I decided to start a new thread about Digital Audio and Dithering from the
thread "Volume and Dynamic Range Question".

There seems to be commonly held misconception that dither is just the noise
we add to digital audio to cover up the real digital noise floor. This is
not true. I obviously can't go into all of the mathematical proofs here, and
that would be pointless anyway, since the information has been published in
many forms in many places by people like VanDerKooy and Lip****z. You can
Google them for more information.

Digital audio has no natural noise floor in the traditional sense. If you
feed silence into an un-dithered, ideal analog to digital (A/D) converter,
you'll get a sequence of binary samples representing zero, forever. If you
apply a very small input signal, you'll get a varying sequence of samples at
the output. The output sequence represents converter step values related to
the converters resolution, correlated in some way (but not musically or
audibly) to the input signal. Not until the signal significantly exceeds the
minimum converter resolution will the coded converter output samples begin
to represent the input signal as recognizable audio. When you run an A/D
converter at the lower limits of its resolution, the result is gross
intermodulation distortion, harmonic distortion and noise modulation.

From a perceptual standpoint, the previous paragraph can be put this way:
With zero input, the noise floor of an ideal A/D converter is infinite. When
the input signal amplitude happens to exceed the converter's minimum
resolution, the noise level jumps to some value dictated by the converter's
resolution (about -96dB in the case of a 16-bit converter). Furthermore, the
noise will be quite different in amplitude and spectral makeup from the
input signal. In other words, highly distorted, and probably not
recognizable by humans as the original input signal.

Dither adds a noise floor to a system that has no natural noise floor. In a
properly dithered system, a pseudo-random signal with a specific spectrum
and probability density (the likelihood of an occurrence a given amplitude)
keeps the converter always switching between adjacent bits. Because the
dither is wide-band and random, it doesn't matter whether the converter
represents it accurately or not. The output is also wide-band and random. It
isn't an accurate representation of the original dither, but two random
sequences are still just random sequences. The input sequence can be
numerically designed to produce the desired random output sequence.

The magic happens when you add the audio signal to this system. As we
already pointed out, A/D converters are highly non-linear when operated near
their lower resolution limits. This non-linearity gives rise to gross
intermodulation distortion, harmonic distortion and noise described earlier.
But what happens when you intermodulate (multiply) a signal with random
noise? You get the same signal, with noise added. It's not the same noise
you started with, but the noise is random, so it doesn't matter. The
important thing is, the signal is intact, except for the added noise. The
intermodulation products are distributed randomly, so they just add to the
noise slightly. The input signal would amplitude-modulate the dither unless
the dither has triangular probability density. This ingenious trick forces
the dither to self-modulate in the opposite direction as it gets pushed
towards or away from an A/D bit transition. The result is a very smooth
noise floor with the ability to linearly resolve correlated signals (tones,
i.e., musical notes) that are below the noise floor (and below the
resolution of the converter).

By the way, this discussion is all about A/D converters. That's where it
counts. The best D/A converter is only as good as the signal being sent to
it. Dither added after the fact cannot linearize or recover information lost
in an
improperly designed A/D converter. Your only hope at that point is to add
enough noise to cover it up. Fortunately, that noise level is probably
equivalent to that an analog master tape, so I guess it's not so bad,
really.

Having a good understanding of dither helps explain why CD audio, which has
been much maligned for being "so marginal" has been so successful. I tend
not to think of CD audio as marginal, I prefer to think of it as "optimal".
It's very elegant in the sense that it delivers extremely high quality
audio, without spending a single bit more than is theoretically required to
deliver it. We now have the storage densities and bandwidth to extend the
resolution and sampling rates. Aside from being great marketing tools, they
do give the design engineers more tolerance, which means we don't have to
work as hard to "get it right". Although I think CD audio is a great
delivery mechanism, it's probably inadequate for professional studio work,
where uncontrolled audio peaks are common, and mixing puts extreme demands
on dynamic range.