[sdiy] DACs on-board 12 vs external 16 vs 24 bit

[rburnett at richieburnett.co.uk]

>> I seriously doubt 24-bit D-A is within the reach of any synth
>> designer. Or even 22-bit. Not even Yamaha would bother.
> 
> I think Richie didn't mean D-A'ing, but doing internal calculations 
> with
> that bitness, in order to have enough margin for summing and 
> attenuation.

You do the internal calculations to as high a precision as you can.  Or 
to put it another way, you want the word-length of the final output DAC 
to be the limiting factor.  If things are getting heavily truncated 
before they reach the final DAC then that's an additional source of 
quantization noise that can't be removed later.  These days cheap 24-bit 
delta-sigma converters are the norm for audio.

You might do all your internal calculations to 32-bit accuracy and then 
output the final left/right mix to a 24-bit DAC (rejecting the least sig 
8 bits.)  Colin Fraser wisely suggested adding dither before truncating, 
as this turns the horrible fizzy buzzing sound of the quantisation noise 
into "random noise".  Our ears find this resulting low-level 
de-correlated noise much more natural & less objectionable than the 
fizzy correlated harmonics from non-dithered truncation.

Chris, an example of why you might want to use larger word-lengths might 
go something like this.  If I synthesise an 808 bass-drum in an analogue 
modelling synth to 16-bits accuracy it will sound pretty good provided 
it is at full volume, (i.e. the positive and negative peaks come close 
to +32767 and -32768).  Okay, the very tail of the BD decay might sound 
a bit fuzzy when the level drops so low that it's not being represented 
by many bits, but let's assume I make a design decision that we can live 
with 16-bits for the BD provided it's always at *full volume*.  Now I 
have to allow for the fact that most of the bass-drum hits are going to 
be "normal" velocity and some will be accented.  In order to make room 
for the accented hits I need to turn down the "normal" level by say 
12dB.  This means that my normal BD hits now only take up 14 bits of the 
16-bit converters available dynamic range.  Now consider that this sound 
might be one of 16 sounds that could potentially all play at maximum 
volume at the same time in a drum machine.  So the volume needs to be 
scaled back further to allow for the addition of the other remaining 15 
sounds otherwise clipping could occur at the output.  If I were to 
reduce the sound of each voice to 1/16th of its normal level then I 
could be sure that all 16 voices could sound at maximum volume & maximum 
velocity at the same time without clipping.  But scaling the BD sound 
amplitude down futher by a factor of 16 would mean that it now uses just 
10-bits of resolution.  This bass drum on it's own would now sound very 
fuzzy, certainly not warm and "analogue", and the tail of the decay 
would deteriorate into a horrendous fizzy, hissy mush.  Clearly not 
acceptable.

The answer here is not to scale down the levels of the instruments but 
rather add additional resolution to the output to allow you to mix all 
of the 16-bit voices together without clipping.  Eg. 16 x 65536 = 
1048576   Or to put it another way sixteen 16-bit numbers added together 
requires a 20-bit number to hold the result.  It's really just the same 
as "gain staging" in analogue design, except is DSP land the evil is 
quantisation noise rather than thermal noise. *(See note)

Regarding 24-bits for DIY projects, Tom pretty much summed that up 
perfectly...

> The DAC chips to do it are easily and cheaply available. Whether the
> analog side lives up to the promises made by that incredible S/N is
> a different story. But Richie is right - at least if you've got that
> much resolution, you can add a load of waveforms together without 
> having
> to lose bits off the bottom, even if some of that might get lost in the
> noise floor.

The only thing I would add is that the analogue noise floor often sounds 
"natural, fuzzy and warm" and the digital noise floor often sounds 
"hard, brittle, glitchy or fizzy".  That's why we might mix our digital 
audio to 24-bits (to keep the horrible digital artifacts down to 
inaudible levels) even though we know perfectly well that real-world 
analogue limitations mean that we will never actually realise that full 
24-bits of dynamic range in the full strict scientific sense of the 
phrase.

-Richie,

* Note:  This is just a mixing example for demonstration purposes.  In 
practice it is unlikely that all 16 instruments will all sound on the 
same step, all with maximum velocity, all with their volume sliders 
pushed to 100%, and all hitting their maximum peak amplitude at the same 
point in time!  So you might make a design decision to increase the gain 
slightly at the mixing stage to recover some signal-to-noise ratio 
during perceived "normal" usage, at the potential risk of unlikely (but 
theoretically possible) clipping under worst case abnormal usage 
conditions.  In a drum machine each sound is not correlated with the 
other instruments so you can use maths to predict how the average and 
peak levels of numerous mixed voices will increase as more sources are 
added to the mix.  You can contrast this to a VCO that allows you to 
adjust the mix of saw, sine, tri and square in the output, where all of 
these waveforms are at the same frequency and are correlated.