Databases Reference
In-Depth Information
We will begin our coverage with a brief introduction to the concept of vector spaces, and
in particular the concept of the inner product. We will use these concepts in our description of
Fourier series and Fourier transforms. Next is a brief overview of linear systems, then a look at
the issues involved in sampling a function. Finally, we will revisit the Fourier concepts in the
context of sampled functions and provide a brief introduction to Z-transforms. Throughout,
we will try to get a physical feel for the various concepts.
12.3 Vector Spaces
The techniques we will be using to obtain compression will involve manipulations and decom-
positions of (sampled) functions of time. In order to do this we need some sort of mathematical
framework. This framework is provided through the concept of vector spaces.
We are very familiar with vectors in two- or three-dimensional space. An example of a
vector in two-dimensional space is shown in Figure
12.1
. This vector can be represented in
a number of different ways: we can represent it in terms of its magnitude and direction, or
we can represent it as a weighted sum of the unit vect
o
rs in the
x
and
y
directions, or we can
represent it as an array whose components are the coefficients of the unit vectors. Thus, the
vector
v
in Figure
12.1
has a magnitude of 5 and an angle of 36.86 degrees,
v
=
4
u
x
+
3
u
y
and
4
3
v
=
We can view the second representation as a decomposition of
V
into simpler building
blocks, namely, the
basis vectors
. The nice thing about this is that any vector in two dimensions
can be decomposed in exactly the same way. Given a particular vector
A
and a basis set (more
4
3
v
u
y
2
1
1
4
2
3
u
x
F I GU R E 12 . 1
A vector.
