### Re: What is the dynamic range and S/N ratio of CD format?

Posted:

**09 Nov 2019 15:47**the home of the turntable

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https://www.vinylengine.com/turntable_forum/viewtopic.php?f=112&t=114725

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Posted: **09 Nov 2019 15:47**

Posted: **10 Nov 2019 03:20**

Who in the world is not calculating 96 db signal to noise? Certainly not me. I said much earlier in this thread that the SQNR is 96 db.

Posted: **10 Nov 2019 03:23**

Missspoke there. It has nothing to do with finished product CD. Has to do with recording, mixing and editing. For instance increasing the volume of a very weak signal.

A 40 Hz signal covers the span of 1102 samples (44100/400 = 1102.5). If the signal level is low enough, say -50 db, the amplitude of the low frequency wave form does not rise and fall fast enough to prevent large blocks of consecutive identical samples due to the quantization granularity. If the frequency and signal level are low enough it can end up being more like a square wave.

But of course studios etc. don't work in the 44.1 KHz 16-bit realm. Though given the right combination of signal level, frequency and sampling rate it could be experienced even with 192 KHz 24-bit.

Posted: **10 Nov 2019 03:31**

Yes, that's loss of resolution, confirmed by a composer/engineer friend of mine as a real thing back in the day, with emphasis on "the day" since 16-bit production started to disappear in the nineties. I don't think it's dependent on frequency BTW.

Posted: **10 Nov 2019 03:37**

Posted: **10 Nov 2019 04:51**

I believe it occurs at all frequencies below the Nyquist frequency and is exacerbated as the signal frequency goes lower due to the increasing amount of quantization errors per cycle as frequency decreases.

Though this isn't really pertinent to the subject of this thread and thus it was a mistake on my part to injected it.

Posted: **13 Nov 2019 02:33**

Each quantization bit gives 6db of dynamic range. But there are only 15 quantization bits. Not 16. The 16th bit (MSB) provides only that the quantization is positive or negative. Not magnitude.

It seems to me that we probably agree on the SQNR being 96.33 db. Though we may not be in agreement on the formula and the dynamic range.

Seems like your SQNR formula would be...

20Log(2^16) = 20Log(65536) = 96.33 db SQNR

Which even though it arrives at the correct value, would be an incorrect formula because the quantization noise level should be half the LSB value. Thus if truly 16 bits the formula should be...

20Log(2^16/0.5) = 20Log(131072) = 102.35 db SQNR

Were as the correct formula I believe would be...

20Log(2^15/0.5) = 20Log(65536) = 96.33 db SQNR

With the LSB being the smallest value possible (1) and the quantization noise being one half the LSB (0.5), the dynamic range is then 6db less than the SQNR db.

20Log(2^15/1) = 20Log(32768) = 90.31 db dynamic range.

Posted: **13 Nov 2019 23:13**

Seems all of the discussion above is theoretical anyway.

*"Digital audio at 16-bit resolution has a theoretical dynamic range of 96 dB, but the actual dynamic range is usually lower because of overhead from filters that are built into most audio systems." ... "Audio CDs achieve about a 90-dB signal-to-noise ratio."* https://books.google.com/books?id=w0vsd ... &q&f=false

What's interesting is that DSD/SACD only has 1 bit, but sampling of 2.8MHz, and provides (in theory) almost 120db of dynamic range - roughly equivalent to (theorectical) 20bit PCM audio.

P.S. I'm not going to argue about any of the above, but merely found the information while doing a little research on my own. Feel free to pick it apart as I have no vested interest in it.

What's interesting is that DSD/SACD only has 1 bit, but sampling of 2.8MHz, and provides (in theory) almost 120db of dynamic range - roughly equivalent to (theorectical) 20bit PCM audio.

P.S. I'm not going to argue about any of the above, but merely found the information while doing a little research on my own. Feel free to pick it apart as I have no vested interest in it.

Posted: **14 Nov 2019 05:06**

Okay I'll pick little since you granted permission. ;)

The digital data contained on a redbook audio CD has exact non theoretical values and thus an exact actual, not theoretical, dynamic range.

Completely disregards the fact that redbook audio CD data is signed and therefore there are only 15 bits representing magnitude. Thus the redbook audio CD data has a dynamic range of 20Log(max/min) = 20Log(2^15/1) = 90.33 db.jdjohn wrote: ↑13 Nov 2019 23:13"Digital audio at 16-bit resolution has a theoretical dynamic range of 96 dB, but the actual dynamic range is usually lower because of overhead from filters that are built into most audio systems." ... "Audio CDs achieve about a 90-dB signal-to-noise ratio."https://books.google.com/books?id=w0vsd ... &q&f=false

The opening paragraph of this thread is clear that it is about "redbook audio CD (16-bit 44.1 KHz)". But nevertheless DSD/SACD is still bits. Just more of them in a different format. Regardless of which format, generally the more bits, the more accurate the representation.

But this thread is about redbook audio data SQNR and dynamic range. So far it seems we agree on the SQNR = 96.33 db. But not the dynamic range.

sine wave at:

-0db: positive phase magnitude: +32767 and negative phase magnitude: -32767

-6db: positive phase magnitude: +16384 and negative phase magnitude: -16384

-12db: positive phase magnitude: +8192 and negative phase magnitude: -8192

-18db: positive phase magnitude: +4096 and negative phase magnitude: -4096

-24db: positive phase magnitude: +2048 and negative phase magnitude: -2048

-30db: positive phase magnitude: +1024 and negative phase magnitude: -1024

-36db: positive phase magnitude: +512 and negative phase magnitude: -512

-42db: positive phase magnitude: +256 and negative phase magnitude: -256

-48db: positive phase magnitude: +128 and negative phase magnitude: -128

-54db: positive phase magnitude: +64 and negative phase magnitude: -64

-60db: positive phase magnitude: +32 and negative phase magnitude: -32

-66db: positive phase magnitude: +16 and negative phase magnitude: -16

-72db: positive phase magnitude: +8 and negative phase magnitude: -8

-78db: positive phase magnitude: +4 and negative phase magnitude: -4

-84db: positive phase magnitude: +2 and negative phase magnitude: -2

-90db: positive phase magnitude: +1 and negative phase magnitude: -1

That is 15 steps of 6db. 6db x 15 steps = 90db dynamic range.

Posted: **14 Nov 2019 18:28**

How do you define dynamic range?

Posted: **14 Nov 2019 20:22**

Posted: **14 Nov 2019 21:25**

Then its 90dB indeed.

But your definition is maybe not a commonly used one.

wikipedia says:

"the ratio between the largest and smallest values that a certain quantity can assume."

The smallest value is of course zero magnitude.

However, if you relate it to the human ear and start at 0dB, that is related to the minimum threshold a human can hear (commonly set at 20µPa)

So there 0dB always relates to a registred signal.

So whether its 90dB or 96dB is more a matter of what definition you give to it.

Posted: **14 Nov 2019 21:43**

Posted: **14 Nov 2019 22:45**

GuidoK wrote: ↑14 Nov 2019 21:25Then its 90dB indeed.

But your definition is maybe not a commonly used one.

wikipedia says:

"the ratio between the largest and smallest values that a certain quantity can assume."

The smallest value is of course zero magnitude.

However, if you relate it to the human ear and start at 0dB, that is related to the minimum threshold a human can hear (commonly set at 20µPa)

So there 0dB always relates to a registred signal.

So whether its 90dB or 96dB is more a matter of what definition you give to it.

Zero is the absence of value and makes no sense for calculating dynamic range.

Try this with min being zero.

20Log(max/min) = db

Ratio of 1:0 is infinity. Ratio of 0:1 is infinitesimal.

Can you please provide an example or instance of using zero as the minimum for calculating the dynamic range of something?

For redbook audio data. Signed 16-bit linear PCM...

The maximum magnitude is 32768 and is referenced to -0db.

The minimum magnitude is 1. Zero is the absence of magnitude. Silence (0) amplified by infinity is still silence (0). Just like 0 multiplied by infinity is still 0.

No. It is a mater of using the correct definition and values for calculating.

Using an incorrect definition resulting in using incorrect values yields incorrect results.

Though even if we use zero as the minimum the magnitude range is 32768 (32768 - 0). Which is 32768/1. Which results in 20Log(32768) = 90.31 db. Though that is an incorrect means of obtaining the value for the log function. As can be demonstrated by 1-0 = 1. Log(1) = 0.

Posted: **14 Nov 2019 22:46**