Exaggerated Claims about A to D Sampling Rates

[Richard Davey]

Wiki Audacity Sample Rates page http://wiki.audacityteam.org/wiki/Sample_Rates makes claims which are exaggerated.

The page suggests:

Analogue Frequency (kHz) ---------- 20**
Sample Rate Required (kHz) -------- 44

This implies about 2 samples per full period of the analogue wave. I doubt this will record the source anywhere near accurately.

I would like at least 8 samples per analogue source period – even that’s sketchy.
At this rate,

Analogue Frequency (kHz) ---------- 5.2* ----------- 20**
Sample Rate Required (kHz) -------- 44.1 ---------- 160

* 5.2 kHz is under 4 octaves above middle C.
** 20 kHz is between 5 and 6 octaves above middle C.

So, to convert 20 kHz analogue we want at least 160 kHz sampling rate.
Comments please.

[steve]

This is an interesting and complex issue, with surprises.

The guys that worked this all our originally were Harry Nyquist and Claude Shannon. They developed what became known as the Nyquist–Shannon sampling theorem (frequently referred to as the "Nyquist Theorem"). This theorem is fundamental to much digital audio technology, and digital signal processing in general.

The basic idea of the theorem is, that a band limited signal may be exactly and uniquely represented by a series of equally spaced sample values, provided that the sample rate is at least double the band limit upper frequency. In other words, to exactly reproduce a signal that has an upper frequency limit of 20 kHz, requires a sample rate of at least 40 kHz.

In practice, the limiting factor is the effectiveness of the band limiting filter. The CD standard of 44.1 kHz was agreed by the industry as the minimum sample rate for full (20 kHz) audio bandwidth, though that was possibly a bit ambitious with the filter technology available at that time. The video standard of 48 kHz was arguably more realistic at that time.

Over the past 50 years or so of digital audio development. there have been huge advances in digital filter technology, which allows practical AD/DA conversion that is very close to theoretical limits.

Here's an interesting demonstration using Audacity:

This is a 21 kHz sine wave, amplitude 0.8. The sample rate is 44.1 kHz.
It looks nothing like a 21 kHz sine wave.

firsttrack002.png (10.37 KiB) Viewed 1154 times

The above track clearly shows the limitations of a "dot to dot" representation of the waveform.

Fortunately, Audacity does not use a dot-to-dot representation internally, but can properly extrapolate a band limited signal - the only signal that can pass through all of the dots without exceeding the Nyquist limit of half the sample rate. We can show this by asking Audacity to resample this track at a higher sample rate.

In this image, I have resampled to 192 kHz:

firsttrack003.png (10.1 KiB) Viewed 1154 times

Note how it now looks like a 21 kHz sine wave.
Note also that the amplitude is less than the original 0.8. This is because 21 kHz is so close to the frequency of the band limiting cut-off filter that it has been attenuated by the filter.

[steve]

An excellent series of videos about the subject, from one of Audacity's early contributors (now a developer at Xiph): https://wiki.xiph.org/Videos/Digital_Show_and_Tell

[Richard Davey]

Massive thanks, Steve, for your reply with great graphics.
You have shown me that my statement "Exaggerated Claims about A to D Sampling Rates" may be wrong.
I bow to your greater knowledge.

[steve]

Bigger bows to Shannon and Nyquist for deducing and proving such a counter-intuitive theorem which would eventually be revealed to be such an important discovery

[joe sixpak]

Wiki Audacity Sample Rates page http://wiki.audacityteam.org/wiki/Sample_Rates makes claims which are exaggerated.

The page suggests:

Analogue Frequency (kHz) ---------- 20**
Sample Rate Required (kHz) -------- 44

This implies about 2 samples per full period of the analogue wave. I doubt this will record the source anywhere near accurately.

I would like at least 8 samples per analogue source period – even that’s sketchy.
At this rate,

Analogue Frequency (kHz) ---------- 5.2* ----------- 20**
Sample Rate Required (kHz) -------- 44.1 ---------- 160

* 5.2 kHz is under 4 octaves above middle C.
** 20 kHz is between 5 and 6 octaves above middle C.

So, to convert 20 kHz analogue we want at least 160 kHz sampling rate.
Comments please.

==================
=========================================

The Nyquist sampling theorem says if you sample at >2x the max of a band limited signal you can reproduce it exactly.
We proved this in my graduate class in engineering in the 60s. It is also in Wagner's textbook.
NOTE that at exactly 2x the theorem fails to counterexamples. This is a common error in technician level textbooks.

The problem comes with engineering which is not perfect like mathematics.

You lose accuracy by digitising with a finite number of bits. You lose accuracy with jitter changing when you sample.
You lose accuracy with both the A/D but also the D/A recreation. You lose accuracy with the microphone not being perfect.
You lose acccuracy with the preamps and other electronics not being perfect. You lose accuracy with some effects that are applied.

In practice a sample rate of 44.1 will give you 20KCps at excellent quality as proven by all the CDs that use it.

[kozikowski]

44.1 will give you 20KCps at excellent quality

Correct, but as I posted several times now, it does it through tricks, errors and sloppiness, not through accuracy. In the same sense, MP3 sounds just fine to 100% of people who hear it, but you shouldn't be doing commercial production in either one. The errors in both will catch you up.

That's why studios use 96K 24-bit. I agree much of that is overkill, but not all.

Koz

[steve]

not through accuracy

Modern software converters have terrific accuracy. See here for a comparison of sample rate converters: http://src.infinitewave.ca/
This one shows the passband for Audacity with default settings:

Audacity203.png (2.19 KiB) Viewed 942 times

and sweep response:

Audacity203-sweep.png (65.9 KiB) Viewed 942 times

and linear phase response:

Audacity203-phase.png (2.19 KiB) Viewed 942 times

[joe sixpak]

44.1 will give you 20KCps at excellent quality

Correct, but as I posted several times now, it does it through tricks, errors and sloppiness, not through accuracy. In the same sense, MP3 sounds just fine to 100% of people who hear it, but you shouldn't be doing commercial production in either one. The errors in both will catch you up.

That's why studios use 96K 24-bit. I agree much of that is overkill, but not all.

Koz

Agreed. But it depends what your needs are for the final project. Not every project needs the absolute best that is possible.

Personally when I looked at this some years ago, 192 seemed to be about the fastest that was reasonable considering the jitter and quantization;
and it would have given slightly better quality but you might need DVD or SACD to really appreciate it.

I have seen 384 gear in the past but dont know how much anyone uses that except for bragging rights