Bitrate reduction by rounding instead of truncating and dithering

Question

From my understanding, dithering is used to mask the systematic error that one introduces by reducing the bitrate simply by truncating (i.e. chopping off) the least significant bits. This makes sense to me.

But what if instead of truncation one uses the lower bits to round instead, wouldn't that be even better for quality? So say you reduce the bit rate from 24 bit to 16 bit, use the lowest 8 bit of the 24bit sample to decide if a one is added to the truncated value (with e.g. round-to-even for the half-way point 128). That would seem like the most "correct" way to me, as it is the mathematically closest representation of the original signal in the lower bitrate.

And by using rounding instead of truncation, there may possibly be less need for dithering afterwards (when reducing to otherwise near-transparent bitrates like 16bit or above, as obviously rounding vs truncation won't matter much for easily audible quality-reducing reductions). Since the maximum amplitude of the error signal by rounding is only half of that of truncation I assume the introduced noise is more quiet, which might be quiet enough to not be noticeable at 16bit and possibly preferable over the added dither noise. So my question is, are there publicly available studies that compare the quality of truncation vs rounding (with and without dithering)?

Answer 1

The purpose of dithering in audio processing is to hide quantizing noise. In effect the dithering moves hearable noise to frequencys where the ear is less sensitive.

Rounding has no effect on the frequency profile of the quantizing noise and gives the same effect as simply truncating.

Answer 2

Rounding gives exactly the same sort of quantisation errors as truncation does.

You can see this because rounding is effectively the same as adding 0.5 (scaled appropriately) and then truncating.  The exact transition points shift, but the basic pattern remains.  (Rounding does avoid a systematic error averaging -0.5 — but that's effectively a DC shift, which inaudible.)

It's worth seeing the Wikipedia page for Dithering, which discusses this.  It uses the following values as an example:

0.8  1.6  2.4  3.2  4.0  4.8  5.6  6.4

Truncation gives:

0    1    2    3    4    4    5    6

While rounding gives:

1    2    2    3    4    5    6    6

The quantisation noise is not identical, but still has the same character, resulting in ‘additional content at discrete frequencies created by the regular and repeated quantisation error.’

The page then goes on to look at how you might vary the rounding to avoid that — and so derives dithering.

This may be clearer in its image examples.  In particular, truncation or rounding are illustrated in the non-dithered example:

(The flat areas show how much detail is lost.)

Whereas the following version uses dithering — you can see how much more detail is preserved, despite having the same bit depth as the image above:

The effects on digital audio are perhaps less obvious, but you can see how the fairly regular, repeating errors in the first image might correspond to prominent low-frequency noise, and those in the second to noise spread more evenly across much higher frequencies where it would be much less noticeable.