DigitalVideoAndHDTVAlgorithmsAndInterfaces.pdf

Figure 16.1 Cosine waves less
than and greater than 0.5f S ,
in this case at the fractions
0.35 and 0.65 of the sampling
rate, produce exactly the same
set of sampled values when
point-sampled – they alias.
Symbol conventions
used in this figure and
following figures are
as follows: cos 0.35 ωt
ω = 2π
fS
−1
[ rad⋅ s ]
1
tS
=
fS
cos 0.65 ωt
sampled
Sampling theory was originally developed to describe
one-dimensional signals such as audio, where the signal
is a continuous function of the single dimension of
time. Sampling theory has been extended to images,
where an image is treated as a continuous function of
two spatial coordinates (horizontal and vertical).
Sampling theory can be further extended to the
temporal sampling of moving images, where the third
coordinate is time.
Sampling theorem
Assume that a signal to be digitized is well behaved,
changing relatively slowly as a function of time.
Consider the cosine signals shown in Figure 16.1 below,
where the x-axis shows sample intervals. The top waveform
is a cosine at the fraction 0.35 of the sampling rate
fS ; the middle waveform is at 0.65fS . The bottom row
shows that identical samples result from sampling either
of these waveforms: Either of the waveforms can
masquerade as the same sample sequence. If the
middle waveform is sampled, then reconstructed
conventionally, the top waveform will result. This is the
phenomenon of aliasing.
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