"From Lagrange to Shannon . . . and Back: Another Look at Sampling"
IEEE SP MAG, September 2009- Paolo Prandoni and Martin Vetterli
If I ever get the chance to teach DSP this article has a unique perspective to incorporate. Instead of starting with continuos signals, sampling them and then interpolating them to get back to continuos land, teach from the point of view that you start with discrete signals. Most of the signal we deal with every day (online) are discrete to begin with.
They suggest to introduce interpolation through the use of Lagrange polynomials, discuss implementation of such a system and why you would want to do this way (it's implementable!). But the catch is, to improve the quality of the interpolated signal, continue to increase the order of the Lagrange polynomial and then prove that when the order is infinity that the Lagrange polynomial converges to the sinc function, the ideal interpolator!
It was this idea that caught my fancy. Further they set the stage for an interesting discussion of Hilbert spaces, completeness, convolution and inner product. Overall a nice little read.
