# 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.

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:

93C62

Keywords: