete samples in order to reconstruct the continuous signals can be based upon a smart handling of the data histogram via Fourier techniques. This may seem just a practical, heuristic approach, but our presentation makes it rigorous, by clarifying the connection between the discrete and continuous sides through distribution theory (briefly outlined). Finally, we state some recent results about reconstruction from non-uniform sampling.