EEL 3135: Signals and Systems
Dr. Fred J. Taylor, Professor
Topic: Aliasing
Lesson Number: 7 (Section 42)
Sampling Modalities
In Section 41, Shannon’s sampling theorem was introduced. The sampling theorem established
the minimum rate at which a continuoustime signal can be sampled and reconstructed from its
sample values.
The lower bound on the sample rate was found to be f
s
>2f
max
where f
max
is the
highest frequency in the signal being sampled (f
max
/2 called the Nyquist frequency).
In reality, there are three possible sampling modalities.
They are:
•
over sampling f
s
>> f
max
/2 (common case)
•
critical sampling f
s
= f
max
/2
•
and undersampled f
s
< f
max
/2
The reconstructed output varies depending on the sampling strategy used. In addition, practical
considerations, such as the minimum or maximum sample rates supported by an analogto
digitalconverter (ADC), influence the sampling decisions.
Aliasing
Whenever Shannon’s Sampling Theorem is violated (
i.e
., under sampled case), a phenomenon
called
aliasing
occurs. Aliasing is a phenomenon that manifests itself as a corruption of a
reconstructed signal or image. Aliasing occurs when a reconstructed signal
impersonates
another signal or image. To illustrate, consider as a youth seeing your first Hollywood western.
The camera was following a moving stage coach at a 30 frames/s rate. While the stagecoach
wheel was moving at a leisurely clockwise rate of say 12
°
per frame, the viewer would see the
wheel spin at a rate of one revolution per second in a clockwise direction as reported in Figure
1. Then a danger appears under a sea of arrows and the background music is translated to a
minor key. The stagecoach was now rushing forward at full speed; wheel’s spinning at a rate
near 30 revolutions per second which translates to a rotational rate is 348
°
degrees (348
°
=360
°

12
°
) per frame. From the viewer’s perspective it would appear to the wheel is turning at a rate of
12
°
per fame backwards as suggested in Figure 1! This is a physical instantiation of aliasing.
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EEL 3135: Signals and Systems
Dr. Fred J. Taylor, Professor
Figure 1: Image aliasing experiment. Slowly moving stagecoach (top) and rapidly moving
stagecoach (bottom).
In this supplement, aliasing will be studied using a number of strategies in this supplemental
note.
All are based on the requirement that a signal, reconstructed from its sample values, is a
baseband signal.
This means that the reconstructed signal’s frequency is assumed to be
restricted to the baseband
range f
∈
(f
s
/2, f
s
/2).
The principal object of this note is to provide you
with the tools needed to determine the frequency a reconstructed signal.
Consider the lowly cosine wave:
(
29
(
29
t
t
x
0
cos
ϖ
=
1.
sampled at a rate
f
s
Sa/s. Assume that the sinusoid's frequency is known to be
ϖ
0
=2
π
f
0
=
2
π∆
0
+
m
2
π
f
s
where
∆
0
is a baseband frequency
∆
0
∈
[
f
s
/2,
f
s
/2], and
m
is an integer. Then
at the sample instances
t
=
kT
s
:
[
]
[
]
(
29
(
29
(
29
(
29
s
s
s
kT
mf
kT
k
k
x
π
π
ϖ
φ
2
2
cos
cos
cos
0
0
+
∆
=
=
=
2.
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 Spring '08
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 Digital Signal Processing, Aliasing, Signal Processing, Dr. Fred J. Taylor, Dr. Fred J., Fred J. Taylor

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