How to Analyze the Frequency Content of a Signal
Author: John M. Cimbala, Penn State University
Latest revision: 11 October 2007
Introduction
•
When the frequency content of a signal is
known
, or at least
approximately
known
, it is easy to choose the
sampling frequency – we choose a sampling frequency larger than twice the maximum expected frequency of
the signal (using the Nyquist criterion).
•
In this learning module, we discuss what to do in situations in which the frequency content of a signal is
unknown
(as in many typical laboratory situations). In particular,
how then can we avoid aliasing errors
?
Apply a low-pass (anti-aliasing) filter
•
The best way to eliminate aliasing errors entirely is to
filter
the signal
before
taking the digital data.
o
In particular, a
low-pass filter
is used; thus, a low-pass filter is often called an
anti-aliasing filter
.
o
Here is how this anti-aliasing-filter technique works:
±
A low-pass filter lets low frequency components of the signal
pass
through (that’s why it is called a
“low-pass” filter), but cuts off frequency components above some specified
cutoff frequency
.
±
An
ideal low-pass filter
abruptly
cuts off the high frequency content of the signal –
all content of the
signal at frequencies greater than the cutoff frequency is removed
.
±
Ideally then, the cutoff frequency of the ideal low-pass filter is set to one-half of the sampling
frequency to avoid aliasing,
f
cutoff
=
f
s
/2 (following the Nyquist criterion).
±
That way,
all frequencies that would have produced aliasing errors are removed from the signal
before the digital data are sampled, and therefore no aliasing is possible
.
o
Unfortunately, real-life low-pass filters do not cut off high frequencies abruptly; instead, the attenuation
of higher frequencies falls off rather slowly with frequency.
o
Because of this, most real-life data acquisition systems employ a cutoff frequency
several times smaller
than the sampling frequency to ensure proper anti-aliasing.
•
Low-pass filters are discussed in more detail in another learning module.
Sample at different sampling frequencies
•
Even without a low-pass anti-aliasing filter, we are often able to analyze the frequency content of the signal
by
sampling at several different sampling frequencies, and comparing the corresponding frequency spectra
.
o
Principle
:
If we sample at two different sampling frequencies, and the peaks in the frequency spectra
appear at
different
frequencies, we can be sure that aliasing errors are occurring
.
o
In this case,
the sampling frequency must be continually increased until the peaks in the spectra
do not
change
, and
the aliased peaks can be consistently explained
. An example is provided below.
•
In many practical applications, such as measurement of turbulent fluctuations in a fluid flow, there are