All digital oscilloscopes digitize analog waveforms in one way or another. And every EE knows the old rule of thumb: when you digitize an analog waveform, you’ve got to take at least two samples in each cycle of the highest frequency present in the incoming signal. If you fail to sample often enough, the scope creates aliases—low-frequency components in the digitized data that don’t exist in the original signal and can’t be removed from the data. Well… many EEs are wise enough to know that, sometimes, shopworn rules of thumb don’t apply. First off, the statement as it appears above is ambiguous and its glib wording encourages unwarranted assumptions. It leads most EEs to conclude that sampling at more than twice f –3-dB (the instrument’s –3-dB frequency) is the bare minimum. However, intentional undersampling can be a shrewd system-design strategy; with it you can shift or translate downward the frequency of a repetitive signal in a manner closely related to mixing or heterodyning in the analog domain.
As to aliasing, if the sampling rate, fS, just barely exceeds 2*f –3-dB, you may not see aliases if the original signal’s energy content at f –3-dB is low enough. Yet with the same filtering and sampling rate, aliasing can be a problem if the signal’s energy content at f –3-dB is sufficiently high. In other words, to avoid problems, you have to know the frequency characteristics of the signals you will be measuring. And you also need to know the gain-vs-frequency characteristics of the amplifier and other signal-conditioning circuits that precede the sampler and digitizer.
For many years, the manufacturers of digital scopes seemed to agree that, for best results at affordable cost, the maximum sampling rate should be ~10 times f –3-dB. However, as customers pressed for ever more bandwidth—and more bandwidth per dollar, the fS/f –3-dB ratio has repeatedly been reduced to the point where a few scopes now on the market spec fS/f –3-dB=2. Part of the reason this reduction has worked is that the scope manufacturers have been refining their waveform-reconstruction algorithms to produce smoother and more plausible displays from datasets that contain fewer samples per cycle.established that in certain cases, you can safely operate a scope using a sampling-rate, fs, to signal-frequency, fSIGNAL, ratio that is less than 2, even though a ratio of just 2 is commonly accepted as the bare minimum. Ratios of ~2 can be acceptable if the amplitude of input-signal components whose frequency is in the vicinity of 0.5fS are no larger than 0.5 LSB (where LSB is the amplitude of the digitizer’s least-significant bit— 2-8Vfs, that is ~0.4% of the digitizer’s full-scale input-voltage span for an 8-bit-resolution digitizer). If you use a lowpass filter ahead of the scope input to reduce the amplitude of the 0.5fS components to the point where they don’t cause noticeable aliasing and if, absent the filter, the amplitude of the 05fS components would drive the output of the scope’s input amplifier to fully span VFS, you will want to attenuate the 0.5fS components by a factor of 29=512. If the filter rolls off at 6 dB/octave, the filter’s -3-dB frequency should then be fS/512. In other words, if you start out with a signal that contains large-amplitude components at 0.5fS and you use a simple one-pole filter to ensure well-behaved transient response, you are going to have to accept a -3-dB frequency of only about 0.2% of fS. This rather extreme example makes the point that the -3-dB frequency must sometimes be only a tiny fraction of the sampling rate.
In the next part of this series, we’ll discuss the pros and cons of a technique—whose use is optional on some scopes—that allows sampling of repetitive signals at extremely high effective rates. The different manufacturers have different names for this technique, which at least one company calls RRS (random-repetitive sampling). The inclusion of RRS had fallen out of favor for several years because the major scope companies felt that real-time sampling replicated the captured waveforms so well that RRS’s added cost wasn’t justifiable. Now, it may be making a comeback.
When you select RRS (random repetitive sampling) in a scope that provides an RRS mode, your instrument operates at the opposite end of the sampling-rate spectrum from scopes that, at their fastest acquisition rate, take just incrementally more than two samples/cycle of signal components at the -3-dB frequency. RRS allows you to take hundreds and even thousands of samples/cycle of signal components at—and even above—the -3-dB frequency. This high effective sampling rate can make visible signal details that you may not be able to see at the scope’s fastest real-time sampling rate, even if, at that rate, the scope takes 10 or more samples/cycle of components at its -3-dB frequency. That’s the good news; the bad news is that RRS works only with repetitive signals and not all signals repeat. Nevertheless, despite having to be repetitive, the signals need not be periodic as long as the scope can trigger reliably on them.
Nearly all digital scopes use reconstruction filtering to connect the dots that represent individual samples. Without reconstruction filtering, if the sampling is sparse, a term that you can apply to waveforms sampled fewer than 10 times per cycle of their highest significant frequency component, figuring out what the dotted display is trying to show you can be frustrating and daunting. Connecting the dots, even with straight lines, can make it significantly easier to interpret the display. Connecting the dots with curved line segments can improve the picture further. But with the RRS display’s hundreds or thousands of samples per cycle, the dots usually overlap, making even straight-line connections unnecessary.
There is much, much more to the display technology of modern digital scopes, and understanding it—and all of the other nuances of DSO design—can help you to use the instruments with maximum effectiveness. It is our goal—over the next months, and maybe longer—to cover these nuances in some depth. However, this isn’t supposed to be a one way street. We’ll understand what needs clarification only if you tell us. Correct out errors. Suggest topics to cover. Furthermore, we don’t intend to restrict the discussion to scopes. You use many other kinds of instruments: waveform. generators and spectrum analyzers, to name only two, and we hope to spend some time on them as well. But we need your feedback to make this series of blogs as effective as we believe it can be.
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