Graphics Reference
In-Depth Information
and All of
R
1
2
,
2
]
we want to study functions on the interval
If instead of functions on
H
=(
−
(
2
]
of length
M
, we can make analogous definitions. The definition of
the Fourier transform gets an extra factor of
−
M
/
2,
M
/
1
M
; the limits of integration change to
±
M
/
2, and instead of using the function
e
k
,for
k
∈
Z
,wemustuse
e
M
(
t
)=cos
2
t
+
i
sin
2
t
.
k
M
π
π
k
(18.52)
M
The Fourier transform now sends
L
2
(
2
(
M
Z
)
, that is, functions
−
M
/
2,
M
/
2
)
to
on the set of all integer multiples of 1
M
. Thus, as the interval we're considering
gets wider and wider (i.e., as
M
increases), the spacing between the frequencies
involved in representing functions on that interval gets narrower and narrower.
It's natural to “take a limit” and consider what happens when we let
M
/
→∞
.
It turns out that in addition to the Fourier transform defined for
L
2
(
−
M
/
2,
M
/
2
)
,
we can define a Fourier transform for
L
2
(
R
)
.
For
f
L
2
(
R
)
, we define
∈
F
(
f
):
R
→
R
by the rule
)=
∞
−∞
F
(
f
)(
ω
f
(
x
)
e
ω
(
x
)
dx
,
(18.53)
where
e
ω
(
x
)=cos(
2
πω
x
)+
i
sin(
2
πω
x
)
.
(18.54)
We can think of
F
(
f
)(
ω
)
as telling “how much frequency
ω
stuff there is in
f
,” but this is a little misleading; it's perhaps better to say that
F
(
f
)(
ω
)
says “how
much
f
looks like a periodic function of frequency
ω
.”
F
(
f
)(
ω
)=
0for
|ω| >ω
0
, we say that
Just as in the case of finite intervals, if
f
is
band-limited
at frequency
ω
0
.
Before we leave the subject of Fourier transforms, there's one last topic to
cover: If we consider a periodic function
h
of period one, then
h
is definitely not
in
L
2
(
R
)
, because it doesn't tend to zero at
, so the integral of Equation 18.53
won't generally converge. On the other hand, the corresponding integral
over just
one period
of the function is the one used in defining the Fourier transform on an
interval, Equation 18.46. Thus, we can use the interval formulation to talk about
Fourier transforms for periodic functions as well.
Roughly speaking, if we truncate a periodic function
f
of period one by setting
f
(
x
)=
0for
±∞
M
,the
L
2
(
R
)
transform of the resultant function tends to be
concentrated near integer points, and its value there tends to be proportional to
M
.
As
M
gets larger, the concentration grows greater, until in the limit, the
L
2
(
R
)
transform is zero except at integer points, where it's infinite. By dividing by
M
,at
each stage we can convert the infinite values to finite ones, and they look just like
the
L
2
(
H
)
transform of a single period of
f
.
|
x
| >
We've already seen (in Figures 18.34 and 18.35) the Fourier transform of one row
of a natural image. The rapid falloff of
grows is typical; in general,
you can expect the Fourier transform to fall off like 1
F
(
f
)(
ω
)
as
ω
a
/ω
for some
a
>
1. For a
