Mathematics and Statistics Seminar Math and Statistics Department Bear Hall 207

Friday 1/21 Bear Hall 219 3:00pm
Thirty minutes before the talk coffee and cookies will be available in Bear Hall 211 at
2:30.

Speaker
Mark Lammers
UNCW

Abstract:   We will begin by  recalling  that  certain functions f(x) can be represented by evaluating (sampling) the  function at points  xk and and using these values  f(xk)  in a sum of translates of another function g(x). For example if xk =k/l

Oversampling occurs when we sample the function f(x) more than we need to, i.e., we evaluate the function f(x) at  more xk then we need to in order to have a representation as above.  A goal of digital signal processing is  to have an approximation of the function  in terms or a sequence of 0 's and 1's.  One  way to do this for functions like those above is to give a binary representation of f(xk) but it turns out this is relatively difficult and costly to do.   An alternative (and very popular) approach is to oversample the signal and apply what is known as a Sigma Delta scheme to our samples.   We will present this scheme here and use the Fundamental Theorem of Calculus to show that we get a good approximation of our function.

In  the second part of the talk we will generalize this Sigma Delta Scheme for use in representing a two dimensional vector.  Oversampling here consists of representing a vector in terms of a finite spanning set of vectors that is greater in number then the minimum required to span. Such a set of vectors is referred to as a frame.  We will use a very special frame that  is obtained by taking equally spaced point on the unit circle as in the picture below, i.e.  roots of unity in the complex plane.