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44 KHz (CD) not enough !? (Nyquist etc.), plethora of distortion frequencies?
zephirus
post May 11 2003, 17:40
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Remarks and conclusions added May 12 2003 - 1:55 PM, and edited May 14 2003 - 08:35 AM :

My dubious claims unfortunately had a very short life span due to the very successful enlightenment efforts of tigre, 2Bdecided, KikeG and mrosscook.

In short: I failed to come up with evidence that cd quality (I mean 44.1 KHz digital sampling) is somehow problematic. It basically was a story of using the wrong tools, jumping to the wrong conclusions, and not having enough of a clue about signal processing.

Nevertheless, I tried again to make less daunting claims that the 44.1 KHz digital sampling rate is not enough to represent all signals less than 22.05 KHz correctly.

And again my claims had a very short life span. This time due to further enlightenment efforts by DonP, 2Bdecided, KikeG, mrosscook and SikkeK.

The conclusion: Arguing against the technical specification of cd quality (44.1 KHz/16 bit) should not be tried by someone that severely lacks in signal processing clue (like me).

If the cd sound quality is perceived as suboptimal, it may have more to do with poor recording, poor mastering, and suboptimal reproduction equipment (i.e. cd-player and sound system/headphones).

What one still could try are listening tests:

Such tests would need to be done with one and the same high end hardware for all signals and all tests (preferably with 192 KHz resolution, with 20-24 bit, and with a DAC that is perfectly shielded and outside of any system that is rich of EM signals, like a computer, and has a near perfect analog circuitry). And when testing the 192 KHz signal against the 44.1 KHz signal, the latter would need to be a digitally downsampled version (to 44.1 KHz), which was upsampled to 192 KHz again. Using the best available algorithms (Cool Edit may do a resonable job here).

And still, asking the test persons for audible artifacts would most likely not work at all. It might be more rewarding letting them rate how the music "felt" (e.g.: more or less "relaxing" for music that should be "relaxing" but is rich in high frequency content nonetheless). This could be done in a way that is scientifically sound and statistically relevant.

My original post:

____________

I have to admit: This 44.1 KHz topic more or less has been discussed to death already. It also seems likely that the following problem has been discussed on Hydrogenaudio several times as well (but I had no luck with the search function).

The 44.1 KHz sampling rate (CD quality) seems to create an infinite number of "mirrors" at its harmonics. These in turn create a complex set of distortion frequencies for every frequency in the analog source.

The strongest "mirror" is at at 22.05 KHz (44.1 KHz/2). But the problem can easily be demonstrated with the one at 11025 Hz (44.1 KHz/4) as well: if one creates a sine signal of 11025-1000 = 10025 Hz in a sound editor (e.g. Audacity, using a 44.1 KHz sampling rate) and plots the spectrum, then two additional frequencies are shown: one at 1000 Hz and one at 22050-1000 = 21050 Hz. More distortion signals can be seen if the FFT resolution is increased above 1024.

The general problem seems to be that a sampling frequency of 44.1 KHz does not guarantee that frequencies below 22.05 KHz are represented faithfully (as is mostly believed). Instead it probably more or less only guarantees that in the resulting complex signal the source frequency is significantly stronger than the numerous distortion signals.

Of course, the remaining question is if these distortions are audible (they resemble pretty much amplitude modulation). I cannot really test this with 44.1 KHz since I donīt have a 96 KHz soundcard. But the example with 11024 Hz surely looks rather disturbing (when looking at the waveform) and doesnīt sound very clean as well.

Did anyone do any respective (blind) listening tests?

zephirus

PS:
The following example is very audible: When using a sampling frequency of only 2000 Hz (instead of 44100 Hz) and creating a sine frequency of 750 Hz (well below the Nyquist limit of 1000 Hz) then the result sounds pretty ugly (itīs some kind of mixed signal of 750 Hz, 250 Hz and 1250 Hz).

This post has been edited by zephirus: May 19 2003, 15:49
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KikeG
post May 14 2003, 14:04
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From zephirus:
QUOTE
A continuous signal of 21800 Hz (with 44.1 KHz sampling, -1.5 dB, 0.5s duration) looks very much amplitude modulated in Cool Edit.

This is because the visual interpolation CE does is not perfect, at high frequencies it doesn't interpolate well enough.

QUOTE
An ideal 192 KHz upsampling filter will create a correct (not modulated) signal regardless (the Cool Edit upsampling does a pretty good job here as well in highest quality mode).


That would be a more proper interpolation, but takes some time to compute well.

QUOTE
But now (before upsampling) letīs silence 0.0000-0.0017 and 0.0023-0.0100. What remains is a small snippet between 0.0017 and 0.0023 (with silence around it).
Without the context around this short snippet, no filter on earth (or in the mathematical domain) should be able to know if that short snippet is meant as a low amplitude signal at around 21800 Hz or a full amplitude signal at exactly 21800 Hz (the upsampling filter will go for the wrong interpretation, and "smears" the signal as well).


This snippet has an only, non-ambiguous, interpretation, given that it doesn't contain any frequencies over 22050 Hz: the one that the accurate upsampling filter does. You say that it smears the signal: well, that is a consecuence of assuming that there are no components over 22050 Hz. If there were no smearing, the signal would have components over 22050 Hz, and it would be impossible to sample and reproduce it properly with a sampling frequency of just 44100 Hz, because it would violate Nyquist theorem.

QUOTE
So it seems: The Nyquist theorem only works for long, continuous signals, but not for short ones. Which are distorted well below the Nyquist frequency. Even with mathematically perfect filters.


No, it is not because of Nyquist theorem, it works always without exception. The problem is on the signal itself: any time-limited signal has infinite frequency components, and the reverse: any frequency-limited signal has infinite duration, from a mathematical point of view only infinitely periodic signals have finite frequency components.

Nyquist theorem says that in order to perfectly sample a signal, it has to be frequency-limited to below fs/2. That signal, in theory, will have infinite duration, it will "ring" forever (infinitely periodic) if it is totally frequency-limited. In real world there are no perfect filters, and since the frequency limiting is not perfect, the time-ringing is limited too: that is the cause of temporal smearing. The sharper (and more ideal) the reconstruction filter, the more time smearing you will get.

This is a problem that happens no matter how high the sampling rate; it is just a question of degrees. How much smearing do you consider acceptable, taking into account human hearing? (smearing=frequency limiting). What is the acceptable duration, amplitude and frequency of the ringing due to the frequency-limiting, taking into account human hearing? (the ringing is of a frequency equal to the filter cutoff, and the duration and amplitude depend on the sharpness of the filter and the amplitude of the signal at the cutoff frequency).

I think that these issues, using a 44.1 KHz sampling rate and typical DAC filters, are not a problem considering human hearing limitations.

Edit: it seems that 2Bdecided was faster than me, but it's basically same explanation.
Edit: some minor modifications done in order to make it more understandable.

This post has been edited by KikeG: May 14 2003, 14:15
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Posts in this topic
- zephirus   44 KHz (CD) not enough !? (Nyquist etc.)   May 11 2003, 17:40
- - tigre   44.1KHz is enough to represent everything below 22...   May 11 2003, 22:59
- - mrosscook   The most recent thread to flog this issue is here....   May 12 2003, 04:32
- - 2Bdecided   QUOTE (zephirus @ May 11 2003 - 04:40 PM)I ha...   May 12 2003, 12:08
- - zephirus   QUOTE (tigre @ May 11 2003 - 01:59 PM)The add...   May 12 2003, 13:07
- - zephirus   QUOTE (mrosscook @ May 11 2003 - 07:32 PM)The...   May 12 2003, 13:14
- - KikeG   QUOTE (zephirus @ May 12 2003 - 01:07 PM)Iīm ...   May 12 2003, 13:33
- - tigre   QUOTE (zephirus @ May 12 2003 - 04:07 AM)......   May 12 2003, 14:16
- - zephirus   QUOTE (KikeG @ May 12 2003 - 04:33 AM)QUOTE (...   May 12 2003, 14:23
- - zephirus   QUOTE (tigre @ May 12 2003 - 05:16 AM)OK. Fin...   May 12 2003, 17:12
- - zephirus   QUOTE (2Bdecided @ May 12 2003 - 03:08 AM)QUO...   May 12 2003, 18:22
- - zephirus   While my original dubious claims obviously had a v...   May 12 2003, 20:54
- - _Shorty   I thought that's what over-sampling DACs were ...   May 12 2003, 21:42
- - zephirus   QUOTE (_Shorty @ May 12 2003 - 12:42 PM)I tho...   May 12 2003, 22:39
- - Doctor   An oversampling DAC still needs to lowpass the sig...   May 12 2003, 22:54
- - Doctor   The distortions you believe in are probably either...   May 12 2003, 22:57
- - KikeG   QUOTE (zephirus @ May 12 2003 - 08:54 PM)The ...   May 13 2003, 07:54
- - Canar   QUOTE (KikeG @ May 12 2003 - 04:33 AM)If you ...   May 13 2003, 08:10
- - tigre   To approximate the waveform between two samples S[...   May 13 2003, 11:15
- - 2Bdecided   zephirus, You were right about the 1kHz tone in th...   May 13 2003, 11:24
- - tigre   Nice link you provided, 2Bdecided. So it seems li...   May 13 2003, 11:42
- - Canar   QUOTE (2Bdecided @ May 13 2003 - 02:24 AM)Can...   May 14 2003, 07:35
- - zephirus   Doctor, KikeG: Thanks for your explanations! ...   May 14 2003, 12:23
- - SikkeK   I think your snippet has alot of frequence compone...   May 14 2003, 12:36
- - DonP   QUOTE (zephirus @ May 14 2003 - 06:23 AM)The ...   May 14 2003, 13:00
- - morelli   it's all too complex for me to understand. but...   May 14 2003, 13:06
- - zephirus   QUOTE (SikkeK @ May 14 2003 - 03:36 AM)I thin...   May 14 2003, 13:39
- - 2Bdecided   Someone do a search. I'm too lazy. But I'v...   May 14 2003, 13:40
- - KikeG   From zephirus: QUOTE A continuous signal of 21800 ...   May 14 2003, 14:04
- - DonP   For an example of this frequency vs time resolutio...   May 14 2003, 14:26
- - zephirus   QUOTE (DonP @ May 14 2003 - 04:00 AM)A pure s...   May 14 2003, 14:40
- - mrosscook   Zephirus, Harry Nyquist formulated his sampling t...   May 14 2003, 15:07
- - DonP   QUOTE (zephirus @ May 14 2003 - 08:40 AM)...   May 14 2003, 15:21
- - budgie   QUOTE (mrosscook @ May 14 2003 - 06:07 AM)I...   May 14 2003, 16:07
- - zephirus   QUOTE (2Bdecided @ May 14 2003 - 04:40 AM)Aft...   May 14 2003, 18:07
- - ye110man   QUOTE (DonP @ May 14 2003 - 06:21 AM)The Nyqu...   May 14 2003, 21:46
- - DonP   QUOTE (ye110man @ May 14 2003 - 03:46 PM)i th...   May 14 2003, 22:22
- - KikeG   QUOTE (budgie @ May 14 2003 - 04:07 PM)20 bit...   May 15 2003, 00:03
- - budgie   QUOTE (KikeG @ May 14 2003 - 03:03 PM)I don...   May 15 2003, 08:51
- - dillee1   Prove of Nyquist theory always use stationary sign...   May 17 2003, 10:44
- - DonP   QUOTE (dillee1 @ May 17 2003 - 04:44 AM)Prove...   May 17 2003, 13:11
- - zephirus   QUOTE (KikeG @ May 14 2003 - 05:04 AM)This sn...   May 17 2003, 17:37
- - zephirus   QUOTE (mrosscook @ May 14 2003 - 06:07 AM)Har...   May 19 2003, 15:05
- - KikeG   If I could have access to some 24/96 music in wav ...   May 19 2003, 15:48
- - budgie   QUOTE (zephirus @ May 19 2003 - 06:05 AM)A ma...   May 19 2003, 15:58
- - 2Bdecided   QUOTE (zephirus @ May 17 2003 - 04:37 PM)QUOT...   May 19 2003, 16:02
- - KikeG   QUOTE (2Bdecided @ May 19 2003 - 04:02 PM)If ...   May 19 2003, 16:39
- - Pio2001   QUOTE (KikeG @ May 19 2003 - 06:39 PM)I have ...   May 19 2003, 20:00
- - Pio2001   QUOTE (zephirus @ May 19 2003 - 05:05 PM)The ...   May 19 2003, 20:16
- - budgie   QUOTE (2Bdecided @ May 19 2003 - 07:02 AM)P.S...   May 20 2003, 11:12
- - 2Bdecided   QUOTE (budgie @ May 20 2003 - 10:12 AM)You ca...   May 20 2003, 17:34
- - 2Bdecided   QUOTE (KikeG @ May 19 2003 - 03:39 PM)I don...   May 20 2003, 17:35
- - zephirus   QUOTE (2Bdecided @ May 19 2003 - 07:02 AM)QUO...   May 20 2003, 17:44
- - zephirus   QUOTE (KikeG @ May 19 2003 - 06:48 AM)If I co...   May 20 2003, 21:48
- - mrosscook   @2Bdecided -- You mentioned in your post that you ...   May 21 2003, 00:04
- - budgie   2Bdecided: Nice post, but it proves nothing, sad ...   May 21 2003, 08:33
- - KikeG   David (2Bdecided): I find your report interesting...   May 21 2003, 10:18
- - 2Bdecided   budgie, I agree that this has no relevance in the...   May 21 2003, 10:24
- - KikeG   QUOTE (zephirus @ May 20 2003 - 09:48 PM)The ...   May 21 2003, 10:25
- - 2Bdecided   QUOTE (mrosscook @ May 20 2003 - 11:04 PM)@2B...   May 21 2003, 10:34
- - 2Bdecided   QUOTE (KikeG @ May 21 2003 - 09:18 AM)David (...   May 21 2003, 11:02
- - KikeG   QUOTE (2Bdecided @ May 21 2003 - 10:34 AM)As ...   May 21 2003, 11:13
- - KikeG   QUOTE (2Bdecided @ May 21 2003 - 11:02 AM)Whi...   May 21 2003, 11:20
- - 2Bdecided   QUOTE (KikeG @ May 21 2003 - 10:13 AM)Then, a...   May 21 2003, 11:30
- - mrosscook   2Bdecided, Do you recall whether the setup of Demo...   May 21 2003, 15:15
- - DickD   With 24-bit there's enough thermal noise in th...   May 21 2003, 19:45
- - DonP   QUOTE (2Bdecided @ May 21 2003 - 05:30 AM)Now...   May 21 2003, 20:36
- - Pio2001   QUOTE (KikeG @ May 21 2003 - 01:13 PM)Also, I...   May 21 2003, 22:07
- - budgie   QUOTE (2Bdecided @ May 21 2003 - 01:24 AM)Tot...   May 22 2003, 09:09
- - towolf   QUOTE (Pio2001 @ May 21 2003 - 11:07 PM)There...   May 24 2003, 00:43
- - KikeG   QUOTE (Pio2001 @ May 21 2003 - 10:07 PM)There...   May 24 2003, 12:54


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