From Lagrange to Shannon... and back: another look at sampling [DSP Education]

Classical digital signal processing (DSP) lore tells us the tale of a continuous-time primeval signal, of its brutal sampling, and of the magic sine interpolation that, under the aegis of bandlimitedness, brings the original signal back to (continuous) life. This article switches the conventional viewpoint and cast discrete-time sequences in the lead role, with continuous-time signals entering the scene as a derived version of their gap-toothed archetypes. Some well-known but seldom-taught facts about interpolation and vector spaces are brought together and the classic sine reconstruction formula derived naturally from the Lagrange interpolation method are recalled. Such an elegant and mathematically simple result can have a great educational value in building a solid yet very intuitive grasp of the interplay between analog and digital signals.