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Mathematics
and Statistics Seminar
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Math and
Statistics Department |
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Friday
1/21 Bear Hall 219
Thirty minutes before the talk coffee and cookies will be available in
Bear
Hall 211 at
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.
