3: the Shannon Sampling Theorem

In this final set of notes, we would like to end where we began. In the first set of notes, we mentioned that mathematics could be used to show how CDs and MP3s can reproduce our music well, even though the file sizes of MP3s are relatively small. In this set of notes, we attempt to describe how this is possible. Surprisingly, the exponential function will play a key role. We assume familiarity with differentiation and integration theory. We begin our discussion with Fourier series. In the class, we have familiarized ourselves with infinite series. Some of the most interesting infinite series are associated to functions f : [0, 1] → R. In particular, we can encode a lot of information of a function f : [0, 1] → R via a certain infinite series, which is called the Fourier series of f . It is this series that will lead us to an analysis of CDs and MP3s.