The
Nyquist Sampling Theorem
In this note we examine an important limitation you will encounter when
sampling continuous signals. Sampled signals happen in many environments,
including those below.

If you measure data in
the lab and store it in a file  to be analyzed later in Mathcad, Matlab
or Excel, for example  you store samples of a signal (as well as the times
at which the data was sampled).

If you use a telephone,
many calls (all long distance calls) sample your voice signal, then transmit
digital information, and reconstitute an analog signal (one someone can
hear) on the far end of the connection.
If you measure lab data,
and analyze it, you will probably plot the data so that you can see what
it looks like. If you talk on the phone, you want to be able to generate
an accurate copy of the original signal on the far end of the connection.
In both cases there are two important things that happen.

The original signal is
sampled at discrete instants  and they are usually sampled at uniformly
spaced intervals.

The sampled version is
eventually used to generate a copy of the original signal.
The problem that can arise is that the sampling rate determines how well
you can reconstitute the signal  to plot a graph, or to acccurately copy
a voice signal. We need to think about a few ideas first.

The signal we are sampling
has a frequency spectrum. Let us assume that the highest frequency
component in the signal is f_{max}.

If you are familiar with
Fourier Transforms, you may realize that frequency spectra of realistic
signals are not zero above some arbitray frequency. A real signal
will have a spectrum that gets smaller and smaller at higher and higher
frequencies, but which only approach zero asymptotically. We are
really assuming that our signal has negligible components above f_{max}.

To reproduce a signal
with a highest frequency component, f_{max}, the sampling
frequency (the frequency at which samples are taken) must be at least twice
the highest frequency component. In other words, the signal cannot
be reproduced accurately unless the sampling frequency is at least 2f_{max}
 a frequency that is referred to as the Nyquist frequency for the signal.
If the sampling frequency is lower than the Nyquist frequency, that is
referred to as undersampling.
It is important to get a sense of what this means and what can happen.
We have a simulator you can use to demonstrate some of the pitfalls of
undersampling and what can happen when that occurs. Click
here to get the simulator in a separate window.
Once you have the simulator running, click the Start button and examine
the output. Then answer this question.
Question
Q1
Does it look to be possible to reconstruct the original sine wave from
the sampled values (large points)?
We want you to do a few numerical experiments using the simulator.
Experiment
1
Change the frequency to 2 Hz. Observe what happens. Would you
be able to reconstruct the signal from the samples?
Now, let us set some conditions.

Set the sampling period
to 0.5 seconds.
Before you continue, answer
the following question.
Question
Q2
When the sampling period is 0.5 seconds, what is the Nyquist Frequency
 i.e. the frequency limit for the largest frequency in the signal?
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.