Nyquist/Shannon is often held up as the defining theory that proves that digitally sampled audio is flawless (OK so I over stated it, grant me that). All signals sampled below the Nyquist freq can be reproduced. People often do not understand what the theory actually states and also tend to forget the conditions required by the theory.
I try not to forget. In fact, although it's not a strict condition of Nyquist, when you have data stored as integers (as in a fixed-point 24 bit system), you also have limit dynamic range as well as bandwidth, to eliminate QD as well as aliasing.
Here is a NASA paper which explores one of the distortions that occur in sampled signals well below the nyquist cutoff. Take a moment to read:
I read the entire paper. Two excerpts stand out especially:
p.1 said:
]The effect occurs whenever the sample rate and the signal frequency are related by ratios of mutually prime integers. For cost and technical reasons, the waveform display devices omit the required reconstruction steps.
The question of course is to what extent do modern D/A converters omit or address the required steps? I don't know, but it's a good question for Lavry, Putzeys, etc.
p.22 said:
Finally, in this report, no detailed analysis has been done to see if the modulation effects around a Region II node result in extra peaks in the power spectrum indicating signal power is aliased into undesirable frequencies. No claim is made that the Region II distortions result in real signal power being lost from the sampled signal.
That (lack of) conclusion is going to make it very hard to evaluate a specific converter. Because if they had done so, and found a sideband distortion, then it would be straightforward to test a converter and see if it generated that distortion or not, and we could conclude whether its D/A reconstruction had addressed it or not.
It's tougher to test as a digitally-generated test signal will have the resulting peak modulation. In fact that's how they are telling people to test it using Excel or a computer program. So when I generate a 5/63 sample rate signal in Wavelab (4992.45Hz), I can see the peak modulation. On on FFT, it just shows that frequency (which makes me wonder why the authors didn't try that, I can't believe NASA didn't have an FFT analyzer in 2000).
A D/A/D roundtrip yields the same resulting frequency peak on an FFT, with no sideband distortion (other than some 60Hz hum and some noise), with the same peak modulation visible in the waveform. The maximum amount of peak modulation I measured on the original digital signal was 4.8%, or about -0.4dB. The incoming wave was roughly the same, 6% peak modulation, or -0.5dB.
This actually occurs with just about any frequency. I Googled around a bit more, but couldn't find much other discussion; perhaps the terminology of that paper is a little different from what developers use, which would frustrate a search.
As an aside, I don't think I ever suggested it was acceptable to push errors down to -60dBFS. Clearly it is not, and the sample files I posted will show that. We need to go at least -110dB, and ideally more. The QD errors in the digitally generated 24 bit waves were -150dBFS; in reality those can't exist because they will be dithered by A/D converter noise (they could exist at the output of a synthesizer though). Any 16 bit system needs to self-limit its dynamic range by adding dither at a sufficient level to eliminate its otherwise audible QD errors.
