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The oscilloscope at my university states both its sampling frequency and the maximum frequency that it can detect. However, the maximum frequency is just 1/10 of the sampling frequency! Nyquist's theorem states that all frequencies up to half the sampling frequency can be reconstructed.

What kind of problems are the oscilloscope constructors expecting?

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  • \$\begingroup\$ You need a more thorough answer then this, but the important component I do not see noted is your probe. What frequency can your probe handled. In most of my university classes it was not mentioned and often I used probes without a noted limit, but the probe acts as a filter also. \$\endgroup\$ – Kortuk Dec 10 '11 at 21:50
  • \$\begingroup\$ As @Kortuk mentioned, probe impedance is a big deal ;) \$\endgroup\$ – tyblu Dec 10 '11 at 23:17
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There are a few reasons for this:

  1. Nyquist's theorem applies to reconstruction of sinusoidal signals of infinite duration from jitter-free, perfectly accurate samples. Real measurement device clocks have jitter and fixed frequencies, real samples have measurement error and real signals are not infinite sinusoids.

    • Jitter is the difference between a sample's recorded measurement time and the actual measurement time. When the display overlays several periods of a signal to create a picture, jitter makes the trace spread out or smear. Other factors will do this as well.
    • The period at which a device samples is not an exact half-multiple of the original -- it's the sampling frequency, and it's not going to change in relation to the input frequency.
    • Sinusoidal reconstruction is sensitive to measurement error and noise near Nyquist's rate. I'd really rather not do any \$\frac{d(freq.)}{dV}\$ right now, but there it is. This error is reduced by averaging samples, which reduces the effective sample rate.
    • Real signals are more than a single tone. They carry information, noise, and Christmas Spirit. A single-frequency sinusoid measurement is of little value, since that was never the original signal. It'd be like expecting anyone who looks at the Orion constellation to immediately interpret a hunter with a club.
      ==> ???
  2. The measurement device (DSO) uses several staggered-clock, lower frequency parallel processes to achieve its impressive sample rate. Not all steps can be done in parallel, however, which can introduce bandwidth bottlenecks. These are largely a thing of the past in high-end equipment with the development of special-purpose ASICs, and fast GPUs and memory.

  3. Several DSO manufacturers have found it more profitable to develop and manufacture a single or only a few high-end circuits, then introduce limitations such as lower frequency clocks and anti-aliasing filters for their mid and lower-end offerings, instead of developing and manufacturing a different design for each target consumer. The 'scope you were looking at may indeed be originally designed to measure higher maximum frequencies than stated, but is somehow handicapped.

Though I am far from an authority on the subject, I have heard the "10X" rule of thumb enough times to be repeating it here: an effective sample rate of at least 10X the signal frequency is required for intelligent reconstruction and analysis. As the listed sample rate on your school's 'scope is exactly that, I imagine the actual sample rate, taking into account the above considerations, is several times higher yet, but it all boils down to 10 samples of limited jitter and measurement error.

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  • \$\begingroup\$ Fast GPUs? Did you mean CPUs? \$\endgroup\$ – Connor Wolf Dec 11 '11 at 14:41
  • \$\begingroup\$ @FakeName, I was thinking about the grunt number crunching going into good interpolation and FFT algorithms, which boils largely down to either the GPU integrated into a CPU, or discrete GPUs. I've only looked inside of half-a-dozen low- to mid-range 'scopes, but have spotted one or two discrete GPUs in most of them. I would bet that they have a bus going directly to the custom ASIC in high-end units. Of course, the development of CPUs is probably what has pushed that of GPUs, so fast CPUs would be correct too! \$\endgroup\$ – tyblu Dec 11 '11 at 23:54
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Often you want to see more than just the frequency, you also want to see the shape of the signal. For example, in digital signals it is often important to see the transitions between the two logical states.

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If one samples a signal at rate R which contains frequencies higher than F/2, those frequencies will appear as frequencies below R/2. In particular, for any integer k, any frequency f in the range kR to (k+0.5)R will appear as f-kR, and any frequency in the range (k+0.5R) to (k+1)R will appear as (k+1)R-f. Except in those very rare cases where one might actually want this behavior, it is necessary for sampling devices to filter out any frequencies above R/2. It's a lot easier to design a filter which will filter out 99% of a signal above a certain frequency while keeping 99% of the signal below a frequency that's 1/10 that, than it would be to design a filter which would filter out 99% of the signal above a frequency while keeping 99% of the signal at half that frequency.

Further, the behavior of a scope is apt to be much more intuitive if its frequency sensitivity rolls off somewhat gradually at frequencies approaching the sampling rate, than if frequencies up to some limit appear full-strength and frequencies that are slightly higher suddenly disappear.

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I don't have much theory to post, but I think that the following screenshots may help illustrate the problem. We're seeing square waves at various frequencies as seen by a Realtek ALC 268 sound card with 16 bit resolution and 96 kHz sample rate. The sound card example here is done on purpose because sound cards have very limited input capabilities.

First we have 1 kHz:

1 kHz square wave

As we can see, the shape looks pretty nice with almost no distortions.

Next we have 2 kHz: 2 kHz square wave

Here we see a bit of distortions.

In order to keep the length of this post reasonable, I'll skip some steps.

Next we have 8 kHz:

8 kHz square wave

Distortions here are obviously visible.

18 kHz:

18 kHz square wave

Even more distortions and the square wave is slowly starting to look triangular.

Here we have 28 kHz:

28 kHz square wave

We're getting spikes now.

30 kHz:

30 kHz square wave

We've lost the signal shape by now. Frequency's still good it seems.

38 kHz:

38 kHz square wave

It's even worse now. We're losing the flat bottom of the signal.

45 kHz:

45 kHz square wave

We're seeing only spikes by now.

48 kHz:

48 kHz square wave

We're at half of the sample rate now and the frequency is getting bad.

Finally we have 60 kHz:

60 kHz square wave

What we see is now barely related to what we have.

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  • \$\begingroup\$ Is this caused by the large frequency content of a square wave? A 45Hz sine wave sampled by this card could be reconstructed, at least theoretically. \$\endgroup\$ – Astrid Dec 11 '11 at 13:59
  • \$\begingroup\$ @Astrid I'm not sure. Could be. I'll do some research and see if I can find something out. Also I don't think it's actually relevant. In general case, you're using scope to see the signal which shape isn't known to you. You can of course expect a certain signal, but the scope is there to show what is actually happening and beyond a certain point the clear picture is lost. Basically same thing from tyblu's answer, section 1.4. \$\endgroup\$ – AndrejaKo Dec 11 '11 at 14:48
  • \$\begingroup\$ You bet, @Astrid. The 48kHz example is basically at the Nyquist frequency, and, if reconstructed with sinusoidal interpolation instead of linear, would show a 48kHz sinusoid. The 60kHz example would be reconstructed as an alias of 96kHz and 60kHz: a 36 kHz sine wave. \$\endgroup\$ – tyblu Dec 12 '11 at 0:14

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