Digital Signal Processing Reference
In-Depth Information
Figure 3.9. It sure looks the same
as when the wheel was rotating
once per second, or 1 Hz. In fact,
the wheel is moving 1
1
/
8
revolution
with each strobe
ash.
Wheel rotating counterclockwise nine times per second, or 9 Hz
Wheel rotating backward (clockwise) once per second, or -1 Hz
Figure 3.10. Now we stopped the wheel and started rotating backward at
1
/
8
revolution with each strobe
ash. Notice
7
/
8
revolution with each strobe
ash.
that this appears exactly the same as when we were rotating forward at
3.1 Nyquist Sampling Rule
To prevent all this, we have to come up with a sampling rule,
or convention. What we are going to agree is that we will always
sample (or strobe) at least twice as fast as the frequency of the
signal we are interested in. And in reality, we need to have some
margin, so we better make sure we are sampling
more
than twice
as fast as our signal.
Consider what happens when we start the wheel moving
slowly in a counterclockwise direction. Everything looks fine until
we reach a rotational speed of 4 Hz. At this point the dot will
appear to be alternating on either side of the circle with each
strobe flash. Once we have reached this point, we can no longer
tell which direction the wheel is rotating
it will look the same
rotating both directions. This is the critical point, where we are
sampling at exactly twice as fast as the signal. The sampling speed
is the frequency of the strobe light (this would be analogous to the
ADC sample frequency), eight times per second, or 8 Hz. The
rotational speed of the wheel (our signal) is 4 Hz.
If we spin the wheel any faster, it appears to move backwards
(clockwise), and by the time we reach a rotational speed of 8 Hz, it
appears to stop altogether. Spinning still faster will make the
wheel appear to move forwards again, until it again seems to
move backwards and the cycle repeats.
To summarize, whenever you have a sampled signal, you
cannot be completely sure of its frequency. But if you assume that
the rule was followed
e
e
that the signal was sampled at more than
