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The questions about modulation of sine waves near the Nyquist limit seem to be common and repeated. The image URL here has a plot that, I think, explains how the "beating" comes about due to adding in frequencies that are solely above the Nyquist limit, and thus how filtering them out removes any "beating" one observes in raw, unfiltered data from a DAC (before the anti-imaging filter).Start at the BOTTOM. That's the waveform you see at the DAC output, before the anti-imaging (reconstruction) filter. In this example, the beating is quite obvious.
Now, going from top down, let us see how this staircase wave was constructed. In the TOP graph, we see the original sine wave (cyan) and the first two images. (magenta and yellow). The black waveform is the SUM of the three. You notice how already, at this point, we have a lot of "beating" happening, and that also, we've very carefully and quite specifically added ONLY sine waves ABOVE the anti-imaging filter cutoff.
The middle three rows add more of the images, adding 2 sets (4 total sine waves) each time. (c,m,y,g in order, lowest to highest image)
The final row is the sum of the first 1000 images. I didn't bother to show the other 900+ very high frequency sine waves.
So, very clearly, shown in simple pictures, the result is clear, the waveform that shows the modulations at the output of the DAC arises from OUT OF BAND signals.
The graph, btw, is Copyright J. Johnston, 1999, 2000.
Thanks jj. And for more info than anyone should ever want, see the previous thread on CD sampling:
I appreciate the thought that you probably meant by including the link, however, I would just as soon forget that thread altogether. In fact, I am so embarrased by the whole affair that I have decided to refrain from further posts; I will enjoy from a distance, thank-you. Much praise to Werner, jj, and others who contributed and ultimately did an admirable job of explaning where I could not. I am humbled.
Sorry, I had read some of the technical stuff and there's some good information there. Honestly I didn't read the whole thing, but now I see the sparks.
I wasn't one was I? I wasn't intending to spark anything. I was trying to find out how a reconstruction filter given infinite time can retrieve a real 20kHz signal with a real world DAC. Apparently, it can't.
If you mean 20kHz signal with 44.1kHz sampling.No problem. And you only need a few milliseconds of delay, even.
Great, can you show me the reconstruction of a real world 20kHz signal (ie:naturally damped)? I'd love to see that
You appear to be assuming that because a damped sine wave with a 20kHz carrier has a 20kHz carrier, it has only frequency content at 20kHz.This is not true, going back to the "convolution theorem" discussed below, or somewhere around here, you must convolve the frequency shape of the damping signal (e^-at) with the sine wave. The shape of the damped sine wave, especialy if it has a very sharp attack, is quite wide, very likely more than 4khz. This means that the FREQUENCY CONTENT of a "damped 20khz sine wave" is WELL ABOVE 22.1 kHz.
So, it has out of band signals. You can, of course, filter that signal BEFORE you sample it, and prevent any aliasing, and remove the out-of-band information at the same time. This will change, somewhat, the shape (mostly the attack) of the damped sine wave. This is not wrong, this is not distortion, it is REMOVING COMPONENTS ABOVE THE NYQUIST LIMIT.
Just because the sine wave is at 20kHz does not, repeat NOT mean that the signal spectrum consists only of 20kHz, and in fact it does not consist only of 20kHz unless the sine wave is of infinite duration.
For anything other than infinite duration, you must convolve that line at 20khz with the shape of the time window, be it a naturally damped signal, a guassian window (which has the narrowest time/bandwidth tradeoff, i.e. it's the case where t*f = 1, where t is time resolution and f is frequency resolution, and we're talking about one sigma in each case, please see Morrison for a complete reduction of this case), a Hann window, a Hamming window, Blackmun, rectangular, Kaiser, ... what-have you.
All windows that are not identically a single constant over all time introduce spreading of the spectrum. This is why you can only recover something at exactly half the sampling frequency with infinite delay, btw, because that is the only time that a sine wave exactly at the Nyquist bound will have NO signal above it.
But, please, understand this, you can get arbitrarily close to that bound.
You can not, though, get closer than that tf> =1 relationship, although you can, if you must, redefine that relationship to have a different error criterion (some people have done that, but all it does is create confusion). That means that if the signal is of length 't' you never get any signal closer than fs/2 - 1/t resolved, because it will necessarily have out of band components, and they will get removed in the input antialias filter.
Is this getting through?
you weren't talking about an infinite amount of iterative steps shown by the few in your reconstruction example ( which I found informative because prior I thought they just used Fourier splining techniques to reconstruct the signal). You were talking about an infinite storage array of data. So, how close to real is the last octave ( fs/4 - fs/2 ) in a real world sample?
It is interative, but the reason I stopped at 1000 aliases is that at that point, the energy in an alias is very small, each alais (due to the step-function nature of the DAC output) is smaller and smaller...There reason I didn't plot the last alias is that it's so small you can't see it on the scale of +-1.
When I say a sinewave must be of infinite length to have ONLY frequency content at the frequency of the sinewave, that's a simple, outright statement.
The problem with your example of a damped sine wave is that it has out-of-band content, i.e. content ABOVE the Nyquist limit, at least for most examples. (It would be possible to construct one that did not encroach, to a given error level, the area above 20khz, and that example would in fact work fine. Demonstrating it would be tougher, because one oculd not merely plot sine waves as aliases, but whole spectra as aliases (yes, they could be turned into a time waveform), and the nature of the signal wouldn't be nearly as obvious.
So, the last octave is FINE. That's what the example shows. It shows that any in-band signal is FINE. Period. (of course given competant filter design, which may not be an entirely safe assumption ;_(
how much?
If you mean a damped 20khz tone that has no out of band components, that's easy, and it's old.Use a guassian impulse with a 10 millisecond sigma. That will mean that its 1-sigma bandwindth is 100 Hz. That means that at 1khz, it's down more dB than you care to talk about.
Now that's not an e^-at damping, but please, just PLEASE go to Morrison or any other Fourier Analysis book, it will show you that there are (literally) an infinite class of what you ask for.
I'm not sure what you're asking for here.
I have only had esperience wwith Fourier in the analog domain. My digital experience was restricted to Guassian splines. Maybe I ought to dust them off to check if there was a Fourier chapter.As far as your designing something, I was referring to the competant filter in your statement ....
So, the last octave is FINE. That's what the example shows. It shows that any in-band signal is FINE. Period. (of course given competant filter design, which may not be an entirely safe assumption ;_(
Somewhere around 1978 or 1979 I designed a set of antialiasing filters that makes the requirements for CD seem like a drunken dance party (this would be the IEEE publication of the "Commentary Grade Coder" by Johnston and Crochiere).Yeah, I have. I've also done both analog and digital antiimaging filters for 44.1 -> 88.2 and direct conversion.
The thing is, the "how" is pretty well known, but people try to CHEAP OUT.
And there in often lies a big ((&*( problem.
I'd like to buy a better one but haven't heard one yet better than the ones with ring DACs & tube powered analog sections... but they were lacking too. Unless, that was the limitations of the tubes?What do you recommend? Or, do you tweak brand-X device into the right configuration reguardless of which manufacturer once you consider units > $1k?
I'm wondering if you haven't developed a preference for a bit of distortion on signal peaks.I have no idea if it's true, but a preference for a bit of distortion on peaks (not clipping!!!!) can result from a preference for 'more dynamic' sound.
Does that sound at all possible? It's awfully hard to guess from this distance.
good guess over this deaf medium for I've known people who have, but what I'm refering to is a perceived temporal lag associated with rendering a bell-curve envelope rather than an exponential damping envelope. Plus, an incapability to reproduce piercing sounds that were in the original signal, while not contaminating other sounds with enhancements that weren't there originally.does this help?
More than that, I'd have to hear what you mean...That is the problem with this medium.
the unfortunate thing is once you have, it'll haunt you as it does me. Then, you'll really get pissed at me. A scenerio I'm more than used to by now.
I pretty much gave up on higher education in...I think it was the eight grade...Algebra I. I sat behind Debby Stewart, who (San Jaun Capistrano Junior HS, in Orange County, Southern California, early sixties), had the most beautiful long blond hair, and huge..."Tracts of Land".This was a time of Earthquake Drills (where they rang the "warning" alarm five or ten minites after the tremblor) and standing up to answer the teacher's questions. A time of tempered steel boners--the classic "all day sucker" kind of pain. Now I forever associate the language of $cience, with hard-ons.
So jj, please, if you have even an inkling of mercy left, please post that link to your resume here and now, so I can gaze upon that smirking, hairy face, and fantasise about Debbie once more.
as our president says, "I feel your pain."
!txet oN
The two people who did explain it have both been around the audio and sampling arenas for quite a while, and have done this more than once before.It's hard to beat a collective 50+ years of experience.
The following two URL's (one standard way, one in text), show:In text URL: This shows 8 sampled 20kHz sine waves, sampled at 44.1 kHz. The 8 different phases are each 45 degrees advancing, from top, down. The fact that the 8 phases are entirely visible is obvious, I think.
In standard link URL: This is a blown up picture of 0-7 degrees advancing. Notice that while the peaks don't seem to move much (because they don't) the details around zero are definately different, and you can see them evolving as the phase rotates.
So, we can show, visually, via rather insensitive displays, that phase IS preserved.
How's that, Werner?
In_text URL is:
Advancing Phase by 45 degrees
Now if we can get the AA moderators to put this info into the FAQ..._Spike
Now that you're busy and have shown that you have
access to a Matlab-like thingy, how about showing everyone
that phase is maintained, even for frequencies close to
fs/2, e.g. by modelling a 1kHz + 20kHz stimulus,
sampling it at 44kHz, and reconstructing it? And this
twice, for different phase relations between the 1k and the
20k?I can't: this company has 1 (one) Matlab license for
800 people :-)
A bit more complicated, but that's a good idea.You guys need more Matlab licenses :_)
... with Macsyma & Mathematica around.However, if phase wasn't maintained the square wave would've never resolved. It would've oscillated instead because the HF correction waves would've been misaligned (out out phase with lower frequency base signal).
If however, your program is up & running, could you be so kind to run your program on a simple 44.1kHz sampling on a naturally damped 20Khz signal? I'd be very interested in how this is resolved.
BTW, if this is all done for oversampling, how does over sampling work so much better?
Depending on how the exponential pulse is generated, it's most likely to have components well above 22.05 kHz in the original... So, it doesn't obey the Nyquest requirement.It's also true that the extra components of that, while analytically calculable, are not as simple as one might think, and are broadband...
Oh well, live & earn. <> It was that precise signal how my doubts with Nyquist were raised initially. I just simplified it to a constant 20kHz signal for acedemic purposes, knowing full well no DAC can use more than a half second of data. Then, I saw the mirage of dawn in the possibility of the reconstruction filter, only to find out it was analog which made me doubt the fidelity of anything above 15.58kHz once again (increasing skepticism with frequency of course until 22.05kHz & then decreasing once again like a octave band of error). However, the fact it can be digitally done for upsampling gives me hope that an infinite XO make this band vanish. But, then again, you'll most likely be at the mercy of a FFT's accuracy. Ah, once more into the breech ...
That the damped sine wave has out-of-band components. As we all recall, we can NOT reproduce those, but they are also above 22.05kHz, too.So, their presence is not very likely to be audible unless you're very young, never rode in a car, bus, train, subway, airplane ...
The modern world does wonders for our hearing :-(
It's problems migrate down to the audible region unless you've degraded your hearing down to a 15kHz upper limit. And, yes I can hear 20kHz thank you, but I do know that I'm one of the few. I do forget this latter point from time to time. Running out of crowded A/V stores that have alarm systems armed or loud CRT noise, are my only reality checks.
Unless you hear above 20kHz. I can still hear the (&(*&(* ultrasonic alarms, too.And I'm old and creaky, earwise.
The problems don't migrate down to the audible range, because you filter the signal BEFORE it gets sampled.
At least I think that's the misconception we're facing here.
I'm sure they analogy 2nd-order reject above fs & digitally reject everything above fs/2 in an infinite low-pass filter & down sample. Anything less would be absurd. As it is right now, I haven't heard any voicing of any note above 10kHz. The real texture has been removed & was wondering what hope there is for it's return in upsampling. I was hoping that it incorporated an expert system now that computation rates make real time analysis was doable.
I can't argue with what you hear... BUTThere aren't any "notes" above 10kHz in any instrument I know of, only overtones (harmonics). A above middle 'C' is 440. The highest A on a piano is 8x that, give a bit (a piano doesn't do it exactly right because of string stiffness). A flute can get 3 octaves above middle 'C' (262 Hz, give or take). That's a bit over 2K. A piccolo and a garklein go an octave above that, to a bit over 4K. That's about it for instrument sounds. Human voices don't stand a chance, and if you've ever been near a garklein going full tilt, well, you will appreciate what that 'C' amounts to. (ooooh, my head!)
Now, that's for fundamentals. Of course overtones go beyond that, or we'd not need any more bandwidth than a telephone.
I think you need to consider what it is you're not hearing (I'm not saying that it's nothing) but there just aren't "notes" up there.
Even things like cymbal rings, etc, don't have fundamentals up there, but of course they do have ring tones, which are NOT harmonic (which is why they sound like bells, of course).
I'm not sure what you mean by 'they analogy', but no, I don't. Any real filter does NOT have infinite cutoff. See the relationship expressed earlier that points out that t*f> =1, where t is time resolution and f is frequency resolution. That, alone, sets an absolute limit of how much time delay you must have vs. how close to fs/2 you can get, and that is a theoretical (and generally unachievable) limit.
In practice, the bandwidth is set to 20kHz. The filters start rolling off there and are down by 22.05kHz. So, there is no "analogy" or 'assumption' here, only some practical (and working) engineering.
I forgot to use the insert key & typed over prior text in an attempt to consolidate mmy thoughts into a cohessive bundle.BTW, thanks for your help & persistance
Why is the harsh grain of brass horns & piercing cymbals removed even when recorded in DDD? Aren't these part of the sound? Isn't the striking impulse of a cymbol a major part of its sound? If so, its associated frequencies are notes, aren't they?
But that's even less crucial than the ever present inadequate reconstruction of the voice of the wave which has components in the last octave. It seems like everything above fs/4 sounds like a smooth bell.
If one does a convertor with no upsampling or oversampling, i.e. one runs the DAC at the sampling rate with no digital filtering, one uses something like a 13th order Jacobi Elliptic filter with a passband of 20/22.05, and a final rejection of 96+ dB.These filters are a pain to make.
As to what you hear, we'd have to examine the CD and perhaps the player in question. Mine does fine on bright or dark brass, etc, and makes bells crunch JUST like they sound in reality (they make me wince in both).
Sorry, what you hear (in your system, or someone else's) is not as easy to discuss as the mathematics and the processes surrounding it, to say the least.
I hear that with any external DAC I've auditioned, both home & out. I've got a POS TT whose only claim to fame is in that octave. Reguardless of the price, I've yet to hear a DAC that operates there. The ring DAC was cool though.
the ONLY reason for oversampling is that it allows part of
the reconstruction filter to be executed in the digital
domain, where it is
1) cheaper
2) "much more possible" to attain very high orders, and
if you want with in-built phase linearity over the
audible bandDigitally you can make 20-40th order linear phase low-pass
filters. If you do that in the analogue domain, you'll
end up with a truck load of (sensitive) components that
will trash the signal you're trying to push through it.
Of course the new fashion is back to basics: R2R-DAC, no oversampling,
low-order analogue filtering, and hoping that the amp
and speakers downstream won't choke on the HF garbage
(Audio Note, 47Labs).
(Note: you can't do this with one-bit DACs, as these require
massive oversampling to obtain their resolution in the
audible band).
Reconstruction is usually analog? No wonder people are worried about phasing. What do they do? XO @ 22.05kHz & feedback the highpass in the inverted input of the op-amp? Thus, the phase discrepancy would be eventually eliminated by recursively converge onto the unphased answer. However, this would mean the accuracy of the HF would heavily dependent on the order of the XO, while orders above 2nd would normally sound more artificial albeit musical in any other situation. Are you saying this reconstruction circuit removes this artifact? I'm sure the digital circuit can resolve this quicker w/o phasing in an infinite-order XO. My only concern is that the upsampler doesn't simply line-double. I feel uneasy about replacing the original data. I'd feel much better if it were merely augmented.
Going back to theory: reconstruction is by definition an
analogue operation, for the simple reason that you want
to reconstruct when you need a clean analogue output (another
reason of course is that when still in the baseband-sampled
digital domain you can not reconstruct at all, as the
only frequency band your digital 'reality' knows is from
DC to fs/2, and the useful signal itself sits smack in
that band).Oversampling expands the bandwidth of the digital realm
you're in, so that you can move a part of the reconstruction
job in.
Your remarks on feedback, opamps, and convergence are entirely
irrelevant since the theory of sampling simply does not
discuss implementation details. What if I chose to use
a non-feedback triode circuit with passive filters?
And anyway, there is no funny convergence in opamps, not
at these low frequencies...> reconstruction circuit removes this artifact? I'm sure the digital circuit > can resolve this
> quicker w/o phasing in an infinite-order XO.I'm not sure what you're talking about, but I feel confident
that neither are you :-) But OK, analogue filters always
marry a specific amplitude response to a specific phase
response (BTW, 'phasing' is not a verb belonging to the
dictionary of electronics!). In the digital domain you can
cheat bigtime, as you can control time: through the
use of memories and buffers you have access to the past
and to the future of the signal, which is neat. So in the
digital domain you can say, within some constraints, I want
THIS amplitude response, and I want THAT phase response
to go with it. However, this does not imply that all
digital audio filters do this, as many are just 1:1 translations
of their analogue cousins, but of higher order.Now what you're referring to above, alluding to digital
reconstruction being better than analogue, is very simply
exactly what every oversampling CD-player since 1983 has been
doing. So there is nothing new here.Again, the new 'upsampling' boxes are here because
1) manufacturers want to sell us more boxes
2) there is a limited need for them in the scope of DACs
that have been designed around an 88.2kHz or 96kHz
fs, which now are confronted with an installed
base of software sampled at 44.1kHz: you have to
translate between formats.Upsamplers are no magic, no recipe for instant bliss. They
are a tool for a very limited job. And if something upsampled
turns out sounding a tad better, then that is just because
the new DAC you're going into may inherently sound a bit
better. But it might as well turn out the other way,
the upsampling process killing much of the original
information, especially if you move between non-integerly
related formats, like between 44.1k and 96k.All else is marketing.
Listen, back in the early days of the dCS pro-grade upsampler,
they themselves stated that upsampled CD, played through the
Elgar DAC sounded a bit better, but that they were at a
loss as to why precisely. It was just a lucky coincidence.
> Upsamplers are no magic, no recipe for instant bliss. They
> are a tool for a very limited job. And if something upsampled
> turns out sounding a tad better, then that is just because
> the new DAC you're going into may inherently sound a bit
> better. But it might as well turn out the other way,
> the upsampling process killing much of the original
> information, especially if you move between non-integerly
> related formats, like between 44.1k and 96k.
>
> All else is marketing.Well, having lived with an upsampling DAC for a couple of days now. It's early yet, but, so far, I think I can say that upsampling isn't a magic bullet that will transform all your CDs into sonic bliss. There seem to be pros and cons sound-wise.
BTW, looking at your website, you seem to have pretty good taste in music, you should rent a cottage over in Music Lane :-).
-Joe.
for those of us who don't have them yet, we need a serious review… ideally from a tube & vinyl fan but any will do in the absence of unaffiliated ones. My real suspicion is the loss of the original data stream instead of the augmentation of a line-doubler to 88.2kHz. However, the digital reconstruction has merit. It seems like the sound may be perceived to be “processed” to “clarity”. Care to comply with your impressions?
I'm still listening. I have another upsampling DAC to try also. I'll probbaly have something concrete to post in about a week or so.-Joe.
I'll explain how I got lost so you know whence I come.This apparent misconception was triggered by both the expressed need to do this quickly & infinitely in the analog domain. Thus, since the reconstruction needs to separate the fs/2 + f frequencies from fs/2 - f . To me, this means XO. Since the analog ( we are still working analogicly? ) XO phases the attenuated octaves, & as you approach fs/2 the results should get more critical, I figured the outputs ( that if it weren't for the aforementioned phase ) would cancel cleanly. However, because of the phase, they'd combine to create a reduced amplitude discrepancy requiring another round of adjustment which'd serendipidously remove the phase problem simultaneously. It looked as if the successive graphs in jj's "Graph of Sum of Aliases" were an illustrated case of feedback as the differences were minimized & converged onto the answer in successive steps. Now, if it weren't for me being completely full of it, I thought I understood.
FWIW, I took a digital control systems class 2 decades ago which gave me a rudimentary familiarity with the Z-plane. I was quite taken with the "cheating" nature of aproblematic infinite-order bandpasses & I realise the accuracy attained by an active control when you know the future precisely because the digital delay distorted time & made the present's output the temporal equivalent to the past's input.
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