Databases Reference
In-Depth Information
Because of orthonormality,
1
i
=
k
x
i
,
x
k
=
0
i
=
k
and
x
k
=
α
k
By repeating this with each
x
i
, we can get all the coefficients
y
,
α
i
. Note that in order to use this
machinery, the basis set has to be orthonormal.
We now have sufficient information in hand to begin looking at some of the well-known
techniques for representing functions of time. This was somewhat of a crash course in vector
spaces, and you might, with some justification, be feeling somewhat dazed. Basically, the
important ideas that we would like you to remember are the following:
Vectors are not simply points in two- or three-dimensional space. In fact, functions of
time can be viewed as elements in a vector space.
Collections of vectors that satisfy certain axioms make up a vector space.
All members of a vector space can be represented as linear, or weighted, combinations of
the basis vectors (keep in mind that you can have many different basis sets for the same
space). If the basis vectors have unit magnitude and are orthogonal, they are known as
an
orthonormal basis set
.
If a basis set is orthonormal, the weights, or coefficients, can be obtained by taking the
inner product of the vector with the corresponding basis vector.
In the next section we use these concepts to show how we can represent periodic functions as
linear combinations of sines and cosines.
12.4 Fourier Series
The representation of periodic functions in terms of a series of sines and cosines was used by
Jean Baptiste Joseph Fourier to solve equations describing heat diffusion. This approach has
since become indispensable in the analysis and design of systems. The work was awarded
the grand prize for mathematics in 1812 and has been called one of the most revolutionary
contributions of the last century. A very readable account of the life of Fourier and the impact
of his discovery can be found in [
181
].
Fourier showed that any periodic function, no matter how awkward looking, could be
represented as the sum of smooth, well-behaved sines and cosines. Given a periodic function
f
(
t
)
with period
T
,
f
(
t
)
=
f
(
t
+
nT
)
n
=±
1
,
±
2
,...
we can write
f
(
t
)
as
∞
∞
2
T
f
(
t
)
=
a
0
+
a
n
cos
n
w
0
t
+
b
n
sin
n
w
0
t
,
=
(4)
0
n
=
1
n
=
1
This form is called the
trigonometric Fourier series representation
of
f
(
t
)
.
