When acquiring low-data-rate signals such as temperature, engineers sometimes forget that low-data-rate sampling does not guarantee an uncontaminated recording. Regardless of the sample rate employed in digital data acquisition, signals entering the sampler above half the sampling frequency will be reflected and added into the band below to distort and corrupt the signal of interest.
The Nyquist Theorem states that to accurately reconstruct a signal recorded as a digital sequence, it must have been sampled at more than twice the highest frequency component in the signal. Sampled energy above the Nyquist frequency (defined as half the sample rate) cannot be distinguished from sampled energy below the Nyquist frequency.
Figure 1 illustrates how two different, equal-amplitude waveforms sampled at 1 Hz produce indistinguishable results. In this case, a 0.2-Hz sinusoid is 0.3 Hz below the Nyquist frequency, while a 0.8-Hz sinusoid is 0.3 Hz above it. A 1-Hz sampling of these two sinusoids produces identically sampled data.
Sampling data can be understood graphically as an effect that separates the energy across the entire frequency spectrum into intervals of half the sample rate, then mirrors and adds (aliases) this information into itself. When reconstructing the signal from this sampled information, an infinite number of replicas of that information is created and mirrored at half sample rate intervals. Figure 2 shows the frequency domain view of a sampled signal after analog reconstruction that illustrates the mirroring effect across integer multiples of the Nyquist frequency. Although a well-behaved (magnitude) spectrum is shown, one cannot know from the reconstructed waveform alone whether this data has been corrupted by energy aliased from the frequency spectrum above the Nyquist frequency.
Is signal conditioning required when using a sensor that is naturally band limited? From the model below, it is clear that noise entering the system before or during sampling can corrupt the data acquisition process even if the sensor is well conditioned.
If “In-band” is defined as the desired signal band extending up to Nyquist frequency (half the sample rate), and “Out-ofband” as all other energy, then corruption of the In-band signal can result from:
- Out-of-band signals coming from along the signal path, either from the analog sensor or from noise picked up from the environment.
- In-band or out-of-band energy causing clipping, analog distortion, or modulation.
- In-band noise obscuring the desired signal.
Installing properly designed anti-aliasing filters (limiting the maximum frequency content to less than half the sample frequency) just before the data sampler (A/D converter) will effectively combat corruptive, out-of-band noise entering along the signal path. While this takes care of corruption arising from item 1 above, additional protective measures may be needed to prevent data corruption arising from the non-linear phenomena addressed in item 2, and in-band noise from item 3.
All electronic components have a limited range over which they remain linear. Beyond that, most passive analog components gradually will become more non-linear. On the other hand, active components such as A/D converters and opamps become non-linear very rapidly when their limits are exceeded.
A-to-D converters (data samplers) are integrated semiconductors that require power to operate. These circuits use reference voltages and timing circuits that must remain stable in order to get a good measurement. Two of the most common problems associated with problematic A/D converters are clipping caused by over-ranging the A/D converter (or opamp circuits), and sample modulation caused by a noisy power supply, poor circuit layout, or unstable clock.
Using Fourier analysis, a clipped sine wave can be decomposed into a sine wave (of the original frequency) plus harmonics of that sine wave. In essence, clipping redistributes some of the energy from the original (fundamental) sine wave into new, higher frequency sine waves at integer multiples of the fundamental. One can infer from earlier discussions that when a clipped signal is sampled, aliasing will occur. Therefore, the harmonics of the fundamental above the Nyquist frequency are folded back into the base-band signal, corrupting it. The only solution is to ensure that input signal to the converter never exceeds the limits of the converter.
The multiplication of two sinusoids produces both a sum and difference of those sinusoids. So, if one multiplies (or modulates) a band of information, primarily encompassing frequencies from 0 to 0.3 Hz with a 1-Hz sinusoid, the resulting band of information could look like Figure 2. Hence, aliasing is a form of modulation.
If sampling a temperature measurement near a strong energy source like engine ignition, an RF or microwave transmitter, or power lines, these sources can couple into electronics. Transient sources like automatic door openers or nearby machines also should not be dismissed. To be sure of immunity, an acquisition system should be tested with dummy sensors of equivalent impedance to see how much contamination there is, compared to what is expected.
This article was written by Cal Swanson, Senior Principal Engineer, at Single Iteration, a division of Watlow Electric Mfg. For more information, visit http://info.ims.ca/5654-121.