Graphics Reference
In-Depth Information
define (f*g)(x). Although f and g were not continuous functions, we see that f*g is. Its
formula is
1
2
(
)( )
=
Œ
[]
fgx
*
x
,
x
01
,
1
2
Œ
[]
=-
1
xx
,
12
,
=
0
,
elsewhere
.
Two easily checked properties of convolution are:
(1) (commutativity)
f*g = g*f
(2) (linearity)
f*(g + h) = f*g + f*h
21.7.3 Theorem.
(The Convolution Theorem) Let f, g:
R
Æ
R
be two absolutely
integrable functions. Then
(1) FT (f*g) = FT (f) · FT (g).
(2) If additionally either the Fourier transform of g converges or both f and g
belong to L
2
(
R
), then FT (f · g) = FT (f)*FT (g).
Proof.
See [Apos58] and [Seel66]. For weaker conditions on f and g, see [Widd71].
What Theorem 21.7.3 says is that multiplying functions in the spatial domain cor-
responds to doing a convolution in the frequency domain and vice versa.
Although we will not elaborate about this here, similar results hold in two dimen-
sions and also for the discrete case.
21.8
Signal Processing Topics
We are ready to apply what we have learned so far. Image enhancement or recon-
struction are two of the major applications of the Fourier transform. This leads us to
the basic problem in sampling theory, which is to determine how many samples one
must take so that no information is lost. Recall our discussion in Section 2.6.
Definition.
A function f(x) whose Fourier transform F(u) vanishes outside a finite
interval is called a
band-limited
function. If F(u) = 0 for |u|>w, where
{
()
=
}
w
=
inf
c F u
0
for
u
>
c
,
then w is called the
cutoff frequency
of f(x).
Figure 21.9 shows a key idea behind reconstructing functions. The functions are
shown in the left column and their Fourier transforms in the right column. In (a) we
have the signal f(x) that we are trying to analyze. As we can see from its Fourier trans-
form F(u) on the right, it is a band-limited function that vanishes outside the inter-
val [-w,w]. The sampling function s(x) in (c) is applied to (a) to get (e). What you see
in (e) is all that we know about the real function in (a). It is the function sampled at
intervals of width d. The question is whether we can reconstruct (a) from (e). If the
Fourier transform of (e) is as shown in (f), then we cannot. It is periodic with period
