DESIGN AND DEVELOPMENT OF MEDICAL ELECTRONIC ...

224 SIGNAL CONDITIONING, DATA ACQUISITION, AND SPECTRAL ANALYSISSAMPLING RATE AND THE NYQUIST THEOREMWithout doubt you have heard that according to the Nyquist theorem, a signal should besampled at twice its frequency. Yes, and no. There is no data acquisition concept that ismore quoted and less well understood than the Nyquist theorem. Let’s explore what thistheorem actually implies for proper data acquisition. Nyquist stated that any bandwidthlimitedsignal can be reconstructed from its digitized equivalent if the sample rate is at leasttwice the highest-frequency component.Signal components with a frequency above half the sampling rate are aliased and showup in a reconstruction as a component with a frequency at the difference between its realfrequency and the sampling rate. This effect is commonly seen in films of moving cars,where the wheels seem to be rotating impossibly slow, or even going backward. The aliasedrotation frequency is caused by the slow shutter rate of the camera relative to the fast rotationalspeed of the wheels’ rims. For a dramatic demonstration, pay attention to the apparentbehavior 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 componentsat frequencies above half the sampling rate. Assuming that to sample a signal ofapproximately x hertz you simply need to select an A/D rate of 2x just doesn’t cut it. Theonly ways of making sure are to supersample 2 the signal and to apply antialiasing filtering.Perfectionists insist that all signals to be sampled must be low-pass filtered to preventaliasing. In reality, however, many signals can be sampled fast enough that they are naturallylow-pass filtered by the response of the sensor or by the process being measured. Forexample, temperature changes in the body occur so slowly that sampling a temperaturesensor even once per minute suffices to eliminate aliases by supersampling. Despite this,care must be taken that power line noise or other high-frequency interference does not contaminatethe sensor signal by using appropriate shielding, differential amplification, and/ora simple RC low-pass filter. A good rule of thumb to avoid aliasing when an antiliasingfilter is not used is to supersample at a sampling rate of at least 10 times the highestexpected (unfiltered) signal component.Ten times supersampling can be unachievable when your application involves theacquisition of high-frequency signals. Here, the use of antialiasing filters is unavoidable.The ideal antialias filter would be a sharp low-pass filter that passes all frequencies belowits cutoff at half the sampling frequency and totally eliminates any components above thatfrequency. As we saw in Chapter 2, however, real-world filters do not yield a perfect stepin the frequency domain, and they will always allow through some components above theircorner frequency. This means that, in practice, sampling must happen at a rate higher thantwice the filter’s cutoff frequency. Please note that the antialiasing filter must be an analogimplementation—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 itstates that all the information needed to reconstruct the signal is provided by samplingat least at twice the highest signal frequency, it does not say that the samples will looklike the signal. Figure 5.12 shows a 48-Hz signal that is sampled at 100 Hz—fast enoughaccording to Nyquist’s theorem—barely more than twice per cycle. It is clear from Figure5.12c, though, that if straight lines are drawn between the samples, the signal looksamplitude modulated (although the signal’s frequency is correctly reproduced). Thiseffect arises because each cycle is taken at a slightly different part of the original signal’scycle. Many engineers would take the modulated signal as an indication that it was sampledimproperly. On the contrary, there is enough information to reproduce the original2Most engineers have heard the term oversampling applied to data acquisition. Although it is intuitive that samplingand playing back something at a higher rate looks better than a lower rate—more points in the waveform forincreased accuracy—that’s not what oversampling usually means. In fact, oversampling usually refers to outputoversampling and it means generating more samples from a waveform that has already been digitally recorded.