Claude Shannon emphatically disagrees. When sampling at twice the highest audible frequency they are mathematically equivalent.
That theorem also has some technical constraints that actual noise does not match, such as band limited and Fourier transforms that are zero outside some bound. Actual music does not match those. It can be approximated with those.
For example, suppose you have a pure sine wave, sample it with enough density to make it mathematically reproducible, then play back those quantized samples on a piezo making square waves - it sounds pretty good (and can be indistinguishable to most ears), but it is not the same waveform.
As you know it's then passed through an ADC and stored as a sine wave, cause no one is mastering inaudible square waves on a reel for kitsch value.
>It's then passed through an ADC and stored as a sine wave.
After an ADC it is not stored as a sine wave. It's stored as quantized values, thus the 'D' in Analog-to-Digital-Converter.
>cause no one is mastering inaudible square waves on a reel for kitsch value
Pretty much all audio processing is now done digitally, which is the same as square waves - each jump in discrete digital value is a step function. When you push it through properly engineered output devices the squareness is smoothed somewhat, but still has frequency ringing because it is square edged.
Take a good speaker, take something that can grab audio spectrum far beyond audible, and look at the output. There is stuff far outside human hearing coming from the speaker because of these square waves. Naively, this is because the Fourier transform of the square waves have high frequency ringing, and this is because the playback has sharp edges. (See this [1] for some related info for example).
Also Nyquist-Shannon is about frequencies, not about amplitudes, which are also quantized. Physical devices making sound have a (up to quantum level) continuum of possible amplitudes. Quantization necessarily loses this forever. For example, take A0, the lowest standard piano note, freq ~27.5. Sample a pure sine wave at 60 hz, in 8 bit audio. Now record this tone going from no sound up to very loud, very smoothly, over some time. The 8-bit audio will necessarily have less smoothness to it, since it is 8 bit audio. It perfectly matched your Nyquist-Shannon claim, yet it fails to reproduce what you hear. Take 16 bit audio - better. Take 32-bit or floating point audio, better again. And so on.
I agree that really well engineered systems can push the errors outside human hearing, but to claim they reproduce the same signals is incorrect.
The gist of this is: to get the most accurate reproduction of the original, merely sampling at 2x the top human freq is no where near state of the art.
An counter-intuitive example: to get the best quality and most accurate output, one needs to add noise to the input. The reason is that due to quantization, if some input signal is between possible quantized output values, adding noise (usually Gaussian, of std dev ~sqrt(step size)) makes that signal trigger both high and low quantized values in proportion to the intermediate value, making the output playback the step square waves as close to approximating the original as possible. The entire field is full of stuff like this.
For reference, I've worked on stuff like this on and off for decades, having written libraries used by others, designed high end audio simulation software (think raytracer for audio in physical settings to help design stadiums), written articles, and produced hardware in a company I own. I am quite familiar with all sorts of audio processing.
[1] https://electronics.stackexchange.com/questions/156197/can-c...
Maybe back in the 1980s on some of the early consumer digital equipment; but those problems were solved in the early 1990s by oversampling in the DAC, and then using some basic analog filtering far above the human hearing range.
IE, a consumer DAC will oversample a 44.1khz signal to (example) 705.6khz in the digital domain; and then use a very gentle analog lowpass filter to deal with the ultrasonic distortion. At that point the difference between the original analog signal and the one coming from the DAC is approximately as accurate as if there was no DAC in the first place. (Granted, some people can hear up to 27khz, which is why some people like 96khz sampling rates.)
You're writing about the sampling end. I said the physical speaker creates high frequencies on playback based on material properties of the device - and I gave a decent reference where you can read the discussion on it.
No amount of filtering at the sampling end will remove physically created noise due to the physical playback membrane that moves air to create sound waves.
I am fully aware of using bandpass filters during sampling. I use them all the time to remove things I don't need before doing things like wavelet transforms to pull music information out of the result. And I often design things up front based on the physical playback mechanism if I know it ahead of time. Or if the hardware (such as embedded devices) will only sample at certain rates, or certain bit depths. Knowing as much about the entire audio path up front helps design each and every piece of the complete signal path.
Here's a simple example: basic speakers are an electromagnet coil - apply voltage V and the membrane jumps to a position. Different values for V make different positions.
Quantized playback, going through an DAC, will create distinct voltage levels. 8 bits will give 256 such levels. 16 bits, 65536 levels.
When that hits a speaker, the speaker membrane jumps to that level. There is some noise with inertia and momentum and point to point, but the end effect is the same - the speaker trying to make a square wave edge. There is no uniformly smooth movement from position to position - only jumps.
This can be seen by putting a mirror on the speaker, and bouncing a laser off it to a large wall, and record the wall in high speed - you see jumpy movement. Fiddle sometime with a pure tone sent at various bit depths to a speaker and watch the laser.
Now, these movements create frequencies in output not in the original analog signal, not in the digital signal, but purely as a physical artifact. And they depend on the playback device - all sorts of work and research is spent on speaker tech, materials, reproducible construction, and on and on, to make the output physical waveform as uniform and smooth as possible over all the possible input voltage jumps and frequencies desired. But all are imperfect, similarly to how all physical lenses (well, except 1-1 and flips) must distort images. It is all about the tradeoffs.
Chris Montgomery (of Ogg / Speech / xiph.org & RedHat) did a series of videos going into this in considerable depth. I encourage you to watch them.
Again, in different words:
Nyquist assumes infinitely precise samples. It's math, not computer sampling. This never happens, since samples are quantized. Having samples at the proper number of Hz is useless unless enough precision is there, and non-infinite precision implies the original signal is never reconstructable. We're dealing with computers, not the real numbers.
Take a pure sine wave. Sample it mathematically. Quantize those values. Now what sin value reproduces those quantized values? None. Never. They are rational numbers - it is mathematically impossible to fit a single sine wave to them, since sine is a transcendental function. End of story.
Sine is a transcendental function, so rational inputs (other than 0 in the case of sin) do not (except for input 0 for sin) give rational output. So you cannot sample it to perfection with a digital device. You can approximate it. That approximation matters. Digital sampling takes rational input deltas (sampling rate) and necessarily obtains imperfect samples, since you quantized the actual value of a sine wave. So Nyquist fails.
Yes, for a bandlimited signal of a given frequency, Nyquist lets you reconstruct that frequency given infinite precision. This NEVER happens in practice, since it assumes infinitely precise samples. Montgomery ignores this (and a host of other issues - he's at level 2 of a 100 level tower. People at level 0 see his videos and assume there are only 2 levels to the tower). Bitdepth matters. Nyquist does nothing about amplitude quantization, which is needed. It ignores the path to reconstruction - Nyquist only applies to a perfect (not floating point or integer) reconstruction of the signal. Nyquist does not deal with the fact that the quantized values, when pushed to any physical device used to reconstruct audio, is more like stairsteps than pure sine waves.
Most physical devices performing playback are more stairstep than smooth sine values, so they are not reconstructing the input signal - another issue that matters. Input signals are (nearly, up to quantum level) infinitely precise in amplitudes - output devices tend to be more quantized.
Please read the thread I wrote and think through it. I posted a link to a good discussion, I posted a simple experiment or two you can do, I posted (here) a simple mathematical exercise showing that Nyquist fails for this.
2. DAC output is not stairstep. There’s an analog reconstruction filter that filters out the ultrasonic components, thereby getting back the original smooth waveform. You should read up on DAC design.
In [0], at time 23:50, he states that there is only one band limited signal that passes through each sample point - this requires infinite precision for reasons I explained elsewhere. It's a theoretical idealization that makes math easier, just like frictionless physics, using the ideal gas law, approximating sin(x) by x for small x, and so on. It's nice, but it's not what happens in practice.
> DAC output is not stairstep. There’s an analog reconstruction filter that filters out the ultrasonic components, thereby getting back the original smooth waveform. You should read up on DAC design.
I wrote that the physical playback device, such as a speaker, adds ultrasonic noise due to simple physics. If this were not the case, there'd be very little need for such variety in speaker costs - they'd all just magically reproduce the perfect waveform. But they don't.
As to DACs, they most certainly do not work as you claim - and it's demonstrably impossible as I explained since sine is a transcendental function, and you lost information needed since you don't have infinite precision samples.
Let's pick a common DAC, say an Analog Devices AD5780, datasheet here [1]. Page 18 has the circuit diagram for - a resistor bank. That's a stairstep (minus some physical noise at the transitions). If you look over the previous pages (Vout is the out signal you want to look at), it clearly outputs a fixed, discrete voltage for a fixed input. Every common chip does this.
Care to point to a chip that guarantees "getting back the original smooth waveform"? I'd love to see the datasheet on such a device.
One can design or buy DACs for $500, $1000, and more, but these are not what most people use. And even these exist in such variety because, as you guessed it, they cannot reproduce perfectly original waveforms, otherwise there'd be no need for such cost or variety. They all make tradeoffs and assumptions to cater to specific needs. Sure, they are very good, but they don't reproduce "the original smooth waveform".
As to analog filters providing magic, they too are not what you think - they're making assumptions and deviate from perfection with tradeoffs. I'd guess Analog Devices engineers can say it better than I : "A reconstruction filter is used at the output of the DAC to attenuate image frequencies. However, a physical filter cannot be implemented with ideal stop band rejection extending out to infinite frequency. This is due to component parasitic effects as well as the physical limitations of printed circuit board layout" [2].
The perfect filter for the output is a sinc (yes, with a 'c'). Anything else simply is not the output you desire. But the problem with sinc is it has infinite support, so is not usable in practice. Filtering on a DAC is therefore a finite support approximation to the sinc filter, and introduces error. Read here [3] for more, or look at a textbook on it. [3] also explains quite clearly that in reality none of this ends up exact as you claim.
Your claims are idealizations that are not met in reality.
"You should read up on DAC design". Indeed.
So, have the spec sheet for this perfect reconstruction DAC you claim exists? I'd like to see one.
[0] https://www.youtube.com/watch?v=cIQ9IXSUzuM
[1] https://www.analog.com/media/en/technical-documentation/data...
[2] https://www.analog.com/media/en/technical-documentation/appl...
2. The analog reconstruction filter doesn’t need ideal stopband rejection. An oversampling DAC [0] pushes the image frequencies far beyond the passband, so a gentle analog filter is sufficient to suppress it to the noise floor.
3. Noone said anything about mathematically perfect reproduction. Of course there is quantization noise. Of course there is clock jitter. And so on. But the cumulative effect of these is still way below the detectability threshold of the human ear. And the noise floor of a digital system is still way lower than what’s achievable with an analog one.
[0] https://www.analog.com/media/en/training-seminars/tutorials/...
I don't think you're using the word "original" correctly.
And now we agree - the reproduction is not perfect. That is exactly what I wrote to begin with.
>Noone said that the anti-imaging filter was literally built into the DAC package.
That's right - I did not write that, so don't imply I did. I demonstrated that the DAC chip does not do it, then demonstrated that any possible outside filter cannot do it. This is counter to your claim, but matches what I originally wrote.
>And the noise floor of a digital system is still way lower than what’s achievable with an analog one.
Not true - both can be run easily down to thermal background radiation noise floor, then both need things like liquid cooling and other techniques if you want to go below (physics experiments run into this stuff and have to push both digital and analog signal noise floors vastly below probably any audio systems).
There is no inherent noise floor for either system.
This is silly. Every system has a current noise floor based on whatever the thermodynamics of the system are. If you can push the artifacts of your output below that then they will be indistinguishable from the noise. That’s the point.
Each fixed system has a noise floor, but there is no noise floor forced on all systems. You can design a system to have as low of a noise floor as you desire, digital or analog. Physics experiments routinely do it to detect extremely rare events.
As to thermo, it's almost like you ignored that I already wrote this, which I repeat: "both can be run easily down to thermal background radiation noise floor, then both need things like liquid cooling and other techniques if you want to go below". In common audio, thermal noise is often the biggest component of the noise floor in a system.
Hence the point to cool them. There is no floor except zero if you want to engineer such a system. And it works for analog and digital.
>If you can push the artifacts of your output below that then they will be indistinguishable from the noise.
Nope, you can signal below the noise floor using many techniques, such as spread spectrum techniques. This is used routinely for military and other covert operations. As usual, this is even Wikipedia level knowledge: "Signals that are below the noise floor can be detected by using different techniques of spread spectrum communications, where signal of a particular information bandwidth is deliberately spread in the frequency domain resulting in a signal with a wider occupied bandwidth." [1]
Heck, even everyday GPS signals are vastly below the noise floor, yet we use those signals all the time with cheap hardware. "The strength of received GPS signal transmitted from a satellite to the ground users is about −157 dBm below the noise floor of −138.5 dBm" [2]. This is all common knowledge among people doing even basic signal processing. Tons of everyday tech signals below their respective noise floors. It's not a rare thing to do.
You make many claims that are not true.
So, as to your claim that "the noise floor of a digital system is still way lower than what’s achievable with an analog one" - have a citation in a paper stating this? Have a theorem in a textbook? I'd like to see why you make this claim.
In this context, we were talking about digital audio & the perception thereof by human listeners. Spread spectrum signal processing techniques are not relevant. Driving the noise floor down to the quantum limit by using liquid helium & a bunch of other abstruse techniques is not relevant. The fact that GPS signals are below the noise floor in their radio spectrum is, once again, not relevant.
Do you argue like this in all your interactions with other people?
Actually, this thread was on sampling theory and whether or not Nyquist perfectly recreated original signals. The only mention of anything about human perception in the chain you're replying to is me writing that good engineering can push errors outside human hearing, while pointing out that it is not the original signal. All the other posts in the thread you claim is about human perception are not about human perception. They're about signal reconstruction with one mention of how that applies to hearing.
>Do you argue like this in all your interactions with other people?
No, most of my friends don't make incorrect claims then double down on them when better information is pointed out. They also don't tend to claim an entire conversation is about a different topic than it was.
You: "Up to the Nyquist limit, a digital signal will completely recreate the original signal". No, as explained.
You: "Digitisation does not result in square wave output anywhere in the output chain" - again false, even for audio, as explained above. Thinking that output is a nice set of pure sine waves making the original band limited (which is also an approximation) leads people to flawed thinking. Realizing the how and why of what actually comes out is useful.
You imply pushing things below a noise floor makes them indistinguishable from noise - this isn't true in general, and is not even true in audio. There are even audio products that use signaling below the noise floor. Repeated misunderstanding of what a noise floor is (and isn't) is extremely useful, even in audio. The other examples were to make it clear there is useful knowledge here.
Understanding these nuances is useful for even audio to understand that reconstruction is always a design tradeoff on where you want to put the errors - and that you never are reconstructing the original signal.
I'm pro analog, but the above is a common misconception, usually caused by the bad "popular science" articles on digital reproduction and sampling, which show quantization as little pixelated waveforms etc.
The waveform produced from a sampled digital signal recreates the original perfectly (for the target frequency), you can verify that on an oscilloscope.
The oscilloscope can detect that the pattern of inputs has a frequency, but it's necessarily an approximation at this point. An oscilloscope adds enough noise and error from it's own workings that on a screen, to your eye, for certain sampling parameters, it looks close. But that is not the original signal.
Take a signal, sample it, DAC it, and try using that signal to cancel the original, amplify the result, and run that through an oscilloscope. If the signal were reconstructed, you should be able to get zero.
You don't. Now you see the differences in the reconstructed signal and the original.
1008 AM is a great station available through this webSDR service: http://websdr.ewi.utwente.nl:8901/
The "cold and thin" vs warm and thick lies on the nonlinearities, the subtle saturation you get on tape/vinyl, the patina, even the tactility. The ease of skipping makes it even worse.
And that's before we've got to the cultural changes in attitudes towards music between the vinyl/tape and the cd/streaming eras (where music, which was king, is now just another passtime for teens), nor the changes in musical content.
To hand me this secondhand abstraction contrived by this guy I never met. It's a bit crazy.
Shadow-removed in fact. Which is to say, they removed it without warning, notice or explanation. And from my account-view, it was not removed at all.
And when I asked what's up with that, I was threatened with banning "if I keep it up".
What do you think of that?
Many audiophiles scoff at "digital" music, i.e. audio that is amplified with silicon transistors.
Modern transistors are very fast, so noise that gets through the filters typically sounded like sharp pops and hisses. With older (and slower to respond) analog technologies, unwanted noise gets smoothed into more pleasing hums and modulations.
These days, there are excellent affordable digital options like the O2 amp. But it would be a boring world if we were all the same.
Of course, all of that can be emulated quite faithfully now.
Last I heard (been some years now), record companies have preferred to do lots of inflicting upon the sound, before letting it near any digital format that ordinary folks can buy.