Introduction to Sampling Theory
Authors: Thomas Zawistowski &
The applet that comes with this WWW page is an interactive
demonstration that will show the basics of sampling theory. Please read
ahead to understand more about what this program does. For more information
on the use of this applet see the bottom of this page.
A Quick Primer on Sampling Theory
The signals we use in the real world, such as our voices,
are called "analog" signals. To process these signals in
computers, we need to convert the signals to "digital" form.
While an analog signal is continuous in both time and amplitude, a digital
signal is discrete in both time and amplitude. To convert a signal
from continuous time to discrete time, a process called sampling is used.
The value of the signal is measured at certain intervals in time. Each
measurement is referred to as a sample. (The analog signal is also
quantized in amplitude, but that process is ignored in this demonstration.
See the Analog to Digital Conversion page for more on that.)
When the continuous analog signal is sampled at a frequency
F, the resulting discrete signal has more frequency components than did
the analog signal. To be precise, the frequency components of the
analog signal are repeated at the sample rate. That is, in the discrete
frequency response they are seen at their original position, and are also
seen centered around +/- F, and around +/- 2F, etc.
How many samples are necessary to ensure we are preserving
the information contained in the signal? If the signal contains high
frequency components, we will need to sample at a higher rate to avoid
losing information that is in the signal. In general, to preserve
the full information in the signal, it is necessary to sample at twice
the maximum frequency of the signal. This is known as the Nyquist
rate. The Sampling Theorem states that a signal can be exactly reproduced
if it is sampled at a frequency F, where F is greater than twice the maximum
frequency in the signal.
What happens if we sample the signal at a frequency that
is lower that the Nyquist rate? When the signal is converted back
into a continuous time signal, it will exhibit a phenomenon called aliasing.
Aliasing is the presence of unwanted components in the reconstructed signal.
These components were not present when the original signal was sampled.
In addition, some of the frequencies in the original signal may be lost
in the reconstructed signal. Aliasing occurs because signal frequencies
can overlap if the sampling frequency is too low. Frequencies "fold"
around half the sampling frequency - which is why this frequency is often
referred to as the folding frequency.
Sometimes the highest frequency components of a signal are
simply noise, or do not contain useful information. To prevent aliasing
of these frequencies, we can filter out these components before sampling
the signal. Because we are filtering out high frequency components
and letting lower frequency components through, this is known as low-pass
Demonstration of Sampling
The original signal in the applet below is composed of three
sinusoid functions, each with a different frequency and amplitude.
The example here has the frequencies 28 Hz, 84 Hz, and 140 Hz. Use
the filtering control to filter out the higher frequency components.
This filter is an ideal low-pass filter, meaning that it exactly preserves
any frequencies below the cutoff frequency and completely attenuates any
frequencies above the cutoff frequency.
Notice that if you leave all the components in the original
signal and select a low sampling frequency, aliasing will occur.
This aliasing will result in the reconstructed signal not matching the
original signal. However, you can try to limit the amount of aliasing
by filtering out the higher frequencies in the signal. Also important
to note is that once you are sampling at a rate above the Nyquist rate,
further increases in the sampling frequency do not improve the quality
of the reconstructed signal. This is true because of the ideal low-pass
filter. In real-world applications, sampling at higher frequencies
results in better reconstructed signals. However, higher sampling
frequencies require faster converters and more storage. Therefore,
engineers must weigh the advantages and disadvantages in each application,
and be aware of the tradeoffs involved.
The importance of frequency domain plots in signal analysis
cannot be understated. The three plots on the right side of the demonstration
are all Fourier transform plots. It is easy to see the effects of
changing the sampling frequency by looking at these transform plots.
As the sampling frequency decreases, the signal separation also decreases.
When the sampling frequency drops below the Nyquist rate, the frequencies
will crossover and cause aliasing.
Experiment with the following applet in order to understand
the effects of sampling and filtering.
Instructions for using the Program
The applet is divided into three sections, the Original Analog
Signal panel, Sampled Digital Signal panel, and the Reconstructed Analog
Signal panel. By making choices of the sampling frequencies, you
can see the effects of aliasing in the frequency domain plots. By
making choices of the filtering frequency, you can control what signals
remain when the analog signal is sampled. You can overlay the original
plot on top of the reconstructed plot if you want to see just how different
the results are. You can also use the reset button to return all
values to their original defaults.
To see this run in an independent window, click here:
For more detailed information on sampling and signal processing
in general, consider the following text:
Digital Signal Processing : Principles, Algorithms, and
Applications, by J. Proakis and D. Manolakis, New York: Macmillan Publishing
Send comments to Dr. John Glover at email@example.com