Comment by nullc

This is the core secret of the sampling theorem. It says if you have signals of a particular type (bandlimited) you can do a certain kind of interpolation and recover the original exactly. This is no more surprising than the fact that you can recover the coefficients for an N degree polynomial using any N points on it, though the computation is easier.

It turns out that if you reproduce a digital signal using stair steps you get an infinite number of harmonics— but _all_ of them are above the nyquist frequency. The frequencies below the nyquist are undisturbed. Then you apply a lowpass filter to the signal to remove these harmonics— after all, we said at the start that the signal was bandlimited— you get the original back unmolested.

Because analog filters are kinda sucky (and because converters with high bit depth aren't very linear), modern ADCs and DACs are oversampling— they internally resample the signal to a few MHz and apply those reconstruction filters digitally with stupidly high precision. Then they only need a very simple analog filter to cope with their much higher frequency sampling.