Sampling in Fourier and wavelet analysis

Abstract

The advent of high-speed digital computers has revolutionized many aspects of our lives. The modern technologies we take for granted rely not only on fantastically fast computer hardware, but also on various mathematical technologies. Sampling theory, in its many guises, is among the most important of these. It provides a means by which continuous-time phenomena such as the pressure waves emanating from a voicebox or musical instrument or loudspeaker, or the image of a natural scene formed in a camera, can be dealt with by fast, though finite, machines. Computers can only ever deal with a discretized, finite-length, finite-accuracy “sampled” version of these phenomena, and sampling theory provides answers to important questions such as:

• (1) Is the continuous-time signal being observed amenable to sampling, i.e., can the signal be “captured” by its values taken at discrete, separated points in time?

• (2) If so, how fast, and where, should the samples be taken?

• (3) If the samples form an equivalent discrete-time description of the signal, how should the samples be combined to reconstruct the signal?