i
i
i
i
i
i
i
i
9.5. Sampling Theory 223
Figure 9.45. A commutative diagram to show visually the relationship between convolution
and multiplication. If we multiply
f
and
g
in space, then transform to frequency, we end up in
the same place as if we transformed
f
and
g
to frequency and then convolved them. Likewise,
if we convolve
f
and
g
in space and then transform into frequency, we end up in the same
place as if we transformed
f
and
g
to frequency, then multiplied them.
The Fourier transform of the convolution of two functions is the product of the
Fourier transforms. Following the by now familiar symmetry,
ˆ
fˆg = F{fg}.
The convolution of two Fourier transforms is the Fourier transform of the product
of the two functions. These facts are fairly straightforward to derive from the
definitions.
This relationship is the main reason Fourier transforms are useful in studying
the effects of sampling and reconstruction. We’ve seen how sampling, filtering,
and reconstruction can be seen in terms of convolution; now the Fourier transform
gives us a new domain—the frequency domain—in which these operations are
simply products.
9.5.3 A Gallery of Fourier Transforms
Now that we have some facts about Fourier transforms, let’s look at some exam-
ples of individual functions. In particular, we’ll look at some filters from Sec-
tion 9.3.1, which are shown with their Fourier transforms in Figure 9.46. We have
already seen the box function:
F{f
box
} =
sin πu
πu
= sinc πu.
The function
3
sin x/x is important enough to have its own name, sinc x.
3
You may notice that sin πu/πu is undefined for u =0. It is, however, continuous across zero,
and we take it as understood that we use the limiting value of this ratio, 1, at u =0.