Why do some people say that digital sound is "pieces of the sound"?
Because they imagine that, in taking a series of samples of the instantaneous variations in air pressure, and translating each of these samples into a number before converting it back via a DAC, an amplifier, and a set of speakers, will result in a "stepped" waveform that is composed of a number of square waves and looks something like a staircase instead of like a smooth curve.
Is here any sense to that point of view?
Well, if you had a really primitive D/A converter, yes, it would make sense. If you were working with a very low sampling rate (15Khz, for example), or with a seriously restricted set of available sampling levels (4 bit or 8 bit) that is exactly what you would get. And it would sound terrible. Real Darth Vader stuff.
But provided you have a high sampling rate (the figure usually cited is 44.1KHz) and a sufficently large number of bits with which to represent the air pressure variation levels (16-bits are usually considered sufficient, though some people claim that you can get improved results with even more levels), the view makes no sense at all.
Why not?
Well, we can show this either of two ways. That's why I introduced the topic of Fourier analysis in my previous post. Here is method (a).
a: Looking at the sound as a complex set of sine waves (as we did in the previous post), if you examine the two signals - the original sound, and the digitally recorded image of it - and we ignore transducer error (because no matter which recording method we use, sooner or later we must pass the original sound through a microphone and a speaker) there is one difference and only one difference between the two. This is the millions of tiny "steps" in the digital representation of the sound.
So what we do is we start subtracting the individividual sine waves from both signals until we have subtracted everything except the "steps". Now we continue with the analysis, in the exact same way, and observe the set of sine wave frequencies that the "steps" resolve into.
And right away, we notice something: every single frequency we have added to the signal through the process of digitisation is well above the limit of human hearing. We have added frequencies at 50, 80, 190KHz. And these are frequencies that (a) no human being can hear, that (b) no competently designed power amp will pass through, and (c) no earthly set of tweeters can reproduce anyway. Whichever way you look at it, the "step" theory is hogwash.
Let's try method (b).
b: Considering the two signals as entire signals now (in other words as simply lists of individual variations in air pressure, one of them continuous the other discrete), we again find that the differences are tiny, and are concentrated in those same miniscule "steps". However, when we play the sound back through speakers, as Time observed, the steps are so small and so momentary that the speakers are unable to reproduce them. The speakers, in fact, are unable to reproduce the overall curve (be it analog or digital) in more than a rather approximate way. The physical movement of the speaker does not and cannot faithfully duplicate the extremely rapid accellerations and decellerations implied by the digital "steps", being limited by physical mass and electrical impedence, it simply takes an average figure, never quite catching up to the intended signal. (Or, to look at it using method (a) again for a moment, it can't reproduce the implied extreme high frequencies.)
But let's imagine that some young Einstein has invented an infinitely rigid speaker system with zero mass and zero impedence. What difference would it make to you, the listener, so far as the analogue/digital question goes? Why, none at all. For your ears are not sufficiently sensitive to hear those tiny differences, let alone pick up those impossibly high frequencies. Provided the sampling frequency is at least twice the highest frequency detectable by the ear, there is no discernable difference. Except the hisses, poor slew rate, and restricted dynamic range of the analogue medium, of course.