>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