From Lagrange to Shannon...and Back: Another Look at Sampling

Classical digital signal processing (DSP) lore tells us the tale of a continuous-time primeval signal, of its brutal sampling, and of the magic sinc interpolation that, under the aegis of bandlimitedness, brings the original signal back to (continuous) life. In this article, we would like to switch 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. To this end, we will bring together some well-known but seldomtaught facts about interpolation and vector spaces and recall how the classic sinc reconstruction formula derives naturally from the Lagrange interpolation method. 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.