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9.5. Sampling Theory 225
9.5.4 Dirac Impulses in Sampling Theory
The reason impulses are useful in sampling theory is that we can use them to talk
about samples in the context of continuous functions and Fourier transforms. We
represent a sample, which has a position and a value, by an impulse translated
to that position and scaled by that value. A sample at position a with value b is
represented by bδ(x −a). This way we can express the operation of sampling the
function f(x) at a as multiplying f by δ(x − a). The result is f(a)δ(x − a).
Sampling a function at a series of equally spaced points is therefore expressed
as multiplying the function by the sum of a series of equally spaced impulses,
called an impulse train (Figure 9.47). An impulse train with period T , meaning
that the impulses are spaced a distance T apart is
s
T
(x)=
∞
i=−∞
δ(x − Ti).
The Fourier transform of s
1
is the same as s
1
: a sequence of impulses at all
integer frequencies. You can see why this should be true by thinking about what
happens when we multiply the impulse train by a sinusoid and integrate. We wind
up adding up the values of the sinusoid at all the integers. This sum will exactly
cancel to zero for non-integer frequencies, and it will diverge to +∞ for integer
frequencies.
Because of the dilation property of the Fourier transform, we can guess that
the Fourier transform of an impulse train with period T (which is like a dilation
of s
1
) is an impulse train with period 1/T . Making the sampling finer in the space
domain makes the impulses farther apart in the frequency domain.
01
x
s
1
(x)
0
x
s
1/2
(x)
0
u
s
1
(u )
0
u
s
2
(u )
1
11
Figure 9.47. Impulse trains. The Fourier transform of an impulse train is another impulse
train. Changing the period of the impulse train in space causes an inverse change in the
period in frequency.
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226 9. Signal Processing
9.5.5 Sampling and Aliasing
Now we have built the mathematical machinery we need to understand the sam-
pling and reconstruction process from the viewpoint of the frequency domian.
The key advantage of introducing Fourier transforms is that it makes the effects
of convolution filtering on the signal much clearer, and it provides more precise
explanations of why we need to filter when sampling and reconstructing.
We start the process with the original, continuous signal. In general its Fourier
transform could include components at any frequency, although for most kinds of
signals (especially images), we expect the content to decrease as the frequency
gets higher. Images also tend to have a large component at zero frequency—
remember that the zero-frequency, or DC, component is the integral of the whole
image, and since images are all positive values this tends to be a large number.
Let’s see what happens to the Fourier transform if we sample and reconstruct
without doing any special filtering (Figure 9.48). When we sample the signal, we
model the operation as multiplication with an impulse train; the sampled signal is
fs
T
. Because of the multiplication-convolution property, the FT of the sampled
signal is
ˆ
f ˆs
T
=
ˆ
fs
1/T
.
x u
0
aliasing
aliasing
originalsampledreconstructed
T
T
1
Figure 9.48. Sampling and reconstruction with no filtering. Sampling produces alias spectra
that overlap and mix with the base spectrum. Reconstruction with a box filter collects even
more information from the alias spectra. The result is a signal that has serious aliasing
artifacts.
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Plate I. The RGB color
cube in 3D and its faces un-
folded. Any RGB color is a
point in the cube. (See also
Figure 3.13.)
Plate II. A colored triangle
with barycentric interpola-
tion. Note that the changes
in color components are lin-
ear in each row and column
as well as along each edge.
In fact it is constant along
every line, such as the di-
agonals, as well. (See also
Figure 8.5.)
Plate III. Left: a Phong-
illuminated image. Middle:
cool-to-warm shading is not
useful without silhouettes.
Right: cool-to-warm shad-
ing plus silhouettes.
Image
courtesy Amy Gooch.
(See
also Figure 10.9.)
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Plate IV. The color of the
glass is affected by total in-
ternal reflection and Beer’s
Law. The amount of light
transmitted and reflected is
determined by the Fresnel
Equations. The complex
lighting on the ground plane
was computed using parti-
cle tracing as described in
Chapter 24. (See also Fig-
ure 13.3.)
Plate V. An example of
depth of field. The caus-
tic in the shadow of the
wine glass is computed us-
ing particle tracing (Chap-
ter24). (SeealsoFig-
ure 13.16.)
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Plate VI. “Spiral Stairs.”
A complex BlobTree im-
plicit model created in Er-
win DeGroot’s BlobTree.net
system. (See also Figure
16.28.)
Plate VII. “The Next
Step.” A complex Blob-
Tree implicit model cre-
ated interactively in Ryan
Schmidt’s Shapeshop by
artist, Corien Clapwijk (An-
dusan). (See also Figure
16.31.)
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