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1/d and the repetitions of F(u) overlap. However, if we had sampled more often and
used (g) where we have a smaller sampling width d, then we can get (a) back in two
steps as follows:
Step 1.
Multiply the Fourier transform F(u)*S(u) of f(x)s(x) by the box function
u
w
Ê
Ë
ˆ
¯
()
=
Gu
B
2
to get the function H(u) in (k) which has isolated one copy of F(u).
Step 2.
Take the inverse Fourier transform of H(u) to recreate f(x) as shown in (j).
It follows that if we have a band-limited function, then we can recover a function
by sampling sufficiently often. This is the Whittaker-Shannon sampling theorem.
21.8.1 Theorem.
(Whittaker-Shannon Sampling Theorem) Let f(x) be a band-
limited function with cutoff frequency w, so that its Fourier transform vanishes
outside the interval [-w,w]. Then f(x) can be reconstructed exactly from samples at
intervals d provided that the sampling frequency 1/d is at least 2w.
Proof.
See [Glas95].
Definition.
The frequency w in Theorem 21.8.1 is called the
Nyquist frequency
of
f(x).
If the sampling frequency is not high enough, then we get the phenomena called
aliasing. The two-dimensional situation is similar, but in practice one must sample a
lot more because of limitations of available reconstruction algorithms. Unfortunately,
in real life we usually do not have such band-limited functions, so that all this is of
theoretical value only. Nevertheless, it does show us what is going on and helps us
determine ways that one can mitigate the sampling problems. Actually, there is one
additional complication to the reconstruction of signals. An important implicit
assumption in our analysis so far has been that we took an infinite number of samples.
This is of course an unrealistic assumption. We therefore need to study the problem
further and ask what would be different if we only take a finite number of samples.
See Figure 21.10. In (a) we again show a function that is about to be sampled and
reconstructed. The mathematical effect of sampling only a finite number of times is
to multiply the function in (e) by a box-like function. Suppose that we only sample
over the interval [0,a]. Let
B
x
a
1
2
Ê
Ë
ˆ
¯
()
=
hx
-
.
This “windowing” function vanishes outside of [0,a]. See (g). Its transform H(u) is
shown in (h). Our actual samples, represented by the function h(x)[s(x)f(x)], is shown
in (i) and its Fourier transform H(u)*[S(u)*F(u)] in (j). Since H(u) has components
that extend to infinity, this has the effect of introducing a distortion in the frequency
domain representation of the function that was sampled over the finite interval. The
unfortunate conclusion is that it is in general impossible to faithfully reconstruct a
