Nyquist theory proves it.
The Ogg people have a fantastic video explaining this. Technology Connections on YouTube has a video about it as well (the Ogg video explains things better, imo.)
It says nothing about noise, distortion, dynamic range. In these areas it is impossible to create a "perfect" DAC, although granted the best DACs are indistinguishable from perfect as far as human perception is concerned.
> If a function x(t) contains no frequencies higher than B hertz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart
I took this to mean that it's any continuous function x(t), including amplitude information. I took a quick read through the proof and that's correct as far as I can tell.
Does that not mean that "noise, distortion, and dynamic range", as they are all encoded in the continuous function that is air pressure over time, can be perfectly captured and reproduced? All you need to do is throw out all information outside of human hearing range to have no frequencies higher than B hertz, and that's all you need for perfect reproduction.
If there exists a transformation f(x(t)), then said transformation can also be captured by the same sample, can it not?
Wires without resistance, capacitance and inductance. Resistors without capacitance and inductance. Capacitors without resistance and inductance. Inductors without capacitance or resistance.
While we're at it, semiconductors with perfect linearity and so on and so on..
I'm not going to argue that audiophiles generally achieve much of these, or even that it's especially important for the perceived audio quality.. But, perfectly recording and reproducing anything is still not possible, not in audio, not in video.
Which is an argument in favor of getting the signal into the digital domain as early as possible and maintaining it there for as long as possible.
The audiophile's stereotypical myth of analog supremacy is the assumption that the opposite is somehow true. Weird, but everybody needs a hobby.
There are costs associated with digital that, like with analog, must be managed.
Just because it says 24 bits on the datasheet of the used DAC IC doesn't mean your circuit will output a voltage that represents your input with 24 bits of precision. Designing and layouting PCBs with high precision DACs on them is certainly something where a lot mistakes can be made.
But you are right, if the PCB is done correctly, the signal exiting the DAC would indeed be an truthful representation of the digital data.
That being said: nowaday reaching enough precision for even the most critical listener should be not that hard/expensive.
you are right about bit depth. a jellybean 24-bit ADC has around 8 bits of noise floor, maybe more. To improve that, you sample at an insane rate (making jitter even less of a significant factor in the final audio) and average your measurements, and you can get 2-3 more bits out of that.
my overall point is that these problems you hear from people trying to sell you audio hardware are almost all not problems at all, as far as human ears and human perception are concerned.
This particular proof assumes that the clock is perfect, with no jitter or delay between one place it is used and another. There are more ways for a clock and uses of it to vary than you probably imagine possible.
It also assumes the A/D and D/A converters produce exactly correct results, which none do. We expect them to be close enough.
So in fact the output is as close to the same as the input as anybody cared to ensure, subject to cost, component tolerances, and amount of attention spared.
In practice, it is almost always as close to exact as anybody listening cares about (and better than a phono needle could manage). But there are ways for digital systems to produce bad results, by happenstance, laxness, or just normal aging.
Few ever check the calibration of their equipment.
audiophiles' continuous search for "perfection" is pointless because human ears are not perfect. not by a long shot. I don't care who you are.
just like how your eyes lie to you, so do your ears.
48kHz is not a difficult clock rate to maintain perfectly. at all.
all you need is a DAC of regular every day commodity quality, and an equalizer if you think "warmth" is a quality of vibrating air.
So, even leaving aside gross incompetence (is that smart?), anything that could be sufficiently perturbed thermodynamically, e.g. by age or decay, could throw off your results.
The great advantage of digital electronics is that they make most of the system relatively insensitive to commonly encountered thermodynamic effects, within limits. Most such effects that exceed limits make your thing just not work anymore. (We have all experienced this.) But some can have a more subtle result, some of those without even exceeding those limits.
Fortunately, most of those involve only a few components. Those components are mostly only in your power supply, your amplifier, your DAC, and the clock circuit driving the digital stuff, including the DAC. Of those, the ones that make a difference to sound quality are mostly in the power supply and amplifier, which usually makes sound obviously bad, and the clock, which can make sound subtly bad.
Fortunately, clocks involve very few components and those are not operated anywhere near physical limits, so they rarely go bad. Furthermore, most when they go a little bad don't affect the clock's output in any way that affects what it drives.
Unfortunately, when they do, the effect on the sound you get may be hard to describe beyond "not right". Then, fortunately, you can swap out the whole subsystem with the clock in it to see if that is at fault.
So the most usual danger from a dodgy clock or other source of subtle wrongness is that, as for the case of a needle dragged down a wiggling groove, you might get used to how that sounds, and want it all the time. And that can be hard to match in another system.
Perfect reproduction means error less than some threshold e.
“True” means has enough sampling frequency and accuracy to ensure the error threshold is less than e.
Proof by contradiction: if there's no threshold, then sound fidelity can in theory be infinitely precise. But nature does not work that way - at Planck length distance breaks down, at the width of a carbon atom the resolution of a vinyl surface ends, etc.
Nowadays there exists audio files recorded in up to 96 kHz with 24 bit samples.
So sharp low-pass filters are extremely difficult to implement as analog filters with reproducible characteristics.
Even when they are implemented as digital filters, as in most modern equipment, i.e. the audio ADC uses a much higher sampling frequency and its output signal is interpolated and decimated digitally to 44.1 kHz, it is still difficult to design the very sharp low-pass filter so that it will not introduce any audible artifacts on transient signals (the sampling theorem is based on ideal non-realizable filters).
Raising the sampling frequency to be distant from the 40 kHz minimum value removes all difficulties with the design of the low-pass filters making it easy to guarantee that the digital signal reproduces exactly the analog input.
When the audio CD format has been designed, the digital technology was much less advanced and storing a lot of bits was a far more difficult task than later.
So they have made the trade-off of requiring very expensive analog low-pass filters in order to minimize the amount of bits stored on the disc and transmitted over the digital interfaces, because the digital processing and storage was even more expensive anyway.
Nowadays, it is much more convenient to just use higher sampling frequencies for audio signals.
You will however need to accept delay from the lowpass filter, which is fine for music reproduction.
So while these effects are measurable with very expensive equipment, they are in all metrics orders of magnitude better than any analog equipment.
24/192 is becoming common, and you can even find 384 kHz files for sale (transcoded from DSD recordings.)
Only if your signal is infinitely repeating in time, with no start or end. Otherwise, it's spectrum is not band limited.
Is your ADC input such a signal?
EDIT: This also applies to sigma-delta sampling. Although there you can often get away with a simpler single-pole input filter.
Not only that, your DAC needs to have infinite lookahead to reproduce the same input that was passed to ADC.
It doesn't matter.
What matters is that with a digital signal processing chain you essentially have a knob you can turn to approach mathematical perfection as closely as you wish, at least until you reach the point where the inherent limitations of your analog components begin to matter. And that point is far, far beyond where human hearing can tell the difference.
But with analog signal processing you reach the limitations of analog componentry much quicker, while you are still well within the noticeable range of some humans to tell the difference. This is because the entire chain is analog and thus no part of the chain is immune to information loss or added noise. With a digital chain, only the ADC and DAC stages require high quality analog componentry. Everything in between is pure math.
Neither process is capable of overall mathematical perfection, but digital can approach it much more closely than analog.
I just seem to take issue when Nyquist is getting inwoked. That is a purely mathematical result. It is sufficient (and I think required, too) to plug source into ADC+DAC, measure the output and show that the difference is insignificant. And much less than, say, with tape.