| The Nyquist Sampling Theorem |
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 fmax.
-
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 fmax.
-
To reproduce a signal with a
highest frequency component, fmax, 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 2fmax
- 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. 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?
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