Graphics Reference
In-Depth Information
•F ,f f is an even function, then each c k is a real number (i.e., its imagi-
nary part is 0). For even functions, we can therefore actually plot c k rather
than
|
c k |
; that's what we did in Figure 18.35, although we only plotted it
0.
• Second, if f is real-valued (as are all the functions we care about, the real
value being something like “the intensity of light arriving at this point”),
then its Fourier transform is even, that is, c k = c k for every k . That's why
we showed the plot of c k for k
for k
1
0 in Figure 18.35: The values for k
<
0
would have added no information.
0.5
We have one example of the Fourier transform already: We wrote the square-
wave function s , as a sum of cosines in Equation 18.35. From that sum, we can
read off
0
s ( k )= 0
k even
k = 2 n + 1 .
(18.50)
1 ) n 4
π
(
0.5
n
10
0
10
The plot of s is shown in Figure 18.38.
Figure 18.38: The Fourier trans-
form of the square wave.
18.11.1 Sampling and Band Limiting in an Interval
Now suppose that we have a function f on the interval H with
F
( f )( k )= 0 for all
|
k 0 . Such a function is said to be band-limited at k 0 . The function f can be
written as a sum of sinusoidal functions, all of frequency less than or equal to k 0 .
Since the “features” of a sinusoidal function of frequency k (the “bumps”) are of
size
k
| >
1
1
2 k , the features of f must be no smaller than
2 k 0 . We can say that the function
1
f is “smooth at the scale
2 k 0 .” In a technical sense, f is completely smooth, but
what we mean is that f has no bumpiness smaller than
1
2 k 0 .
Turning this notion around, suppose that the graph of f has a sharp corner, or a
discontinuity. Then f cannot be band-limited—it must be made up of sinusoids of
arbitrarily high frequencies! This is important: A function that's discontinuous, or
nondifferentiable, cannot be band-limited. The converse is false, however—there
are plenty of smooth functions that contain arbitrarily high frequencies.
The set of all functions band-limited at k 0 is a vector space—if we add two
band-limited functions, we get another band-limited function, etc. The dimen-
sion of this vector space is 2 k 0 + 1, with coordinates provided by the numbers
c 0 , c ± 1 ,
, c ± k 0 . (This is the dimension as a real vector space; each number c j has
a real and an imaginary part, contributing two dimensions, except for c 0 , which is
pure real.)
If we evaluate the function f at k 0 + 1 equally spaced points in the interval
H =(
...
2 , 2 ] we get k 0 + 1 complex numbers, which we can treat as 2 k 0 + 2
real numbers. If we ignore any one of these, we're left with 2 k 0 + 1 real numbers.
That is to say, we've defined a linear mapping from the band-limited functions to
R 2 k 0 + 1 . This mapping turns out to be bijective. (The proof involves lots of trigono-
metric identities and some complex arithmetic.) What that tells us is somewhat
remarkable:
If f is band-limited at k 0 , then any k 0 + 1 equally spaced samples of
f determine f uniquely. Conversely, if you are given values for k 0 + 1
equally spaced samples (except for either the real or complex part of one
value), then there's a unique function f , band-limited at k 0 , that takes on
those values at those points.
1
 
 
 
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