[Publius]

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Originally Posted by Roam /img/forum/go_quote.gif Nyquist's theorem gives the sampling rate required to capture information of a given bandwidth onto the recording media. This is the famous sampling rate equals twice the highest frequency axiom which we're all familiar with.

It's not an axiom, it's a theorem.

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Unfortunately, the Nyquist theorem which people endlessly quote without understanding tells you nothing about the requirements of converting the data on the media back into an analogue waveform, that part's covered by Claude Shannon's contribution to the Nyquist-Shannon sampling theorem. It states that at 22050Hz in the case of redbook CD, an infinite number of samples are required to accurately reproduce the waveform, which of course is clearly impossible. In other words, as the frequency of the waveform approaches half the sampling frequency, D/A conversion accuracy goes down the crapper.

Actually, that's the exact opposite of what Nyquist-Shannon says. At fs/2, no reconstruction is possible, but for all frequencies <fs/2 reconstruction is perfect (subject to implementation limitations). Dan Lavry's infamous paper is a good source to read for the nuts and bolts of what can go wrong in reconstruction, but for a modern sigma-delta DAC, this isn't believed to be any problem.

It's easy to show through simulations that a 22.04khz sine wave is completely reconstructed. Whether your hardwave can do that or not is your problem, but I've seen scope plots showing that 14khz sine waves are accurately reconstructed up to several harmonics up, so... exactly what situation are you referring to?

 

[Roam]

Quote:

Originally Posted by Publius /img/forum/go_quote.gif It's not an axiom, it's a theorem.

Yes it's a theorem, but people constantly quote it without understanding, hence for them it's an axiom.

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Actually, that's the exact opposite of what Nyquist-Shannon says. At fs/2, no reconstruction is possible, but for all frequencies <fs/2 reconstruction is perfect (subject to implementation limitations). Dan Lavry's infamous paper is a good source to read for the nuts and bolts of what can go wrong in reconstruction, but for a modern sigma-delta DAC, this isn't believed to be any problem.

Nyquist-Shannon states that at fs/2, an infinite number of samples is required to reconstruct the waveform, which is why it can't be done. Frequencies of less than fs/2 are in theory perfect, however the implementation is anything but, so in practice there is decreasing accuracy as fs/2 is approached.

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It's easy to show through simulations that a 22.04khz sine wave is completely reconstructed. Whether your hardwave can do that or not is your problem, but I've seen scope plots showing that 14khz sine waves are accurately reconstructed up to several harmonics up, so... exactly what situation are you referring to?

Try using something other than a sine wave. Sine waves and any other regular waves are trivially easy for interpolation algorithms & filters to reconstruct even with very few data points. Use a sine wave modulated with random frequency chirps, it gives most interpolation algorithms the fits.

 

[Publius]

Quote:

Nyquist-Shannon states that at fs/2, an infinite number of samples is required to reconstruct the waveform, which is why it can't be done. Frequencies of less than fs/2 are in theory perfect, however the implementation is anything but, so in practice there is decreasing accuracy as fs/2 is approached.

I'll give you that, but what evidence do you have that this is a real issue with high quality (heck, even commodity) audio gear?

The one post I have seen that actually looked at the analog reconstruction implies that this is simply not an issue, at least at 14khz.

'Normalization' of PCM audio - subjectively benign? - Hydrogenaudio Forums

Assuming that reconstruction is not a problem up to the end of the passband, and assuming that the passband is adequate for the ranges of human hearing, there's no problem.

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Try using something other than a sine wave. Sine waves and any other regular waves are trivially easy for interpolation algorithms & filters to reconstruct even with very few data points. Use a sine wave modulated with random frequency chirps, it gives most interpolation algorithms the fits.

Ah, yes, the old "complex signals are too much for nyquist" saw. Explain to me how the chirps are bandlimited? And how exactly you can show that digital sampling adds distortion besides the matters of quantization noise and lowpass filtering?

 

[systemerror909]

Quote:

Originally Posted by Roam /img/forum/go_quote.gif It states that at 22050Hz in the case of redbook CD, an infinite number of samples are required to accurately reproduce the waveform, which of course is clearly impossible. In other words, as the frequency of the waveform approaches half the sampling frequency, D/A conversion accuracy goes down the crapper.

This is consistent from what I remember back when I was an optics major studying the reconstruction of waveforms. I no longer remember the mathematics behind this, but I've always been suspicious of the the redbook format because of this.

I don't know what the limits of human hearing are, but I suspect the 44.1khz standard is at least an order of magnitude off.

 

[joe_cool]

Quote:

Originally Posted by systemerror909 /img/forum/go_quote.gif I don't know what the limits of human hearing are, but I suspect the 44.1khz standard is at least an order of magnitude off.

DAT (digital audio tape) recorders were standardized at 32kHz sampling rate.

 

[hciman77]

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Originally Posted by Seidhepriest /img/forum/go_quote.gif That humans cannot hear well above 22050 does not mean gear can't.

Well, the point is surely what we as humans can hear, we know that our ears are very poor compared to some animals and a lot of measuring kit, but if we cant hear it is pointless to worry about capturing it.

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That humans cannot discern well above 20 KHz also doesn't mean they can't actually "hear" and be affected by such frequencies.

Actually if you cannot detect the presence or absence of a signal above 20K and sufficiently above the ambient noise level it is reasonable to say you cannot hear above 20K (at that volume level and under those conditions), you might experience it in some other way but that is not hearing which is the point at issue.


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Here's an article on tests of how hypersonic audio affects perception: Inaudible High-Frequency Sounds Affect Brain Activity: Hypersonic Effect -- Oohashi et al. 83 (6): 3548 -- Journal of Neurophysiology

An interesting paper, well explained, I just have a few issues with it.

1) The listening tests were not blind in the sense that they took a known ABBA form , i.e the listeners knew that A and B were not the same , thus of course they could tell a difference since they knew they were different a priori. This is a fatal flaw.

2) The graphs are pretty poor but if you zoom in on them you find that the HCS and FRS sound spectrograms are not quite the same below 20K, thus the test is invalid to start with.

3) Oohashi is indeed a PhD, he is a Doctor of Agriculture, he is less qualified to do this neuro-physiological analysis than I am , I have two degrees in Psychology and I would not call myself adequately qualified to do that part. Yet he gets first authorship on the paper, this is strange and a little bit iffy.

4) Oohashi has a patent out on a therapeutic device based on high frequency sound generation, thus he has an interest in the outcome of the study, this is a conflict of interest and my IRB would chuck a proposal for that study straight into the bin.

5) Oohashi designed the super-tweeter, a commercially available product and this gives him another interest in the outcome of the study, see conflict of interest.

6) Clearly the high frequency sound + normal sound, if we can trust the results has a physiological affect, but it isnt in the auditory cortex, it is in the occipital lobe and that houses most of the visual cortex, processing visual signals. Also the effect is to increase alpha wave activity. Increasing alpha wave activity actually results in relaxation. Thus the sound plus hypersonic sound is causing a relaxation response. So the subjects relax more, this means that they are inevitably going to ascribe positive attributes to this stimulus, note also which ones 5/10 are affected, these are entirely predictable. Thus what we have here is pretty much a smoke and mirrors job and nothing to do with hearing.

7) Meyer and Moran (2007) - stripping the High frequency sound from High res recordings by downsampling - 500+ trials, not one subject detected the difference. There are similar AES papers from 1980 that show the exact same effect for filters at 20, 18 and 16K. Let me hunt them down and I will provide exact citations.

8) See also - NHK Laboratories Note No. 486 - Perceptual Discrimination between Musical Sounds with and without Very High Frequency Components

Toshiyuki Nishiguchi, Kimio Hamasaki, Masakazu Iwaki, and Akio Ando
PREPRINTS- AUDIO ENGINEERING SOCIETY

Similar to the Meyer and Moran paper. One out of 36 subjects could detect a difference but when retested could not repeat this feat.

9) Audio Engineering Society Convention Paper 5401, presented at the 110th Convention, 2001 May 12-15, Amsterdam, The Netherlands:

"Detection threshold for tones above 22 kHz," by Ashihara Kaoru and Kiryu Shogo, National Institute of Advanced Industrial Science and Technology 1-1-4 Umezono Tsukuba, Ibaraki 305-8568, Japan.

Suggested that the effect that Oohashi et al got was due to intermodulation distortion. Ultrasonic components can intermodulate with signals in the audible range and produce intermodulation products in the audible range. When the ultrasonic components were sent to separate drivers, intermodulation was avoided, and the DBTs no longer showed that the listeners could detect the presence of ultrasonic components.

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See Digital Sampling and Fidelity Issues - DSPWiki

A serious citation please.


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It won't be "perfectly adequate". Real dynamic range of 44.1/16 is ~80 dB. Anything outside of that is already severely limited. Try creating a 16-bit 44100 Hz wave file in any editor (Cooledit/Adobe Audition, Soundforge, etc.) and see how low the editor will even allow a test A440 sine to go. It won't be capable of generating anything meaningful below -80 dB. It will generate at -96 just fine in 32-bit though.

I just did this in Audacity a 440Hz sine wave created at 16/44.1, I have selected some excerpts from the analysis - so not to waste bandwidth

Frequency (Hz)Level (dB)
86.132813-40.981213
172.265625-33.443314
258.398438-21.920282
344.5312507.525634
430.66406314.968718
516.79687510.345199
602.929688-17.853697
689.062500-30.517448
1550.390625-69.190414
1636.523438-71.023209
2239.453125-80.871658
2670.117188-85.912354
3100.781250-89.921509
3186.914063-90.604042
3273.046875-91.288101
3445.312500-92.530106
3617.578125-93.650513
3789.843750-94.592583
3875.976563-95.095192
4134.375000-96.217117
4392.773438-97.193878
4823.437500-98.116386
5770.898438-99.002975
10077.539063-96.357849
12058.593750-95.272141
13092.187500-94.826103
14039.648438-94.436935
15073.242188-94.079483
16020.703125-93.927109
17054.296875-93.612221
18001.757813-93.369453
19035.351563-93.268044
20068.945313-93.094437
21102.539063-93.111069
21963.867188-93.163643

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This is why, by the way, the better-sounding CD players have oversampling and 20-bit+ DACs.

This is an opinion.

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Many CD players have an SNR of 84-88 dB. Apple IPods have an SNR of 90 dB. Cowon players have an SNR of 95 dB, and it shows instantly. The clarity of an expanded dynamic range is quite obvious. This means, by the way, that the internal DA resolution of such a player is better than 16-bit (a 16-bit device cannot produce a listenable 96-dB dynamic range).

Which CD players have a SNR of 84 - 88db ?, please point me to concrete examples. And first gen players dont count. Even my $60 DVD player does much better than that, so does my $210 5 disc player and a 1990 Onkyo CD player. The last player I had that might even have gotten close to being that limited was my 1984 Marantz CD63 which was a 14 bit player !

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The effect an amplifier has for headphones, supplying enough power to properly drive them, bringing out detail and presence, is similar to the effect 32-bit audio has compared against 16-bit.

Er, this is unrelated to what you said last time which was that a headphone amplifier increases dynamic range which I believe I showed to be misleading.

 

[systemerror909]

I don't know if this is relevant, or even how sound reproduction actually works, but I'm throwing it out here to see what people say.


Nyquist theory applies to all waves, not just sound. Here is an image I'm posting that visually depicts my concerns for the theory.

Image hosted with 4FreeImageHost.com

This is an optical resolution test image. Before it was digitized, it was five perfectly straight lines converging to a point. I've got two points of interest on the image. The max resolution arrow shows the point where the lines reach the maximum sampling frequency. One cycle (black to white) per two vertical pixels. Now the second arrow points to a lower frequency (say of wavelength 2.25 pixels) where the line still appears, but is sorta gray/fuzzy. What gives? The frequency is less than half the screen resolution, and while providing an approximation of what the actual lines might have looked like, is still a visual distortion of the original.

Two things to point out about this model:
1. This model violates one of the conditions of nyquist sampling: in the reconstruction of this image there is not an infinite number of samples, but I believe the same is true for audio dacs.
2. This distortion should be audible, the gray/fuzzy area is located below the maximum frequency.

Does this make any sense?

 

[Publius]

The short answer is that your example corresponds to the use of zero-order-hold NOS DACs.... and zero-order-hold NOS DACs suck.

The eye (unlike the ear) is astoundingly sensitive to high-frequency detail in images. Anytime in real life that you have one object occluding the other, that is effectively a discontinuity at the eye's natural resolution - something like 9000x9000 effective resolution as an upper limit, according to Wikipedia.

The obvious grayness is due to the natural antialiasing effect of the camera's imaging process. Each pixel of the image corresponds to the light covered by a single square, averaged across that square. So naturally if there's a little line and a little white where a pixel's supposed to be, that pixel will turn out gray.

This is exactly like how a non-oversampled zero-order-hold DAC works. You output a sample to the DAC, it will spit out that voltage, and it will stay there. So you get lots of harmonics this way (just like in this image, on an LCD, there are still gross discontinuities between the shades of gray and white). You also get significantly reduced high-frequency response close to the Nyquist limit. This results in the graying effect you're seeing.

An oversampling technique would not have this issue as much, but would instead suffer from ringing near discontinuities, which the eye is also very sensitive to. (That's why JPEG artifacts are so visible.) So instead, some other interpolation algorithm is used for up/downsampling. That's just because the eye is not an ear, and doesn't have anything to do with Nyquist per se. If you had the image sampled at the eye's "Nyquist limit", the ringing may not be a problem.

 

[bigshot]

There is a point where the bandwidth is sufficient to reproduce the sound within the range of human perception. A lot of audiophile energy is expended on achieving sound quality that the audiophile himself can't hear. The irony is that most audiophiles do very little to balance the frequency response of their systems within the band occupied by music. They ignore what they *can* hear to focus on what they *can't*.

See ya
Steve

 

[bigshot]

Quote:

Originally Posted by hciman77 /img/forum/go_quote.gif Well, the point is surely what we as humans can hear, we know that our ears are very poor compared to some animals and a lot of measuring kit, but if we cant hear it is pointless to worry about capturing it.

When it comes to real world audio issues, trying to push subsonic frequencies (down around 20Hz) through an amp at a balanced response is going to require some real horsepower to avoid clipping. There's a real chance that the frequencies you can't hear will mess up the reproduction of the ones you can.

See ya
Steve

 

[Hellenback]

Forget it.