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194 CHAPTER EIGHT
systems above 10 radians per second. We have, after all, eliminated the antialias filter
from the DSP system to illustrate the problems that could occur in its absence. We
should expect problems.
Take a look at the evidence in the following figures. Each chart pair shows the input
sine wave on top and the sampled result on the bottom. These charts were made in a
spreadsheet, which attempted to fit a curve to the sampled data at the bottom. The wave-
form thus reconstructed from the sample data is shown on the bottom of each chart.
It represents what the DSP computer thinks the original waveform looked like (see
Figure 8-2).
The sampling went reasonably well from 3 to 9 radians. Looking at Figure 8-2, it’s
clear the software could not discern the frequency (or the shape) of the input sine waves
with frequencies above 10 radians per second, but something else emerges. The sam-
pled waveform looks increasingly like a lower-frequency signal. Take a look at what
happens in Figure 8-3 as we extend the charts well beyond a 15 radian per second input
signal. The sampled waveforms seem to decrease in frequency from 16 through 21 radi-
ans per second, and then increase in frequency again between 21 and 26 radians per sec-
ond. The sampling system thinks the real waveform is doing something that is is not
doing. This is classical aliasing right before our eyes. The sampling system is being
fooled.
An alias, as defined in Webster’s dictionary, is an “assumed name.” The sampled,
reconstructed waveform at 16 radians per second looks like a waveform only two-
sevenths the same frequency. It’s representing itself as something it is not, hence the
name alias.
We’ve all seen this exact same effect take place with car wheels. At night, under
incandescent lights, look at the hubcaps of a moving car as it slows down to a stop. Pick
a car with many spokes in the hubcap. Because electrical power is at 60 Hz (or 50 Hz
elsewhere), electric lights flash at that frequency. The lights are effectively sampling the
hubcap spokes for our eyes. We can only see the hubcaps when the lights are at their
brightest. As the car decelerates from high speeds, the hubcaps appear to slow down to
zero before the car has even stopped. Then, as the car continues to decelerate, the hub-
caps appear to start moving backwards. This is the exact same effect that we just saw in
our charts about aliasing.
To avoid having the DSP computer fooled in the same manner, pay strict attention to
the Sampling Theorem. Have the computer sample at twice the highest frequency in the
input signals. Further, put an antialiasing filter in the input of the D/A that will filter
out all frequencies above half the sampling frequency.