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is it true, oscilloscope must reach at least 4x observed freq?

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robert.rozee:

--- Quote from: Fungus on September 13, 2022, 12:58:10 pm ---Fun fact: Your ears can't tell the difference between a 10kHz square wave, a 10kHz sawtooth wave and a 10kHz sine wave. They all sound exactly the same!

--- End quote ---

i think your ears might need checking...


cheers,
rob   :-)

wasedadoc:

--- Quote from: robert.rozee on September 13, 2022, 12:38:12 pm ---
--- Quote from: Fungus on September 13, 2022, 12:17:41 pm ---... to infinity.

--- End quote ---

the device generating the square wave did not make use of an form of infinity, either theoretical or actual.

now a square wave can be represented by an infinite sum of sine waves, but it is not made out of them. it is made with just a little fellow switching a light switch: on... off... on... off... on... off... very fast.

the infinite sum of sine waves you talk about only exists in one's imagination.


cheers,
rob   :-)

--- End quote ---
A perfect square wave can be represented by that infinite sum.  In practice no generator can make a waveform that has zero risetime.  It may be close enough to zero to make no engineering difference in the situation under examination but the risetime will be non-zero and in the frequency domain that corresponds to no harmonics above some point.

nctnico:
No. Google 'Gibbs phenomenon'

switchabl:

--- Quote from: Fungus on September 13, 2022, 12:55:55 am ---Imagine a sine wave that you're sampling at exactly 2x frequency.

a) You might sample the signal exactly on the peaks/troughs in which case you'll be fine.

b) OTOH you might sample it exactly on the zero-crossing points, in which case you'll see nothing at all.

You can also get every possible value in between (a) and (b), it's just dumb luck.

If you sample at 99.99999% of Nyquist you'll drift slowly between (a) and (b) and see the amplitude varying on screen ("AM effect").

2.5x Nyquist is the minimum to avoid this AM effect.

--- End quote ---

The familiar version of the sampling theorem as published by Shannon is his seminal 1948 and 1949 papers assumes a function f that "contains no frequencies higher than W cps". At first sight this might seem to include your counter-example (sine at frequency W), however Shannon is writing for engineers, not mathematicians, so he glosses over some details. The proof has an implicit assumption that f is square-integrable which in fact excludes signals containing any pure sines. Note that there are ways to generalize this using more advanced mathematical machinery but you may or may not be content to just claim that there are no pure sines in the real world anyway.

On the practical side, it is also important to realize that the sampling theorem is inherently infinite time. It assumes that you have sampled the signal for an infinite duration. If you take a finite portion of a band-limited signal you are effectively applying a window function and thereby broadening the spectrum beyond its original bandwidth.¹ Together with any sampling jitter this will give you a hard limit that is higher than 2x but probably not by much.

More importantly, the sampling theorem also assumes you are using all the samples for reconstruction all the time but you probably aren't because a) it would use a lot of processing power and b) it doesn't work for real-time applications (the ideal sinc-filter is non-causal and has infinite delay). Depending on the number of taps you use, frequencies close to the Nyquist limit will still alias close to DC. This is what you observe. However, this is a trade-off and the limit of 2.5x is an arbitrary choice of the implementation you are using. You can get a lot closer to 2x if you use more than a couple of samples for interpolation.

¹ It will technically no longer be band-limited at all (you cannot have a signal that is both time- and band-limited).  But let's ignore that here, as long as it drops below our noise floor, we don't really care.

mawyatt:

--- Quote from: tggzzz on September 13, 2022, 07:57:49 am ---Precisely.

A good practical example is the "Tayloe mixer" found in SDR receivers.

Those irritate me. 40 years ago  I built an 4kHz bandpass filter with a Q of ~4000 using 10% capacitors. It was based on the N-path filter concept I found in a 1950s BSTJ paper. The concepts were so counterintuitive that I had difficulty explaining them to analogue/RF engineers.

They continued to fascinate me, I felt sure they would be useful in other ways, but had no professional reason to use them. After retiring I was just gearing up to playing with them again, and found Tayloe had "beaten me to it". Rats.

--- End quote ---

Interesting, we employed the N-path filter concept back in ~1980 to pick out signaling "tones" displaced within multiple 4KHz bandwidth "sections" displaced at microwave frequencies, the filter had a bandwidth of 10Hz. These multiple (thousands) 4KHz sections were down-converted using Nyquist as our "friend" rather enemy, employing Microwave RF Downconversion by means of Nyquist sub-sampling where we knew exactly where each multiple displaced 4KHz section would be and where the tones should be.

So in a way Nyquist can be helpful but not in the DSO discussions here!!

It's amazing that it took another ~35 years to discover the N-Path Mixer, or Polyphase Mixer, which is fundamentally the N-Path Filter without the up-conversion back end section :o

Best,

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