Sampling: What Nyquist Didn’t Say, and What to Do About It
The Nyquist-Shannon sampling theorem is useful, but often misused when engineers establish
sampling rates or design anti-aliasing filters. An article by Tim Wescott explains how sampling affects a
signal, and how to use this information to design a sampling system with known performance.
The assertion made by the Nyquist-Shannon sampling theorem is simple: if you have a
signal that is perfectly band limited to a bandwidth of f0 then you can collect all the
information there is in that signal by sampling it at discrete times, as long as your sample
rate is greater than 2f0 . As theorems go this statement is delightfully short. Unfortunately,
while the theorem is simple to state it can be very misleading when one tries to apply it in
practice.
The difficulty with the Nyquist-Shannon sampling theorem is that it is based on the notion
that the signal to be sampled must be perfectly band limited. This property of the theorem
is unfortunate because no real world signal is truly and perfectly band limited. In fact, if
a signal were to be perfectly band limited—if it were to have absolutely no energy outside
of some finite frequency band—then it must extend infinitely in time.
What this means is that no system that samples data from the real world can do so
perfectly—unless you’re willing to wait an infinite amount of time for your results. If
no system can sample data perfectly, however, why do we bother with sampled time systems? The answer, of course, is that while you can never be perfect, with a bit of work you
can design sampled time systems that are good enough.
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