how is digital sound created?

millercarbon

... Really good expensive DACs have a lot more quality parts and do a lot more processing trying to smooth the signal out by interpolating values in between the stair steps ...

That is completely mistaken but a common misnomer. The only interpolation that is part of the digital audio standard is when it is used for error correction. Because the data on a CD is encoded redundantly and interleaved (and cached in streaming) error correction is actually quite rare.

Within the bandwidth of the system, the Fourier theorem shows us that digital audio can perfectly describe the analog waveform. If you have doubts, watch this. (Kindly note that I'm not claiming digital audio is "Perfect Sound Forever." But if we want the best sound from digital, it's helpful to understand how it works.)

It's odd how many audiophiles refuse to accept this math, which is conceptually simple even if the details are not. Consider that the Fourier mechanism also explains perfectly how the squiggles on an LP can represent a full orchestra.

The only interpolation that is part of the digital audio standard is when it is used for error correction.

There’s no interpolation happening when performing error correction since there’s no ’guessing’. The proper bits are either recovered or the data stream is so corrupted that some errors remain. In the latter case, the player may mute the output or cease playback.

Interpolation is required whenever you increase the sampling rate in digital audio. Most DACs these days, whether some form of multibit resistor ladder or the sigma delta variety, increase the sampling rate to net several benefits such as reducing or ’shaping’ quantization noise or relaxing the design requirements for the analog reconstruction filter.

Here is a great tutorial on oversampling / upsampling and interpolation from Analog Devices:

https://www.analog.com/media/ru/training-seminars/tutorials/MT-017.pdf

dspGuru also has some great information on interpolation:

https://dspguru.com/dsp/faqs/multirate/interpolation/

yage

There’s no interpolation happening when performing error correction since there’s no ’guessing’

You are mistaken. Interpolation is "guessing" by definition (in this context) and interpolation is part of the CD standard.

The proper bits are either recovered or the data stream is so corrupted that some errors remain. In the latter case, the player may mute the output or cease playback.

That’s how a data disc works because it has to be bit perfect. But it’s not how digital audio works at all. The Reed-Solomon error correction code is part of the CD standard and, as MC noted, it’s all part of the fun math that makes digital audio:
" ... whereas subsequent constructions interpret the message as the values of the polynomial at the first k points a 1 , … , a k {displaystyle a_{1},dots ,a_{k}} and obtain the polynomial p by interpolating these values with a polynomial of degree less than k ..."

+1 cleeds! Too many still believe that more expensive DACs do a better job of "smoothing out" the steps between digital samples. Nonsense! My $99 DAC outputs rival those of any four-figure DAC (low noise, low distortion, accurate LSB ...).