The dynamic range of 16 bits - Page 11 - diyAudio
 The dynamic range of 16 bits
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diyAudio Member

Join Date: Sep 2002
Location: Sacramento, CA
Re: Re: Re: ...Yes M'Lud, the evidence...

Quote:
 Originally posted by MBK Unbelievable. We're back at square one. That's what I thought to begin with: we have half the bits for plus and half the bits for minus. So, we have 65,536/1 or 96 dB peak to peak, but only 32,768/1 or 90 dB peak dynamic range.
No, we don't have half the bits for plus and half the bits for minus. We have half the quantization levels for the plus and half the quantization levels for minus. Since the quantization is 16 bits, the total quantization levels is 2<sup>16</sup> or 65,536 which gives us 32,768 levels for positive and 32,768 levels for negative.

And we have a dynamic range of 96dB peak-to-peak AND 96dB peak.

Again, the maximum quantization error will be 1/2LSB. So we have a 1/2LSB error for postitive and 1/2LSB for negative.

So if you're looking at just the peak value, the dynamic range is determined by 1/2LSB. So 32,768/0.5 = 65,536. 20 x log 65,536 = 96dB.

If you're looking at the peak-to-peak values, the dynamic range is determined by 1LSB. So 65,536/1 = 65,536. 20 x log 65,536 = 96dB.

It's no different than if you're comparing the relative levels of two sinewaves. If you're referencing the peak value of one sine wave, you have to calculate based on the peak value of the other. If you're using the peak-to-peak value of one sinewave, you have to calculate based on the peak-to-peak value of the other. If you're using the RMS value of one sinewave, you have to calculate based on the RMS value of the other.

And when you do this, the ratios remain the same and the result remains the same.

Quote:
 And Christer is right, to this theoretical discussion what matters is the definition of dynamic range. DC offset or whatever other technical implementation details don't matter.
The tried and true definition of dynamic range has been the ratio of the noise level to the maximum signal level. This definition works just as well for analogue as well as digital systems.

Quote:
 My conclusions: A) The data have a dynamic range of 96 dB. Can we agree on that?
Sure.

Quote:
 B) But in the context of audio where dynamic range is usually expressed as "LSB" (minimum recordable/encodable/whatever) -to - peak value, we have 90 dB.
In audio, the dynamic range is expressed as the ratio of the noise level to the maximum signal level.

In the digital domain, the noise level is determined by the quantization error. Which again will be no greater than 1/2LSB either plus or minus.

If it makes one feel better to call it "noise" in the analogue domain and "error" in the digital domain, fine. But you're still effectively talking about the same thing.

And I've already explained why the peak-to-peak 90dB figure is incorrect so no need to repeat it here.

se

UGH! When are they going to get the HTML tags working again?

 18th June 2003, 07:02 PM #102 diyAudio Member   Join Date: Sep 2002 Location: Sweden MBK, I was implicitly referring to my previous post where I explained to Fred that he was making the mistake of thinking that one bit can go both positive and negative from a certain value (presumably called 0). I then saw that you made the same mistake in your argument and tried to point that out in a (possibly/hopefully) humorous way. Of course, it was a bit stupid of me, perhaps, to assume that you had read that previous post of mine. BTW, what do you think of the Zobels now after a few days?
 18th June 2003, 07:03 PM #103 diyAudio Moderator     Join Date: Oct 2002 Location: Chicagoland Blog Entries: 2 MBK: Not my definition, the standard one, the ratio between the largest and smallest encodable signals. I'm sure Steve or Christer or someone else with an electronics library to hand will be happy to give you an exact quote. I think that your difficulty is that you inappropriately ascribe something special about the 0 volt level. That's an easy trap to fall into (believe me, I know!) and a harder trap to get out of. Say this slowly, three times: The system is not constrained to be symmetric about a particular value. All values are equally valid. __________________ Remember: life is ten per cent what happens to you, ten per cent how you respond to it, and eighty per cent how good your reflexes are when the Tall Ones come at your throat with their pincers.
diyAudio Member

Join Date: Sep 2002
Location: Sacramento, CA
Quote:
 Originally posted by SY Pohlmann says 98.1 dB. Go figure.
Yes. When you look at it statistically which just complicates the issue even further and brings a lot more math into it. Keeping it simple, we get 96dB. Which one might say is the maximum minimum dynamic range of 16 bits as opposed to the minimum minimum.

se

 18th June 2003, 07:07 PM #105 diyAudio Member   Join Date: Apr 2002 Location: Singapore Steve, actually this just dawned on me while the posts were underway - if you reference to 1/2 of maximum level, you also have to halve the LSB... and we're back to 96 dB.
diyAudio Senior Member

Join Date: Aug 2002
Location: Belgium

Hi,

Quote:
 UGH! When are they going to get the HTML tags working again?
For info:

Those are deliberately disabled by the board for security reasons.
See the "Troubleshooting" section of the forum.

Cheers,
__________________
Frank

 18th June 2003, 07:17 PM #107 diyAudio Member   Join Date: Sep 2002 Location: Sweden Don't know if it helps anybody but one of the things that have bothered me a little, but which I finally managed to answer myself is that it was not obvoius to me that it is fair to think of the quantization error as a noise floor in the same sense as for analog, since if we have no input we have no quantization error and hence noise exists only when a signal is present. (Do I get the prize for longest sentence of the day? ). However, this was obviously wrong, in retrospect, since that presumes we know if there is a signal or not. If all the data is zero, we actually cannot know if this is because there is no signal (well, 0V DC is a signal too, in a sense) present, or if there is a very low level signal that just happens to be lost in quantization error. That is, there is no observable difference, so the quantization error is the noise floor even when no signal is present.
 18th June 2003, 07:17 PM #108 diyAudio Member   Join Date: Apr 2002 Location: Singapore Christer, no offended, was just confused. Actually I wasn't toggling the bit between -1 and +1 for a third dimension but assuming just one-zero. But my error was that I introduced the reference value at ground, but only for the maximum signal level, not for the minimum level. I think I got it now. Zobels: Ha! Yet again the KISS principle strikes. The Zobels did improve the sound but maybe 2/3 of the disturbance remained. Today I now found that I had made an illogical grounding path in my chip amps. This created a ground loop which previously didn't matter since all paths are short im my amp ... but... but ... some weeks ago I added an output resistor to signal ground of my dipole EQ (to get balanced output impedances). I found out by chance of course. And as a result the signal return promptly must have gone to power ground. I believe this created the main fuzz in the system but I can't test before tomorrow (neighbors, police, blah blah . ). That explains of course why headphones sounded much better... Conclusion, never touch a running system.
diyAudio Member

Join Date: Sep 2002
Location: Sweden
Quote:
 Originally posted by MBK Zobels: Ha! Yet again the KISS principle strikes. The Zobels did improve the sound but maybe 2/3 of the disturbance remained. Today I now found that I had made an illogical grounding path in my chip amps. This created a ground loop which previously didn't matter since all paths are short im my amp ... but... but ... some weeks ago I added an output resistor to signal ground of my dipole EQ (to get balanced output impedances). I found out by chance of course. And as a result the signal return promptly must have gone to power ground. I believe this created the main fuzz in the system but I can't test before tomorrow (neighbors, police, blah blah . ). That explains of course why headphones sounded much better... Conclusion, never touch a running system.
Good to hear. So whether real improvement or imagined, it
was a cheap and easy tweak that was interesting to try.
Now if only some more people with possible RFI problems
would try it, to see if there seems to be some correlation in
the results. Well, this is getting off-topic, so that's it for now,
I guess.

diyAudio Member

Join Date: Sep 2002
Location: Sacramento, CA

Quote:
 Originally posted by fdegrove Those are deliberately disabled by the board for security reasons. See the "Troubleshooting" section of the forum.
Were they disabled quite recently? Like in the past few days? Because they were working just fine up until then. At least the tags I've regularly used which are a, i, b, center, img, sup and sub.

se

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