Sampling: What Nyquist Didn’t Say, and What to Do About It
strictly cyclical, which means that we can count on it repeating over and over again. We’ve
alreadydescribedthis in thetimedomain—how each new sample lands a bit farther ahead
on the cycle where it falls than the previous sample did. If wetry to think about this in the
frequency domain we’re still left with an apparent contradiction, however.
What is happening in the frequency domain is the result of a quirk in the Nyquist-Shannon
sampling theorem, and a matching quirk in the frequency domain behavior of repetitive
signals. These quirks work together to let you sub-sample truly repetitive signals and
build up a picture of the waveform, even if your sampling rate is much lower than the
fundamental frequency of the signal.
Thequirk in the frequency domain is that a steady, repetitive signal has it’s energy concen-
trated in spikes around the fundamental frequency and its harmonics. The more steady
the cyclical waveform is, the narrower these spikes are. So a signal that varies over a span
of a few seconds will have spikes that are around 1Hz wide, while a signal that varies over
aspan of a few minutes will have spikes that are only a small fraction of 1Hz wide.
The quirk in the Nyquist-Shannon sampling theorem is that the spectrum of the signal
doesn’t have to becontiguous – as long as you can chop it up into unambiguous pieces, you
can spread that Nyquist bandwidth into as many thin little pieces as you want. Sampling
at some frequency that is equal to the repetition rate divided by a prime number will
automatically stack these narrow bits of signal spectrum right up in the same order that
they were in the original signal, only jammed much closer together in frequency – which
is the roundabout frequency-domain way of saying that you can sample at just the right
rate, and interpret the resulting signal as a slowed-down replica of the input waveform.
So in the end, we are taking advantage of these quirks to reconstruct the original wave-
form, even though our sampling is otherwise outrageously slow.
There is a huge caveat to the foregoing approach, however: this approach only works
when the signal is steady, with a known and steady frequency. In the example I’m using
it to analyze the voltage of the North American power grid. This works well because in
North America, as in most of the developed world, you can count on the line frequency to
be rock solid at the local standard frequency. This approach would not work well at all on
any unstable power source, such as one might see out of a portable generator or from a
power line in a less well developed area.
Even in a developed country, this approach has the drawback that it simply will not ac-
curately catch transient events, or rapidly changing line levels. Transient events (such as
someone switching on a hair dryer) havetheir own spectra that adds to theseries of spikes
of the basic power; rapidly changing line levels act to spread out the spikes in the power
spectra, as does changing line frequency. Any of these phenomena can completely destroy
the accuracy of the measurement.
The bottom line is that things aren’t always as black as your reading may paint them—but
you have to interpret your theorem carefully to make sure you don’t fall into a trap.
Nyquist and Band Limited Signals
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