viber6,
It really takes an expert to properly explain resolution and bit depth, hopefully there is someone qualified on this string that can more accurately explain why your rational is incorrect if I get it wrong. But in my limited understanding I will try and partially explain. For the record, you have already stated that you "do not understand one solution (Leedh) the problem of loss of bits at low levels." And that you "don't see how any digital volume control is better than analog at even moderate levels." So, I would appreciate it if you don't criticize my admittedly partial explanation of this issue with your limited understanding. I will happily stand corrected from an expert for any mistakes that I make in my calculations...
One problem is that your rational does not allow for upsampling.
It is quite easy to convert a 16 bit depth resolution to a 24 bit depth. This involves only simple multiplication. It is true that the first increase in bit depth does not in any way increase resolution... at that level. At full output the analogue conversion redbook standard is +/- 2 volts so 65,536 digitally represents 2 volts at 16 bit and 16,777,216 represents 2 volts at 24 bit. They are equal. When the conversion from D to A occurs, both result in exactly the same instantaneous voltage level, without any complex DSP applied. To make this conversion, you merely multiply every sample by 256. No loss or gain occurs, only a difference in numeration.
But since bits are relegated to either 1 or 0, you can't divide them. What the increase in bit depth allows is for division to occur without loss. You can then very accurately lower the volume significantly while maintaining excellent resolution. 16 bit resolution gives 65,536 levels at 2 volts, 24 bit resolution gives 16,777,216 levels at 2 volts. You can then mathematically lower this amplitude by 256 times while still maintaining a higher bit depth than maximum redbook level of 16 bits which is actually only used at 100% output (effectively never). Algorithms can deal quite well with any rounding errors as you reduce volumes i.e halving the volume with a digital number that is odd. This results in a potential maximum error of 1/16,777,216 of 2 volts error... significantly below the threshold of hearing.
It is true, that the information available does not increase resolution at the full level in the upsampling, but what it does is then allow volume adjustment with more increments without rounding errors. I.e. digital level 4000 in 16 bits (.122 volts output) becomes level 1,024,000 in 24 bits (also .122 volts output). You can then reduce the volume 250 times for an lowered volume value of .000488 volt output level volts with a digital level of level of 4,096. So, 1/250 of the output while resolution is still effectively higher than with what you started (4000 increments below vs 4,096 levels below but at 1/250 of the volume.
Digital editing has been utilizing these mathematical formulas for decades. In studios, every track is recorded for a maximum resolution then during mixing lowered to an appropriate level. So when implemented properly, it is actually far easier to digitally reduce volume with minimal sonic impact than it is to do it in the analogue realm. If this type of dsp were not readily possible, quality digital mixing would be nearly impossible and while some might argue that digital anything is inferior to all analogue, there are other threads for that discussion as Jay only works in digital. While you can accurately argue that information is still lost in this math, the information level lost is far lower than the base threshold set by redbook 16 bit depth.
In summary, the theoretical number of volume levels available through high bit depth digital is far higher than with what can be reasonably achieved in the analogue realm. Perhaps not at 16 bit, but certainly at 24 bit. Once the volume adjustment has been achieved, you can then reduce the bit depth back to 16 bits if the DAC can't handle the higher bit depth but all high end DACs I am aware of, built in the last 30 years can handle these higher bit depths if not even more at 32 bits.
I hope this satisfactorily answers your "question".
