SIGNAL CONDITIONING, DATA ACQUISITION, AND SPECTRAL ANALYSIS - Design and Development of Medical Electronic Instrumentation - page 224

SAMPLING RATE AND THE NYQUIST THEOREM

Without doubt you have heard that according to the Nyquist theorem, a signal should be
sampled at twice its frequency. Yes, and no. There is no data acquisition concept that is
more quoted and less well understood than the Nyquist theorem. Let's explore what this
theorem actually implies for proper data acquisition. Nyquist stated that any bandwidth-
limited signal can be reconstructed from its digitized equivalent if the sample rate is at least
twice the highest-frequency component.
Signal components with a frequency above half the sampling rate are aliased and show
up in a reconstruction as a component with a frequency at the di ff erence between its real

frequency and the sampling rate. This e films of moving cars,

where the wheels seem to be rotating impossibly slow, or even going backward. The aliased
rotation frequency is caused by the slow shutter rate of the camera relative to the fast rota-
tional speed of the wheels' rims. For a dramatic demonstration, pay attention to the appar-
ent behavior of the wheels of a speeding stagecoach in virtually any old western movie.
The way of preventing aliasing is to ensure that there are absolutely no signal compo-
nents at frequencies above half the sampling rate. Assuming that to sample a signal of
approximately x hertz you simply need to select an A/D rate of 2 x just doesn't cut it. The
only ways of making sure are to supersample 2 the signal and to apply antialiasing

ff ect is commonly seen in fi

fi filtering.

Perfectionists insist that all signals to be sampled must be low-pass filtered to prevent

aliasing. In reality, however, many signals can be sampled fast enough that they are natu-
rally low-pass

fi

filtered by the response of the sensor or by the process being measured. For
example, temperature changes in the body occur so slowly that sampling a temperature
sensor even once per minute su

fi

ces to eliminate aliases by supersampling. Despite this,
care must be taken that power line noise or other high-frequency interference does not con-
taminate the sensor signal by using appropriate shielding, di ff erential ampli fi cation, and/or

a simple RC low-pass fi filter. A good rule of thumb to avoid aliasing when an antiliasing

fi filter is not used is to supersample at a sampling rate of at least 10 times the highest

expected (un ltered) signal component.

Ten times supersampling can be unachievable when your application involves the
acquisition of high-frequency signals. Here, the use of antialiasing

fi

fi filters is unavoidable.

The ideal antialias fi filter would be a sharp low-pass fi filter that passes all frequencies below

its cutoff at half the sampling frequency and totally eliminates any components above that

frequency. As we saw in Chapter 2, however, real-world

ff

filters do not yield a perfect step
in the frequency domain, and they will always allow through some components above their
corner frequency. This means that, in practice, sampling must happen at a rate higher than
twice the

fi

fi filter's cutoff ff frequency. Please note that the antialiasing fi filter must be an analog

implementation—it is too late to use digital filtering once you have done the sampling.

The other common misunderstanding about the Nyquist theorem is that although it
states that all the information needed to reconstruct the signal is provided by sampling
at least at twice the highest signal frequency, it does not say that the samples will look
like the signal. Figure 5.12 shows a 48-Hz signal that is sampled at 100 Hz—fast enough
according to Nyquist's theorem—barely more than twice per cycle. It is clear from Fig-
ure 5.12 c , though, that if straight lines are drawn between the samples, the signal looks
amplitude modulated (although the signal's frequency is correctly reproduced). This
e

fi

erent part of the original signal's
cycle. Many engineers would take the modulated signal as an indication that it was sam-
pled improperly. On the contrary, there is enough information to reproduce the original

ff ect arises because each cycle is taken at a slightly di ff

2 Most engineers have heard the term oversampling applied to data acquisition. Although it is intuitive that sam-
pling and playing back something at a higher rate looks better than a lower rate—more points in the waveform for
increased accuracy—that's not what oversampling usually means. In fact, oversampling usually refers to output
oversampling and it means generating more samples from a waveform that has already been digitally recorded.