Mathematical problems suggested by Analog-to-Digital conversion

Thursday, October 30, 2008 - 12:15pm - 1:05pm
EE/CS 3-180
Ingrid Daubechies (Princeton University)
In Analog-to-Digital conversion, continuously varying functions (e.g.
the output of a microphone) are transformed into digital sequences from
which one then hopes to be able to reconstruct a close approximation to
the original function. The functions under consideration are typically
band-limited (i.e. their Fourier transform is zero for frequencies
higher than some given value, called the bandwidth); such functions
are completely determined by samples taken at a rate determined
by their bandwidth. These samples then have to be approximated by
a finite binary representation. Surprisingly, in many practical applications
one does not just replace each sample by a truncated binary expansion;
for various technical reasons, it is more attractive to sample much more
often and to replace each sample by just 1 or -1, chosen judicously.

In this talk, we shall see what the attractions are of this quantization
scheme, and discuss several interesting mathematical questions suggested
by this approach.
This will be a review of work by many others as well as myself. It is also a case
study of how continuous interaction with engineers helped to shape and
change the problems as we tried to make them more precise.
MSC Code: