Admittedly I have forgotten a lot about signal processing, but I'm not sure I agree with everything you're saying.
Roughly, dither convert harmonic distortion to noise.
That much is clear.
If we just add
noise (random) to signal, distortion gone but noise will by modulated
by signal (correlate with signal). This noise not fully random.
So, let's consider the sequence a[n] = rand() / (float)RAND_MAX -
0.5.
How is each value in this sequence not "fully random" and completely uncorrelated to all other values? AFAICS, this sequence is white noise and if you add it to your audio, you get no modulation.
This is rectangle dithering.
Certainly the probability distribution of each a[n] is uniform between -
0.5 and 0.5, so it is rectangular.
To prevent this modulation and get pure white (fully random) noise (it
much better for human ears) not correlated with signal, we consider
previous dither value.
If I'm not mistaken, we are in effect getting a new sequence:
b[n] = a[n] - a[n - 1]
And we are using this sequence for our dither. Now, this is in effect applying a high pass filter to a[n], so if a[n] was white noise, b[n] will not be.
And, most certainly, each element of b[n] is not independent from the previous one. For example, if you have b[n0] = 1, you know for a fact that a[n0] =
0.5 and a[n0 - 1] = -0.5. Therefore, b[n0 + 1] = a[n0 + 1] - a[n0] = a[n0 + 1] - 0.5 < 0. Therefore, b[n0] = 1 implies b[n0 + 1] < 0; these two elements are correlated and b[n] is not white noise.
What am I getting wrong here?