[ChrisLomont]

>Montgomery talks about quantization noise. He doesn’t assume infinite precision.

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...

[3] https://en.wikipedia.org/wiki/Reconstruction_filter

[aaaaaaaaaaab]

1. Noone said that the anti-imaging filter was literally built into the DAC package.

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/...

[ChrisLomont]

You: "Noone said anything about mathematically perfect reproduction." Also you: "thereby getting back the original smooth waveform".

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.

[pja]

> 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.

[ChrisLomont]

>This is silly. Every system has a current noise floor based on whatever the thermodynamics of the system are.

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.

[1] https://en.wikipedia.org/wiki/Noise_floor

[2] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3292143/

[pja]

You are very insistent on proving yourself right by changing the topic to one where your assertions are correct instead of dealing with the discussion that everyone else was having. It’s a neat rhetorical trick, but kind of annoying in a context where everyone else is just trying to have a constructive discussion.

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?

[ChrisLomont]

>we were talking about digital audio & the perception thereof by human listeners

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.

[aaaaaaaaaaab]

Ok, very cool, but please get back to the original argument about analog vs digital audio.

[pja]

Yeah, you can‘t just ignore the analogue circuitry that you’re expected to put in place around the bare DAC. It’s part of the design!