Graphics Reference
In-Depth Information
It then turns out that for a continuous
L
2
function
f
satisfying
f
(
2
)=
f
(
1
2
)
,
−
f
(
x
)=
k
c
k
e
k
(
x
)
,
(18.47)
that is, computing the inner product of
f
with each basis element
e
k
lets us write
f
as a linear combination of the
e
k
's. This is exactly analogous to the situation in
R
3
, where a vector is the sum of its projections onto the three coordinate axes. The
only difference here is that the sum is infinite, and so a proof is needed to establish
that it converges.
The
Fourier transform
of
f
is now defined to be the sequence
{
c
k
:
k
∈
Z
. With this revised definition, we see that the Fourier transform of
f
is just
the list of coefficients of
f
when it's written in a particular orthonormal basis.
Such “lists of coefficients” form a vector space under term-by-term addition and
scalar multiplication, and the Fourier transform is a
linear
transformation from
L
2
to this new vector space. Be sure you understand this:
The Fourier transform is
just a change of representation
. It's a very important one, though, because of the
multiplication-convolution property.
The function
f
}
L
2
(
H
)
is often referred to as being in the
time domain,
while
its Fourier transform is said to be in the
frequency domain.
Since one is a function
on an interval and the other is a function on the integers, the distinction between
the two is quite clear. But for functions in
L
2
(
R
)
, the Fourier transform is also in
L
2
(
R
)
, and so being able to talk about the two domains is helpful. We'll sometimes
use “value domain” or “value representation” for the original function, and “fre-
quency representation” for its Fourier transform, because
f
(
x
)
tells us the
value
of
f
at
x
, while
c
k
tells us how much frequency-
k
content there is in
f
.
We mostly won't care about the particular values
c
k
in what follows, but we'll
want to be able to take a big-picture look at these numbers and say things like
“For this function, it turns out that
c
k
=
0 whenever
∈
200,” or “The complex
numbers
c
k
get smaller and smaller as
k
gets larger.” (Recall t
hat the “
size” of a
complex number
z
=
a
+
b
i
is called its
modulus,
and is
|
k
| >
=
√
a
2
+
b
2
.) Because
of this big-picture interest, rather than trying to plot
c
k
for
k
|
z
|
∈
Z
, we instead plot
|
is a real number rather than a complex one, so it's
easier to plot. The plot of these absolute values is called the
spectrum
of
f
, and it
tells us a lot about
f
. (The word “spectrum” arises from a parallel with light being
split into all the colors of the spectrum.)
The Fourier transform takes a function in
L
2
(
H
)
and produces the sequence
of coefficients
c
k
. It's useful to think of this sequence as a function defined on the
integers, namely
k
c
k
|
. The advantage is that
|
c
k
|
→
c
k
. In fact, the sum
k
|
2
c
k
|
(18.48)
turns out to be the same as
1
2
−
2
dx
, which is finite because
f
is an
L
2
2
|
f
(
x
)
|
1
2
function, and thus the Fourier transform
function. This means that
k
→
c
k
is an
takes
L
2
(
H
)
to
2
(
Z
)
. From now on we'll denote the Fourier transform with the
letter
F
,so
:
L
2
(
H
)
2
(
Z
):
f
F
→
→
F
(
f
)
.
(18.49)
Notice that
(
f
)(
k
)
is defined to be
c
k
,the
k
th Fourier coeffi-
cient for
f
. For simplicity, we'll sometimes denote the Fourier transform of
f
by
f
.
We'll often use two properties of the Fourier transform.
F
(
f
)
is a
function
:
F
