I read something rather interesting today. I browsed through the
headphone amp designs available at headwize.com and found this article.

Look at figure 1. The guy explains why 44.1kHz isn't fast enough to
sample a 21kHz signal. Nyquist was completely wrong!

Now.. if you look more closely at the second picture of figure 1, you
will notice a flaw in the "21kHz image-signal" picture. He simply
performs a linear interpolation between every point. (!)

How could such an intelligent guy makes such a mistake? I mean, read
the article.. you will see this guy is far from being stupid.

Am I missing something?!?!

I am in no means qualified to really discuss this... it seemed logical to me though. In what sense do you feel he was incorrect though? How would you draw it differently?

I've never read such a crock of lard in my life. Nyquist would be scandalized. Clearly this character doesn't know the first thing about oversampling, sigma delta modulation, and decimation filtering. Instead, he advocates using a less-than-brickwall analog filter to "warm up" the sound in the passband. Hey - let's add some phase distortion (among other types) to the passband and completely ignore the fact that converters these days make it unnecessary to filter out the narrow gap between the edge of the passband and 22.05kHz. This fellow's project was a museum piece back in the mid eighties when the first 4X oversampling machine hit the stores.

Why can't people who think they know what they're talking about take a bit of effort to read up on DSP fundamentals...

:x

edit: I'm backing out of this conversation while I still can

If I didn't know I was dealing with sines, I could use a quadratic, cubic, Nth order polynomial function to approximate what really was the function (the data between the two points). A linear approximation is really crude.

Knowing I am dealing with sines, well... that's even easier. There is mathematically one and only one way to represent a sine given two points (with only two points you can represent it perfectly.. frequency wise and amplitude wise). I am not inventing anything here. This has been discussed amply.

So why does he say you cannot represent a 21kHz sine (well.. he says anything beyond 15kHz I think) correctly when sampling at _only_ 44.1kHz? His linear interpolation is debatable to say the least.

I am not saying 16/44.1 is perfect. Don't want to go there either. I was just wondering if I was missing his point or something...

Exactly. The shortest distance between two points is not a straight line when dealing with a band-limited signal. By drawing straight lines, he completely misses the point and introduces spanking brand new harmonics to the original sampled signal outside the passband. I think we learned not to draw straight lines on the first or second day of my 4th year Discrete Time Systems course...

:x

Phony. The guy takes the sampling of a 41.5 kHz wave, and since it's like a 2.5kHz wave, he extrapolates backwards and says that's a problem at 21kHz, which it's not.

I see. I would really like to learn about this stuff, starting from a pretty ignorant background - any tips as to a good website or book? I thought his description was pretty easy to follow, too bad it is apparently bulls**t

I don't think Meier adequately (or correctly) described why the ringing occurs in the first place, and therefore I can see why you all are missing the point. It is definitely not because DSP algorithms use linear approximations. His picture is misleading. Series (Fourier, Laplace, etc) are often used to represent analog signals (mathmatically sound by the way).

Some sort of low-pass filtering must be used for digital conversion; you notice CDs cut off around 22kHz (44.1kHz /2, according to Nyquist's formula). Ringing occurs when an infinite series is truncated (discontinuous), and occurs with any practical digital filter. This is known as Gibbs phenomenon and is mentioned in the article. According to Meier this ringing is most pronounced in the case of a hard cut-off (or brick-wall filter). Meier's filter apparently uses a method to ease this hard cut off, and thus lessen the effect of Gibbs phenomena (according to Meier). I have to admit I haven't taken enough signal processing classes to say if it would work or not, or even if common modern digital filters are inadequate in this regard.

Because this ringing is due to the filter, it doesn't make sense to say Nyquist's formula is wrong.

lol

Maybe he just put the linear-approximation diagram on the page to encourage people to buy his $260 Analoguer. The rest of the post might at first seem reasonable, though (to an undergraduate EE student like myself): the more gradual the low-pass, the less ringing present. For people with poor high-frequency hearing such as Meier, high-frequency attenuation won't hurt, so you get rid of the ringing "for free".

But from what I've read in my digital audio books, modern low-pass filters don't suffer from significant ringing, even with relatively sharp filtering. And even then, if the brick-wall filter introduces high-frequency ringing (I think that was what Joe Bloggs briefly mentioned at the beginning of this thread), Meier won't notice it, because he has already admitted that he can't hear high frequencies. What's the point of filtering out something you can't hear? If there is aliasing present in the recording, a high-pass filter won't fix it without removing the frequencies that should be present. Perhaps I'm committing a serious fallacy here, but to me, the Analoguer appears to be pretty useless.

And also, some of this was discussed in an ancient thread back in 2001...

@MTRH: this isn't an FB2K discussion (that was appalling). Try not to post such useless comments.

It should also be noted that this ringing occurs in the frequency domain, not the time domain (Meier's article makes it look the other way). That is, it creates a ripple on the amplitude of the frequency response close to the point of discontinuity, and graduately evens out as frequency decreases. Check out the animation on this page for an illustration of this effect.

Jebus: Meier's explanation for why there is ripple (and what kind of ripple) seems to be incorrect, and does not give credibility to his design.

SometimeWarrior: (off-topic) You go to UCSB? I'm at Cal Poly, computer engineering...

Yes, I'm a computer engineering student at UCSB. They've somehow avoided teaching me what a Fourier series is, after two years. Maybe I should have attended SLO instead?

I'll dig up my math book and look again at your link in a couple of days, thanks. I'll probably end up eating my words...

Don't torture yourself man! We didn't get to Fourier analysis until the end of our second year, and it was not the most exciting thing in the world. I've been avoiding my next signal analysis class like the plague. I avoided looking at the equations on that page; the animation was the part I found interesting.

But hey, I'd have to say that SB just may have more hotter women - but it's a close call. Can't wait 'till it summer weather B)

No no, I don't think it's phony. He sampled a 21kHz wave @ 44kHz (which is okay), but then he interpolated linearly (which is the problem).

Except for that incorrect example though, I think this paper is (mostly) correct.

Question 1: would that 21kHz wave be accurately reconstructed after the 44kHz sampling, if correct interpolation were used (ie: sinc) ? Nyquist says yes.

Question 2: for transients, what is more important: the shape or the curve, or the precise harmonic content ?

The problem occurs when a low-pass filter is introduced, and this is the issue Meier is trying to address. A sine wave at a fixed frequency of 21kHz should be able to be reproduced accurately, and so would a sine wave at a fixed frequency of 19KHz. The real problem is that they may have inaccurate amplitudes. The problem is in the frequency response, and it's not clear if Meier even realizes this. It would be nice if he labeled the axis of his graphs. He describes a picture of a sine wave at a fixed frequency of 21KHz wobbling; this is not correct as I understand it.

So the answer to your question: depends on the quality of the filter. But there is an ongoing debate of whether 44.1kHz is high enough to begin with - so there may be no easy answer.

freakngoat:

I can understand that the ringing problem makes sense, if the ear is more sentitive to the graphical shape of transients than to their harmonic content (which is very possible...).

About sampling, though, I had always thought that any [0Hz .. 22kHz[ tone could be reproduced exactly when using 44kHz sampling and no quantization. Well, it seems that any such tone can be reproduced, but not always accurately ? Or, is his interpolation solely at fault ? Would "sinc" solve that problem ? I don't understand that..

Right: here are the facts.

A brick wall low pass filter should ring. You see his first criticism - that the 21kHz sine wave is amplitude modulated? The ringing removes the amplitude modulation. It's the ringing that fills in the gaps between the sample points, and also maintains the 21kHz wave at the correct amplitude in the regions where the samples appear to be low in amplitude. Without the ringing, you'll get the amplitude modulation. Pick your poison!


This begs an obvious question: if the "correct" filter rings on and on like this, surely that causes problems when the 21kHz sine wave stops. Surely, in this system, it can't just "stop dead"? Correct!

The thing is, if a 21kHz sine wave just "stops", it will generate a click, which has frequencies WAY above 22kHz (the nyquist limit). You can't have a sine wave that stops dead, but has no higher frequency components at the stopping point. It's a fundamental of audio (and other areas too!) - you can even try it practically in Cool Edit! So, nyquist implies that a 21kHz tone can't stop dead, because this would mean it contained frequencies way above 21kHz - i.e. above the nyquist limit.

This means that, the nearer the frequency is to the nyquist limit, the poorer the time resolution is (in terms of cycles taken for the sine wave to decay to some fraction of it's former amplitude) - but this isn't a fault - it's just the physics of bandlimitting the signal.


All he's doing in the article is making a gentle filter. There's no magic. A gentle filter won't ring as much. This means that either:
a) you allow frequencies higher than the nyquist limit through - which will give you apparent amplitude modulation of the 21kHZ tone (he didn't show you that his box still does that, but it may!), or
B) remove frequencies below the nyquist limit. This will give a dull sound. Also, because there's almost no 21kHz signals left after you've killed them with the gentle filter, you're less likely to notice any ringing that may remain.

He's doing both. In a way, this is ideal. With higher sampling rates, you can use a gentler filter, and there will be no ringing or amplitude modulation to speek of. BUT at 44.1kHz, it's a difficult compromise. But it's not magic. You can either:
a) keep all the frequencies up to nyquist, kill all the ones above, and have lots of ringing at the nyquist frequency, or
B) compromise this, reducing ringing, but letting more frequencies above nyquist through, or reducing the amplitude of some frequencies below nyquist that may be audible.


You shouldn't be able to hear ringing at 22kHz - but that's with perfect equipment and old ears. There's a theory that when the energy of an impulse is stretched in time, even though the stretching is only at inaudible frequencies, that it make an audible difference. This is because real audio equipment (especially speakers) produces distortion with all sounds, but not with silence. So, you're adding distortion before and after all sounds. Thinking about how sensitive we are to pre-echo (think about mp3!), and how long a true brick-wall filter will ring for (infinity!) - if the equipment makes this inaudible ringing become audible, then it could be a real problem.

In practice, it's all very subtle stuff, creating very subtle effects. But it's one argument for higher sampling rates - you can use gentle filters and avoid any possible compromise. The best thing at 44.1kHz seems to be to filter gently, starting at around 18kHz, making sure it's nearly dead by 22kHz. Because of the compromise, it makes designing good "sounding" nyquist digital filters as much of an art, as it is a science.


Coming back to the original article, in his box, one of the filters is 2dB down at 10kHz and 4dB down at 15kHz - it's hardly surprising that it makes things sound "less harsh" - try this filtering in Cool Edit and things will sound less harsh too!

Cheers,
David.

NumLOCK: I've just been reviewing my books, and yes, mathematically it can be proven that a 44.1kHz sampling rate is able to recover the exact original waveform with a bandwidth of 22kHz (Nyquist's theorem) so you are correct.

Perhaps the basis for the debate on whether 44.1kHz is high enough has to do with the effect of ringing from the lowpass filters being used. (48kHz would allow a bandwidth of 24kHz and more diminished ringing in the audible range). Or maybe it's because various alterations to the waveform typically being applied these days (effects, equalizers) diminish the original signal less at higher sampling rates. Unfortunately I don't know the answer.

Strictly speaking, all approximation by the analog to digital process is done in the quantization step as long as Nyquist's theorem is satisfied. This is where more bits help.

I don't see any reason why the sampling process would add amplitude modulation to the original waveform. The whole idea is to reproduce the original waveform, and this can be done precisely since 44KHz > 2*21kHz. Certainly the sampling process is not used as a demodulation step. Therefore I can see no reason why ringing would ever be desirable.

Also, you talk about how "long" a filter will ring for. I'm not sure what you mean by this. Do you mean the bandwidth of frequencies which produce noticable ringing? It is my understanding that this ringing is in the frequency domain, not the time domain - however I am going to research this further.

As far as the click at a sudden cutoff, you may be right, but it is not discussed in any of the papers, books, or webpages I have read so far. Since a true brick-wall filter would have a slope of infinity and therefore be non-differentiable at the cut-off point, practical ones do not exist and therefore my be a non-issue. All practical filters have some slope. From what I've read the biggest problem with a very steep filter is the ringing effect. You are right in that if you can't increase the sampling rate some sort of softer filter is needed to reduce ringing. I don't think the answer is to have a filter transisition beyond the Nyquist limit, since this will introduce aliasing errors. My guess is that some sort of gentle filter starting around 20kHz would be good and would not audibly dull the sound.

My question is, what type of filters are typically being used these days for audio? Maybe we're talking about something so slight that it doesn't even make an audible difference. It would be nice to hear some samples.

Just a side note: This is not just a problem in audio, this is also a problem in digital imaging as well, and there are well known filters (Lanczos, cubic) to reduce the same ringing we've been talking about due to Gibbs Phenomenon.

Sampling does exactly this to frequencies near the nyquist limit - if you look at the sample points themselves, rather than the reconstructed waveform. It's what he's trying to show in his first figure.

At the DAC, before the low pass filter, the spectrum extends to infinity (theoretically), each sample having infinite height, infinitely short duration, with an area equal to the amplitude value. That's the theory. In practice, the samples are often reproduced as a staircase waveform - this still has a near-infinite spectrum (just as a "real" square wave has an infinite spectrum). Most importantly, from 22-44kHz, this spectrum is a mirror image of the real spectrum from 0-22kHz.

That's what causes the "amplitude modulation" if you just look at the sample points themselves. For a 21kHz tone, there's also a 23.1kHz alias - without a filter, the two beat - that's what you see in his first figure.



I read this earlier up the thread, but forget to comment - the ringing is in the time domain. Frequency domain wobbles in the filter response have little to do with this discussion. So, by length, I mean duration.



That's because I haven't written any yet Seriously, this is advanced stuff. The ideas come from some backround research which I did for a post doctoral research project which never happened! They're not all my ideas - I spoke to a lot of clever people.



Yes - infinite slope = infinite duration. Impossible. Rather than thinking about making a finite slope, and seeing what the result will be in the time domain, it's easier to take the infinite length time domain filter (a sinc function) - chop it (or window it) down to a sensible length (e.g. a few thousand samples), and see what the result is in the frequency domain. It's the practically the same either way, but I find it easier to think about it this way. You can do it all in Cool Edit with an impulse (drag one sample up), the FFT filter, and the frequency analysis window.



Yes. Though, in the analogue world (I know we're not talking about the analogue world, but anyway...) a filter which drops from 0 to -90dB within 1/10th of an octave isn't called "gentle"!



I can't help you there (how would you listen to them anyway?!) - but you can read...
http://www.dcsltd.co.uk/papers.htm
- the second paper addresses the issue of energy smearing, which is caused by (and, in effect, is) ringing.

"typical" filters are a compromise, and are usually realsed as cheaply as possible. You can download specifications from several companies e.g.
http://www.analog.com/UploadedFiles/Datash...892AD1855_b.pdf



Though the best compromise is vastly different in image processing. Images look OK with the kind of aliasing that would sound terrible in audio. Conversely, audio sounds OK with the kind of ringing that would look awful in images!

We discussed about all this some time ago: http://www.hydrogenaudio.org/forums/index....2957#entry29125

Quoting myself, here are my views:



I think that usual FIR filters at players have quite short pre-ringing, due to the limited number of taps used.

Edit: corrected wrong link.

I'm not an expert in signal processing (yet), but i think the problem is cardinal (sinc) interpolation is practically impossible to do in real-time, since you should know the amplitude of samples which are after the ones you're processing (in the time domain). Someone correct me if I'm saying bulls**t.

Norman

WTF are you guys talking about!!!!!

I thought Chemistry was difficult!!

auldyin

Yeah, that's why FIR filters cause pre-ringing: they begin to output filtered signal *before* the actual signal has come out of the filter. They work realtime by delaying the signal a few ms., they need to take data in advance, in order to work. I don't know if I've explained myself very well...

If fact I used to find chemistry more difficult than these things...

Actually, it was the Shibach EQ thread where you said this.

I did a search for "filter AND ringing" before posting, but I knew I should have also filtered by your username, KikeG. I mean, with 775 posts, I was sure you had addressed this issue before, I just couldn't find where.

Yeah, so did I. I guess it just comes down to what you're interested in.

Anyway, this is a facinating topic. I'll have to come back to it after I take my next signal analysis class (this fall). It's funny though, because this topic relates to a lot of things. We were discussing Fourier series, as well as Nyquist's and Shannon's formulas in my networks class today; I was on top of it.

Yep, true, it seems that I made a mistake when cutting and pasting. I've corrected it, thanks.

He's trying to sell his stuff?

I'd propose that in the future this kind of thread gets renamed or split quicker. It contains a lot of usefull information, but my first reaction on seeing the title was 'sigh' and closing it.

I'm afraid so. The pity is that surely some people bought it.

Either this Jan Meier thinks that he's smarter than Nyquist (when in fact he doesn't understand him), or he is just taking advance of people ignorance, or a mix of both things. I wonder how much he charges for his "analoger" tha tin fact is just a very simple analog delay line.

Be careful with those "audiophile"-style products and explanations!!

No, I'm not an idiot, I don't feel like an audio-guru either, and I don't like to sell equipment using "audiophile" style explanations. Actually, I have a PhD in physics and a profound professional experience in (digital) signal analysis.

Yes, the linear interpolation as shown in the article is a rather simplified one. Not because I don't understand, but because of didactic reasons. "Popular" articles always have to be written in such a way that the layman can understand without a throurough knowledge of the underlying theory. You always have to choose a certain balance between detail and simplification.

However, the beating effect is much closer to the truth than people at this forum seem to think. For a newer version of the article, with a slightly more elaborate discussion on the "beating" phenomenon I suggest people take a look at my home-page: www.meier-audio.com. Simply click at the "analoguer"-button and enjoy.

Cheers,

Jan

I'd say, in case of your article, it is balance between "wrong" and "right" explanation.



I see you have tried to disguise the effect you talk with new fancy graphics and explanations. The fact is that they are still *wrong*. The beating you talk about is a nonlinear type of distortion. Sinc() filtering, which is the filtering used in DACs, won't cause any kind of nonlinear distortion, no matter how short the filter. A short filter will result just in a less sharp filter, but will cause *zero* distortion or beating. Beating is amplitude modulation, which translates into the addition of sidebands to the original signal. The inexistence of this phenomenon can be easily verified with any decent soundcard, just play and record a 20 KHz tone, and analyze it. Result? No beating or sidebands.

As to other assertions you do, good ADCs/DACs introduce minimal phase distortion at high frequencies, just see PCAVTech phase analysis of the LynxTwo, it is close to nothing. Maybe you SACD is one of those esoteric ones that in fact are not close no anything neutral. As to pre-ringing, I explained why it is not important in this case, at a previous post in this thread.




This is just basic theory signal. I guess you need an introductory course, or just some real-world measurements that show any of this.

Edit: added more explanations.

I think the main argument against the analoguer in the posts above was not the linear interpolation in the explanation, but rather that it solves one 'problem' by introducing several others.

PS. I don't know about where you live, but here biomedical engineering is a quite different discipline from physics.

That doesn't wash if the simplification is what introduces the problem you are trying to explain.

Should this apply if the 20 kHz tone is played by a different soundcard (clock shift -> beating) ?

I've analised the output of several CD players and DACs, using a tone 1kHz below nyquist, and an external spectrum analyser (an old HP machine which had 70dB range, and a frequency response from DC to either 100kHz or 200kHz - I forget which). They all showed this effect to some degree. Two of the DACs cost over $1000, so we're not talking cheap junk here!

As you reminded me, we've been through this before. Since we have, we must have discussed (and I'm sure you know) that a shorter filter gives rise to a less steep frequency domain response. The less steep response let's more of the nyquist +1kHz alias through - which beats with the 1kHz below nyquist tone.

Outside the theoretical limits, both sampling and quantisation are totally non-linear. They become linear over a specified range with correct filtering and dither.


FWIW I would rather perform the function of the analoguer using Cool Edit Pro (or a DSP board with 24-bit output) and a good DAC running 2 or 4x (e.g. 88.2kHz or 176.4kHz - it may oversample further internally). Having tried with CEP and the audiophile 2496 (which may not count as a good DAC) I can't say that I hear a difference, but (using a 21kHz sine wave) I can certainly measure one.

I must try ABXing the frequency response of the analoguer - I bet that the high frequency filtering of setting "5" is ABXable, using most normal DACs. This would probably improve the sound of most harsh recordings - but is it really any better than just EQing them to match your own personal taste? That's the fun of Cool Edit et al - we can all be audio engineers in our own bedrooms!


Cheers,
David.

2Bdecided:

We would have to distinguish between issues due to real-world implementation, ideal implementation, and theory. We could split this into 3 levels. In reverse order:

-In theory, when reconstructing sampled information in a pure, limitation-free, mathematical way, without quantization levels, there should be no problems (beating, noise, etc) at all, the reconstructed signal would be identicall 100% to the band-limited original signal. This is what Nyquist says, and I believe that Jan Meier agrees too, at least in his second version of the page.


- Going into a more real-world version of a DAC, we could consider a model that uses quantized, fixed-resolution sampled data, and digital reconstruction filters of fixed length, but that would have no real-wold analog parts inside that could cause any linear or nonlinear distortion or noise by themselves. This is how real-world DACs are, with the exception of the analog components inside them. A DAC of this kind can be perfectly modelled and simulated inside a computer.

Here is where things start to differ. Jan Meier says that in this case, there would be "beating" (nonlinear distortion) due to the use of fixed-length reconstruction filters. False. If you simulate a DAC inside a computer this way, it will still be distortion-free, given that you use dithering as needed. There would only appear quantization noise, depending exclusively on the data size of the samples at the adquisition (ADC) and the reconstruction (DAC) stages. Depending on the implementation of the filter, there would be for sure some images of the reconstructed signal (aliases, although it is not a proper definition of this) remaining above Nyquist frequency, specially if the filter is short and those images are not well attenuated. But, there would be still no distortion below the Nyquist frequency, just reflected attenuated "images" of the signal, above this frequency, due to this possible limitation of the digital filters.


- Now, if we go into a real-world DAC, there is a possibility that due to the non-linear limitations of analog components, that ultrasonic remaining information ("aliases", or better say "images" of the signal over Nyquist frequency) could intermodulate with the reconstructed signal, appearing intermodulation products into the audible band (below Nyquist frequency). This is the only way that this beating can happen, due to analog limitations (linear and nonlinear distortion, noise) of real-world, solid-state, components. But this non-linearity has no direct relationship with the lenght of the filters used at the reconstruction process, or the frequency of the sampled signal, as Jan Meier tries to imply. It can be a non-direct consequence of it, but depends exclusively on the nonlinearity of the system, not the reconstruction sampling process, and will happen equally when playing lower frequency signals.


Now:



I find this quite strange. This is a loopback recording of my $150 card playing two tones of 19 KHz and 20 KHz.



As you see, there are minuscule intermodulation products at around -115 dBFS. However, those are result of the intermodulation of the two played tones, not of the tones plus aliased information over fs/2 (Nyquist frequency). See, the intermodulation products of those -12 dBFS tones fall at -115 dBFS. For sure, the intermodulation products of these tones with the aliased ones will be much lower, since those aliases will be of significant lower amplitude, due to the filtering of these frequencies over fs/2 that any DAC performs.

If I repeat that same measurement with a single 20 KHz tone, I bet that intermodulation products will be minuscule, if measurable at all. Maybe the effect you saw was due to the old HP anayzer you used?



Well, if you use proper dither, the system will be perfectly linear below Nyquist frequency, in the sense of that there will be just quantization noise added, but no distortion at all.

If you have a real tone at 22kHz, and an image of that tone at 22.1kHz, the two will beat, won't they? If they're the same amplitude (unlikely in this case, but just imagine...), you'll see a classic beat waveform, with the amplitude envelope going from minimum to maximum in a regular pattern - in this case 100 times per second. Just what Jan shows on his site.



This is the key to the problem. What you say here is true. "If we look at this very carefully, and if we only consider frequencies below the nyquist limit, then they have not been harmed or changed in anyway." That's true - but in saying it, you are incorporating a perfect low pass filter into your hypothetical viewpoint.

"there would be still no distortion below the Nyquist frequency" = if I have a perfect filter, there is no distortion.

To "see" that there is no distortion, you have to look through a filter. In your viewpoint, you are removing all the components above 22.05kHz; then from this viewpoint, everything looks fine.

(If you use a spectrum analiser, with a long FFT setting, you can see that there are two tones - one above the nyquist limit, and one below - you're saying we ignore the one above - fair enough - but ignoring everything above a certain frequency is imposing a low pass filter on your view. That's not a useful way of looking at it...)


Now, back to reality: Let's say that we can hear 22kHz. I can't. I doubt most people can. But let's say we find someone who can. I think it's quite likely that, if your hearing extends to 22kHz, it'll also extend to 22.1kHz. Therefore, when you listen to the 22kHz tone, reproduced by this DAC which allows a (smaller) 22.1kHz image through, what you'll hear is a rough, beating tone, with a frequency of around 22.05kHz.


There you go - no analogue electronics involved, but a hiedous non-linear effect.





OK - first, I'm not talking about intermodulation - I'm just talking about the presence of the image tones.

Then, second, your looped back sound card doesn't work at a high enough frequency to capture the image tones of itself - the only way to do that it to run one soundcard playing at 44.1kHz, and another recording at (say) 96kHz.



You can see the beating between a 22kHZ and 22.1kHz tone on a 'scope. But I think we're talking at cross purposes - this all goes back to the fact that I think having an image tone is a bad thing (despite it being above nyquist) whereas you don't.

You're right - the effect at 20kHz is probably tiny.

(goes away, looks round the office, opens a box from university)

I've dug out the results of the experiment. I checked 3 DACs on a 'scope, and more using the spectrum analyser. All checked using 16-bit 44.1kHz test signals generated in CEP.

On the scope, I measured the beating, or amplitude modulation, in volts - peak to trough.

Meridian DAC, ref: 1kHz tone Vpp=6.4V
22.0kHz: 1.35V
21.0kHz: 0.32V
20.0kHz: <0.02V

Digilog DAC, ref: 1kHz tone Vpp=7.6V
22.0kHz: 2V
21.0kHz: 0.6V
20.0kHz: 0.1V
10.0kHz: ~0.02V

Sony DAC, ref: 1kHz tone Vpp=7.2V
22.0kHz: 0.96V
21.0kHz: 0.08V
20.0kHz: <0.02V

Then, on the scope, I measured the relative amplitudes of the real and image tones:

SME DAC
20.0kHz: +1.62dB > no image above -78dB
20.5kHz: +1.57dB > 23.6kHz: -50dB
21.0kHz: +1.24dB > 23.1kHz: -27dB
21.5kHz: -0.20dB > 22.6kHz: -13.31dB
22.0kHz: -3.95dB > 22.1kHz -5.09dB

Sony DAC
1.00kHz: +2.21dB
10.0kHz: +2.37dB
15.0kHz: +2.19dB
20.0kHz: +1.98dB > no image above -78dB
20.5kHz: +1.54dB > 23.6kHz: -51.5dB
21.0kHz: +0.12dB > 23.1kHz: -31.01dB
21.5kHz: -2.92dB > 22.6kHz -18.11dB
22.0kHz: -8.11dB > 22.1kHz: -9.45dB

CardD+ sound card
1.00kHz: -4.05dB
10.0kHz: -4.14dB
15.0kHz: -4.22dB
20.0kHz: -4.34dB
20.5kHz: -4.38dB > 23.6kHz: -66dB
21.0kHz: -4.52dB > 23.1kHz: -40.15dB
21.5kHz: -5.62dB > 22.6kHz: -22.03dB
22.0kHz: -9.76dB > 22.1kHz: -11.17dB

Digilog DAC:
1.00kHz: +2.96dB
This DAC has terrible filtering - even the 1kHz tone gives:
~43kHz: -57.8dB
~45kHz: -59.2dB
~87.2kHz: -60.4dB
~89.2kHz: -60.7dB
Similar rubbish for 10 and 20k, then for the main ones:
20.5kHz: +2.62dB > 23.6kHz: -21.59dB
21.0kHz: +1.88dB > 23.1kHz: -12.67dB
21.5kHz: +0.41dB > 22.6kHz: -6.74dB
22.0kHz: -2.05dB > 22.1kHz: -2.69dB

Meridian DAC
1.00kHz: +1.74dB > ~43kHz: -66.9dB (I hadn't checked this high for the other DACs before the one above)
10.0kHz: +1.32dB > ~34kHz: ~64dB
15.0kHz: +1.67dB > 29.1kHz: ~-65.5dB
20.0kHz: +1.54dB > 24.1kHz: -52.7dB
20.5kHz: +1.08dB > 23.6kHz: -29.75dB
21.0kHz: -0.01dB > 23.1kHz: -18.61dB
21.5kHz: -2.03dB > 22.6kHz: -11.21dB
22.0kHz: -5.25dB > 22.1kHz: -6.08dB



At the time, I dumped these numbers into Excel and plotted graphs of the image rejection filter response in each of these DACs - not quite the brick walls I was expecting before I performed these measurements. The Digilog DAC from Musical Fidelity was very old, but both the Meridian DAC and the SME DAC were recent, and generally thought to be excellent.


I did these tests because of an AES paper by Richard Black (I think) who suggested that these image frequencies were a real problem. He doubted that we could hear them, but he pointed out that they are not harmonically related to anything within the true audio spectrum - hence any intermodulation caused by amplifiers or speakers could create some nasty inharmonic components within the audible range. I'm not sure the numbers add up to a huge problem for typical music, but he had a point - it is possible that there is an audible effect.


Cheers,
David.

Not exactly. The output of a DAC goes through a filter (in practice, it's one digital filter with over-sampling plus an alalog filter) that makes sure there's nothing left above 22.05 kHz. Depending on your hardware, it may start cutting at 20 kHz to achieve that (in which case, you'll likely lose the 22 kHz tone), as long as nothing's left above Nyquist frequency. Now for your example, it's quite possible to make a filter that would have its transition band between 22.01 and 22.05 kHz, in which case you'd get the signal correctly. In the case of a normal soundcard, trying to play a 22 kHz tone would probably result in no output at all.

jmvalin,

Have you read the rest of the thread?

Actually, just the rest of the post that you quoted will do - especially the part where a 22kHz tone comes out of a soundcard at only 5.71dB less than a 1kHz tone - that's hardly "no output at all". :;


Hope this doesn't sound harsh - I'm just trying to stop the thread discussion from looping back to the start!


And though it sounds like I think I know all the answers, I actually learn something more about this subject every time it comes around - so I'm not having this discussion with KikeG to say "I'm right, you're wrong" because I've learnt things from what he's said in the past. Just hads to put this bit because it's very easy to sound argumentative in typed things when you don't mean to be.

Cheers,
David.

Well, it has nothing to do with 44.1 kHz not being enough to sample a 21 sine wave. It just shows that the DAC's you analyzed aren't behaving the way a "standard DAC" ought to do. I'm guessing that they just didn't really bother filtering the exact way because you don't really hear above 20 kHz anyway. I don't think it's really worth discussing what each individual DAC does.

The main thing here is that people keep saying "Nyquist was wrong" when they're just completely mis-interpreting the sampling theorem. So far I've seen:

- I use linear interpolation between samples and it doesn't look like the original.
- My soundcard doesn't work with a 22.0499 kHz tone.
- I can sample a 1 MHz tone at 50 kHz and still get perfect reconstruction, Nyquist was wrong. (no you can just compensate the aliasing which is OK as long as you don't have other stuff in the baseband)

this is obviously cannot be a discussion whether nyquist was right or wrong. this discussion is basicly a 'theory vs. practice' discussion..
theory : nyquist wasn't wrong. the sampling function is a one-to-one function iff (if and only if) the sampling-rate is equal or higher than twice the bandwidth of the source signal. in this case (one-to-one), there exist an inverse function.
practice : implementing the inverse function (reconstruction filter) is a problem.


i wonder what's your opinion about an alternative approach :
process your digital signal through a digital upsampler from 44.1khz to 88.2khz (cs8420, for instance, can do this directly on spdif traffic), and then use a lousy anti-aliasing reconstruction filter with cut-off frequency at 88.2khz.


edit : typo

Well, of course you can get almost as close to the Nyquist limit you like by spending more on hardware. As for up-sampling, that's of course the logical thing to do, although you still need a sharp filter at 22 kHz. The only difference is that the filter can be implemented numerically, which is easier. In theory, you could have a transition band of only 1 Hz. The down side is that the (huge) filter would introduce a delay of several seconds.

Anyway, I still don't understand the point of this thread.

This thread deserves to be pinned one and for all times IMHO. What i do understand is that even the most intelligent experts here are slowly coming to the point where i have been at almost 5 years ago ( and i dont understadn at least half of what is said here, being a low-level electrical engineer with bad results in the DSP courses :

Its much easier to drop the old 16/44.1 standard and concentrate on 24/96 instead, to make sure we have a digital frontend that is superior to the human ear under all circumstances, instaed of discussing endlessly if maybe 16/44.1 iss good enough or not, and then tweaking the old standard with expensive equipment to make the best out of it.

F..ck ( censored ), any crappy DVD can hold up to 5 hours of 24/96 recorded Stereo music today, so whats the bloody point in defending the usage of 16/44.1 still ? The necessary ADC/DAC's to record in 24/96 are no miracles anymore, and for low cost equipment you simply downsample to 16/48 and the user of this unit will still be more than happy .....

The mere feeling of being on the safe side does justify the using of higher sampling rates already IMHO, especially with respect to the fact that its technically feasible today !! There is only one reason why DVD-Video with 24/96 has not found broad usage already, and that is because Chesky was the only one to release music on this standard, while Panasonic and SONY are still trying with huge amounts of money to push SACD and DVD-A .. sad but true .... every crappy DVD-player out there ( and there are billions already ) has to be able to play 24/96 audio tracks on DVD-Videos ( somehow ), that makes it even more sad ....

Hey, keep in mind that with Speex, I'm still trying to convince people to switch from 8 kHz to 16 kHz . My opinion is that 96/24 is overkill. While I agree that 44.1 kHz might not be enough (even then, I'm not completely sure), 48 kHz should be enough. It's not only a matter of storage but processing too. In many cases, processing at 96 kHZ is just 100% overhead. As for 24 bits, I think it's usful if you're going to do some processing, but just for playback, I don't think it makes much of a difference when taking into account compression (even at high bit-rate) and the rest of the sound system. I have yet to see a power amp (or even a 24-bit soundcard) with 144 dB SNR.

2Bdecided:

Ok, I must admit I've been trying to explain the wrong thing, and that didn't totally understand what this beating effect consisted of. What mislead me was Jan Meier suggestion of that this was an audible phenomena and for that reason, I assumed it happened below Nyquist frequency. My explanation holds true if you assume that signals above 20 KHz or 21 KHz are not audible. But, as you say:



That's all true, from the point of view of the whole signal, there is distortion. But assuming that our "natural" lowpass filter will filter out all frequencies close to Nyquist frequency and above, it still holds.



True. Doing a short-time analysis of the signal, as our ear does, in this case there would be obvious amplitude modulation.



Now I see. For the reasons explained before I thought this was an intermodulation problem, so I just analyzed below Nyquist frequency.



The measurement data you have provided is very interesting. I have done some quick audibility tests based on the image rejection data, but with audible test tones of 15 KHz instead. According to those, assuming that we could hear above 20 KHz, this beating effect would be audible just near 22 KHz. At 21.5 KHz, the attenuation of the images is of around 15 dB, and 1.1 KHz over the original tone. In this case, my tests suggest (using 15 KHz plus 16.1 KHz "image" tone, both audible by themselves in my case) that the image tone wouldn't be audible, because the beating effect dissapears when the frequencies are distant enough, and because the image gets masked from the original tone.



I think this could only be verified via blind tests. I think that over 16 KHz or so, many of the audible content of music is usually of percusive type (drums, cymbals, transients, etc) and of inharmonic content then, too. On the other side, this beating effect would happen just over, say, 21.5 KHz (being pesimistic), where usually ADCs start rolling off and there's less signal content recorded. I think very plausible the idea that intermodulation products due to high-frequency image signals would be small in comparison with the intermodulation effects on this non-harmonic percusive content over 16 KHz.



Thanks, I'm happy to have helped you on those occasions. That same thing has happened to me sometimes from you in the past too, and has happened in this this particular case.

I still think that, given limitations of human hearing, 44.1 KHz is enough. Maybe for some 0.1% of young bat-eared people there could be a difference using a higher sampling rate, and just on critical material. But even in this case, the difference I believe would be very subtle. Still, I would only be convinced form a rigorous, repeatable, blind test.

KikeG,

Hey - I think we agree!

Cheers,
David.

Stumbling across this discussion, I just had to jump in on some of what's being discussed.


1. Filter ringing / Impulse response

Any filter, may it be analog (electronic), digital or even mechanical, will ring around fc. The steeper the filter, the longer the impulse resonse (ringing) will be. Symmetric filters (only realizable in the digital domain) will have even distribution of pre/post ringing, creating a flat phase response.


2. Audibility of filter steepness

As it has been stated in this thread, it should be impossible to hear filter ringing with fc beyond human hearing. This is INCORRECT. It might be tempting to think so. The human hearing cannot hear steady-state sines over 20kHz. This has been concluded in millions of hearing tests all over the world. However, when listening to impulse responses, we have to take into account what is analyzing these sounds. The human ear has its own set of filters, analysis windows. Analyze a long enough impulse response with an auditory model and you will find that these filters are indeed triggered. This is why we can hear steepness of filters even when fs=96kHz and fc=47kHz.


Cheers,

Martin Saleteg
N i n j a F X