| The Nyquist Sampling Theorem |
When you are done, you realize that the highest frequency that you can sample is
1.0 Hz. OK, so let's set the frequency to be right at the limit.
Experiment 2
When the signal frequency is 1.0 Hz and you sample right at the Nyquist rate,
can you reconstruct the signal?
OK, now, theoretically, if you sample below the Nyquist rate you should get
samples from which you can reconstruct the signal.
Experiment 3
Change the frequency to 0.9 Hz. Observe what happens. Would you be
able to reconstruct the signal from the samples?
Now, for the curious among you, let us see what would happen if you had a signal
just over the Nyquist limit.
Experiment 4
Change the frequency to 1.1 Hz. Observe what happens. Would you be
able to reconstruct the signal from the samples?
Maybe you need to be even more curious. To help you along, try this.
Experiment 5
Change the frequency to 2.1 Hz. Observe what happens. Would you be
able to reconstruct the signal from the samples?
Experiment 5 is a real puzzler. It certainly seems like you can
reconstruct a sine wave from the samples, but the obvious signal that you would
reconstruct is not the signal you want - the original signal. It is at a
frequency that is not correct. Actually, you get samples that are
consistent with a lower frequency sine wave, and if you had that data in a file
(just the sampled points, the large dots) you would have no way of knowing that
the signal had not been sampled correctly (i.e. that whoever generated the data
had undersampled.) and you would be justified in thinking that the apparent low
frequency sine wave (see the large dots) was the actual signal recorded.
The consequences of undersampling are that a frequency higher than the Nyquist
limit is aliased into another signal with a frequency within the Nyquist
limit. Aliasing causes errors in computation of the magnitude of frequency
components at frequencies far from the signal that causes the problem.
The conclusion you should get from this is that is is very possible to get
spurious information into a data file if you have not sampled fast enough, and
you should be aware of the possibility of that kind of error.
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