Ah geez, you had to throw in Einstein, didn't you?

As Jinga8 would point out, the answer is in fact "42". Of course, that assumes infinite resolution.

ORIGINAL: bitflipper
tarzier, I am not flaming you. I appreciate your thoughtful contributions to this and other threads. But I suspect (and I won't take it personally if you prove me wrong) you've bought in to the notion that there are exceptions to Nyquist when there are none.

No flame taken. By all means, call me out when I say something stupid. (not "if" but "when" ) But in this case you misunderstood me, sorry I wasn't more clear. UnderTow got my interpretation right, I was taking exception to the word "exactly". Whereas you were mostly talking about frequencies and sample rate, I wanted to emphasize the fact that we still can't get the original waveform "exactly" due to the quantization error which is a result of having "only" 24 bits to work with.

Thus, you can sample a 3 kHz sine wave at a 6 kHz sample rate and reconstruct it exactly except for the quantization error. And with 24 bits, the quantization error is not audible. Agree?

But in your example of a 3Khz wave sampled at 6KHz (assume a perfect reconstruction filter to keep it simple), you DO get an exact copy, because anything other than a 3KHz sine wave would have "illegal" harmonics in it. The key to the Nyquist theorem is that you CANNOT have frequencies higher than half the sample rate. There can be no quantization error because all of the harmonics have been removed.

If that 3KHz wave was square, for example, it could not be reconstructed from 6KHz samples. It would no longer be a square wave because the harmonics would have been removed. You would indeed need a higher sample rate if that was your intent. But if we were only interested in frequencies up to 3KHz, and no higher, then we'd still be OK because for all practical purposes that's what we've got.

Now consider a 20KHz wave sampled at 44.1KHz. If the original wave was square, then we would indeed not be able to either record or reconstruct an exact copy. But by definition, frequencies above 20KHz are irrelevant because they are inaudible. The upper harmonics would have been removed by the anti-aliasing filter on the way in, and we would record a sine wave -- and that sine wave would be accurately reconstructed on playback.

So if your goal is to accurately record frequencies up to 20KHz, Nyquist says unequivocally that 44.1 works. The real question in all of this is: is it necessary to record frequencies above 20KHz?

ORIGINAL: bitflipper

But in your example of a 3Khz wave sampled at 6KHz (assume a perfect reconstruction filter to keep it simple), you DO get an exact copy, because anything other than a 3KHz sine wave would have "illegal" harmonics in it. The key to the Nyquist theorem is that you CANNOT have frequencies higher than half the sample rate. There can be no quantization error because all of the harmonics have been removed.

If that 3KHz wave was square, for example, it could not be reconstructed from 6KHz samples. It would no longer be a square wave because the harmonics would have been removed. You would indeed need a higher sample rate if that was your intent. But if we were only interested in frequencies up to 3KHz, and no higher, then we'd still be OK because for all practical purposes that's what we've got.

Don't think square waves or harmonics, think noise. Assuming a properly dithered system, the inaccuracies in the copy, also called distortion, is noise.

UnderTow

Ah, yes. Noise is always added, even if it's above 20KHz.

I think we've just about flogged this topic to death by now. It's been fun.

For me the bottom line is I will continue to sample at 44.1, only now it will be with a clear conscience.

I don't know why I'm always drawn to this debate. Been in almost every one of 'em on this board... and quite a few on other boards. I've learned tons. Bought Apogee converters for great AD/DA filters and now almost any sample rate sounds the same in my studio. I will say that I hear a slight improvement at higher sample rates in my studio, but I don't hear that improvement on the final 44/16 CD. It's not worth the cost in resources on my DAW and I HATE waiting for bounces and exports at high sample rates.

And then there's this:

http://forum.cakewalk.com/fb.asp?m=53631

Try that test in your studio. This null test shows that it is possible for a 44.1khz file to contain the same AUDIBLE information as the 88.1khz file. The data that is not nulled against the imported 44.1khz file is still there, but either your system can't reproduce it, or you can't hear it, or both!

The quality/ability of the converters at different sample rates is a different issue.

Regarding Nika Aldrich, I recall that he posted here in a lengthy thread on Dither about a year ago, which some of you may have even participated in. I remember him entering the thread midway or so, writing quite a bit on the subject and from that point on some of us just took to reading and asking him questions.

I don't think I've seen him around here since.

ORIGINAL: bitflipper
But in your example of a 3Khz wave sampled at 6KHz (assume a perfect reconstruction filter to keep it simple), you DO get an exact copy, because anything other than a 3KHz sine wave would have "illegal" harmonics in it.

Not quite. If you had an infinite number of bits, then you do get an EXACT copy. But you don't have an infinite number of bits. And since the sine wave that is exactly half the sample rate is even trickier to conceptualize, let's go simpler. a 1 kHz sine sampled at 6 kHz--although the numbers aren't really important right now.

Let's assume that it is a perfect 1 kHz sine wave, no extra frequencies/harmonics/noise. You then sample it at 6 kHz. Most likely the amplitude of the sine wave at the given moment in time represented by the sampling clock (not the peak amplitude, but the continuous amplitude) will not correspond exactly to one of the quantization steps of the digital representation. (not one of the sampling steps, one of the quantization steps--bits, not sample rate) The amplitude has to be rounded up or down to the nearest sample value. That is the quantization error and it then becomes a part of the sampled signal. It is a form of noise, a form of distortion, a form of harmonics--non-harmonic harmonics. But I'll call it quantization error.

The key to the Nyquist theorem is that you CANNOT have frequencies higher than half the sample rate.

Agreed.

There can be no quantization error because all of the harmonics have been removed.

Disagreed. Quantization error has nothing to do with harmonics of the signal. It has everything to do with the signal itself. Even in the case of the perfect 1 kHz sine sampled at 6 kHz (which has no other harmonics, yes?) there will be quantization error because of the rounding of the amplitude up or down to the nearest sample value. Unless you have infinite bits in which case there's no need to round either way.

Now forgive me for snipping out the rest of your post that I agree with.

...
The real question in all of this is: is it necessary to record frequencies above 20KHz?

I had the pleasure of learning digital audio from Dr. Tom Stockham, and he did all his final work at 16 bit 50 kHz. He thought the 44.1 kHz CD rate was a big mistake due to the difficulty of filter design, but 50 kHz should be sufficient for digital audio. I tend to agree, but since I can't do 50 do I round up to 88.1 or down to 48...? (actually with oversampling converters, the filter problem has been solved pretty well. So I record at 44.1 or 48)

I hope you don't think I'm beating this to death. I'm just trying to clear up misunderstandings. Whether I'm misunderstanding you, or vice versa.

The real question in all of this is: is it necessary to record frequencies above 20KHz?

Based on what has been discussed so far it would be save to answer, it depends (it depends on the quality of your converters, media format, if you own equipment capable of capturing and reproducing these frequencies, available system resources, etc). But now I have another question for you guys: Wouldn't it better to import a stereo file at the original sampling rate to a mastering project of higher sampling rate and bit depth instead of upsampling the wave file for mastering?

I don't understand. Why would your mastering project not be at the same sample rate as the imported stereo files?