About Nyquist, Shannon, filters, and how stuff sounds

[KeithL]

I've come to the conclusion that most people don't REALLY understand what the theory MEANS

I base this conclusion on various quotes like: "According to the theorem, a 44k sample rate should be able to perfectly reproduce anything up to 20 kHz. Since I can hear the differences between a CD and the original, then the theory must be wrong." (I made that specific quote up - based on the gist of what I've heard from several people.)

Unfortunately, that's not exactly what the theory says. What it says is that it's POSSIBLE to perfectly store and reproduce a given signal as long as the sample rate you pick is such that the Nyquiest Frequency (half of the sample rate) is above the highest frequency you wish to record. (So, if you want to store all frequencies up to 20 kHz perfectly, then it is POSSIBLE to do so by using any sample rate of more than twice that frequency. This is why CDs use a sample rate of 44.1kHz, which is slightly more than twice the 20 kHz "limitation of human hearing".)

However, carefully note the highlighted word....

The theorem DOES NOT say that, if you use a sample rate with a Nyquist Frequency higher than what you want to record, you are GUARANTEED to get a perfect reproduction of your original. Instead, it lists that as a minimum requirement. So, what it really says is that, if you want to reproduce everything up to 20 kHz, you must AT LEAST use a sample rate of more than twice that; it doesn't preclude that there might be other requirements.

In fact, if you read a bit further, there are indeed a few more requirement:

1) You MUST limit the frequencies you're trying to store to below that Nyquist Frequency. If you have information at frequencies above that, not only won't it be accurately stored, but it will interfere rather badly with the storage of the frequencies you want. In fact,  if you let anything above that frequency reach your encoder (ADC) at all, it will be "magically" transformed into noise and distortion down in the audio band where you CAN hear it.

2) When you play that signal back, in order to get what you're supposed to, you MUST remove some extra junk that "appears" as a result of the conversion process. Specifically, you MUST apply filtering to limit the bandwidth of your output signal to the same bandwidth you were allowed to have going in (below the Nyquist Frequency for your sample rate). And, again, any junk above that frequency that you fail to remove 100% will usually come back to haunt you in the form of distortion inside the audio band (and especially annoying non-harmonic distortion at that).

It is these two requirements that cause all the "problems", and that explain why digital recordings, as good as they are, still aren't perfect.

By #1, if you were recording a perfect 12 kHz sine wave on a CD, you'd be all set. A sine wave contains only one frequency, and 12 kHz is safely under 22 kHz (the Nyquist Frequency for a CD; half of 44.1k), so you're done. However, unless you've created a perfect sine wave on a computer, nothing much in the real world is that perfect. Most musical instruments produce lots of harmonics, many of them reaching well above 20 kHz. Even things like the room ambiance, the creaking of the chairs, and the background hiss from your master tape, may have some energy above 22 kHz. Well, guess what? Unless you want that out-of-band content to cause nasty audible distortion, you have to TOTALLY remove it. Which means that your audio source MUST be passed through a rather powerful filter before you can record it. Since filters tend to have side effects, which tend to be worse as the filter becomes more aggressive, this is your first trade off - between how well the filter works and how much damage it does.

Now, by #2, you also have to apply a similar filter to the analog audio after you convert your digital recording back to analog. Please note that this extra junk, which has to be filtered out, is NOT some nasty distortion or error in the recording process; IT IS THE INEVITABLE RESULT OF THE CONVERSION PROCESS. Even in an absolutely theoretically perfect recording, this extra junk will be there in the decoded analog audio, and you will have to filter it out. So, once again, we're faced with that tradeoff between a filter that does a really great job, and any side effects it may also produce. (And, yes, SACDs have this exact issue too - even worse than with PCM. Also note that those "NOS filterless DACs" simply ignore this requirement; they're hoping that the limited frequency response of the rest of your signal chain will act as a de-facto filter, and that the distortions caused by the missing filter will be less annoying than the side effects of HAVING a filter.)

Both of these tradeoff were a huge problem in the early days of CDs - because you basically needed a filter that passed everything up to 20 kHz perfectly, yet also removed anything above 22 kHz - also pretty much perfectly. (We're talking about a filter sharper than 96 dB/octave - which isn't at all practical to build - and which tends to have nasty side effects. Poorly designed filters are almost certainly a big part of the "digitalitis" present on many early CDs.)

But what about oversampling....

Oversampling is a neat trick. By using a higher sample rate, you get a higher Nyquist Frequency... which, in turn, lets you use a more gradual filter that takes effect further from the audio frequency range. (Passing everything up to 20 kHz, but nothing above 22 kHz, is rather tricky; passing everything up to 20 kHz, and stopping everything above 40 kHz, is much easier, and can be done with a much simpler filter - which is cheaper to build AND produces less audible side effects. That's oversampling in simplest terms.) You can then keep that higher sample rate or, if you want to record the result on a CD, you can use some really tricky math to convert it back down. By doing this you get to keep most of the benefits of your gentler filter, and do a lot of the nasty bits in nice clean math instead (a slight oversimplification there, but philosophically pretty accurate).

And now, digital filters.... ..... 

Once you get to tricky math-type stuff like oversampling, you're pretty well stuck using digital filters; it's simply not practical to do this stuff in the analog domain. Virtually all digital filters, as part of how they work, do some "time shifting". They take some of the original signal, shift it by a certain amount of time, and add a certain amount of it back into the original; it is by carefully calculating the shift times and amounts of signal from each step that is added back that a digital filter works. The end result, when you're done adding up all the additions and cancellations, is that you end up with what you wanted. There is another variation, where some of the output signal is fed back to the input in calculated quantities, but it has the same flaw - that the results do not happen instantaneously.

When you see those little pictures showing the transient response of the filter in a DAC - the ones with one big spike and little ripples fading off to the left and right - the ripples are the effect of doing things this way. The signal in the center is really very very close to right, then, a little further away, it goes a bit too low, then, a little further away, it goes a little too high - but less so than how far it went too low - in ever decreasing "ripples". Those ripples are almost entirely errors between what the signal really should be and what it is. At this point, I should clarify a few things:

1)  In a properly designed filter, those ripples should be pretty small, and should be so close in time to the main signal that you don't hear them - much.

2) The ripples are part of the accurate signal. If you were to add up the main spike and all of the ripples, both negative and positive, you would end up with a TOTAL that was very close to the perfect result. (Adding the ripples will make it more precise rather than less, because they're sort of corrections to the original signal, just not at exactly the right times. You can't simply "cut them off" without creating even worse errors.)

3) That whole picture thing is very misleading. Seeing how a DAC will reproduce a perfect square wave or impulse is INVALID. Since you MUST limit the bandwidth of any signal you record to below the Nyquist Frequency, your real digital signal can ONLY contain BANDWIDTH LIMITED transients. In other words, your DAC CANNOT ever be asked to play the sort of signal they used as a test signal to make that picture - because it wouldn't be a valid signal to record to begin with. The transient pulse used for that test is a test signal - it provides useful information for the designers - but it can't exist on a real recording.

Also, while you simply can't "just remove the ripples", you CAN use some fancy math to shift them around. On common way this is done is to push most of the ripples from before the main signal to after it - you end up with fewer smaller ripples before the spike, and more, larger ripples after it... and the ones after carry on a bit longer. The end result, when you sum them all up, is mathematically the same. However, since the real signal masks the ripples after it better than the ones before it, the ripples we shuffled to after it are less audible - so, at least in theory, it should sound better that way. That last option is commonly called "an apodizing filter", and quite a few DACs offer it as an option (remember that it isn't necessarily more accurate, but some people think it sounds better). Incidentally, the DACs in our current products do NOT include that option, and we think they sound perfect just the way they are ... but you may see it in some future products

***
OK, if you're read this far, and are now totally confused... I guess you've earned the right to know

THE BIG SECRET 

DACs aren't perfect; Like everything else on planet Earth, they involve some compromise.

The analog output of your DAC is NOT a perfect recreation of the original analog signal - NONE, NOWHERE, AND NOT AT ANY PRICE.

Even though the S/N and THD of most DACs are absurdly good, they do have some other (rather tiny but possibly audible) flaws.

Digital filters are necessary - simply because they have far fewer and smaller flaws than any of the available alternatives. However, digital filters do alter the signal slightly. The signal coming from your DAC is not an EXACT reproduction of the original. Odds are that the frequency response and distortion really are so close to perfect that no human being could hear the spots where they aren't, but there are indeed tiny areas where the signal, while correct in power amplitude, may be slightly wrong in time. These errors are mostly caused by the way in which digital filters work, and can be seen if you look at oscilloscope pictures of the transient response of various DACs. Obviously, if you look at the transient responses of several DACs, or of different choices of filters on a single DAC, and they are different, then they CAN'T all be right.... right?  (The math boys may fall back on proving that the amplitude responses of all of them, taken over time, are actually precisely right, but clearly the waveform has been changed a little bit.)

HOWEVER, the important point to remember here is that the errors introduced by the digital filters on a modern DAC are FAR less audible than the errors introduced by any other ways of reproducing a signal - like the hiss and nonlinearity distortions of magnetic tape, or all the mechanical resonances and mechanical and electrical nonlinearities of analog vinyl. If you really find those few little extra ripples scary, then you really shouldn't look at an oscilloscope picture of a phono cartridge trying to "ride" a square wave, and you'd better take a few stiff drinks (or a hit of your anesthetic or tranquilizer of choice) before comparing frequency response graphs or THD numbers....

But, if you were just looking for an explanation how different DACs could sound different, even though "all the numbers are the same", then there you have it.

You're not crazy, the differences are really there, and that's where you can find them. (Unfortunately, at least so far, there is no simple number that describes those "discrepancies", and, unless you know exactly how to interpret them, those pictures show you that there's something going on, but don't correlate to what it sounds like.

If you were really just tired of "them digital guys" always saying that "DACs wuz perfect", then you can consider yourself vindicated. 
(But Nyquist and Shannon were ALSO right )

[audiobill]

Keith, of course there's the quality of the analog stage and the power supply that supports it too..........

[garbulky]

Thank you Keith. This must be the reason I hear differences in DAC's when all of them have insanely low THD values and frequency repsonses 20-20khz +/- 0.1 db. I don't have golden ears nor am I able to tell between 1% THD and 0.0001 %THD. So why do they all sound different? This is a good start. I would like it expanded a bit more
Have you heard of Schiit's Yggdrasil? They are trying to go away from delta sigma and the approximations. Do you have any comments as to the idea?
www.head-fi.org/t/701900/schiit-happened-the-story-of-the-worlds-most-improbable-start-up/3045#post_10944447

[leonski]

Gar,
my 'reasonably' priced Cambridge Audio DACMagic+ has 3 output filters. They don't make much difference with the truncated input you get from either MP3-160 or even 'small dish' audio feeds.
BUT, give the DAC a well recorded redbook CD and you can begin to tell the difference between filters. MOST of the effect these kinds of filters have is in the highest frequencies.

www.audiokarma.org/forums/showthread.php?t=266589

Explanation, sort of, RE: DACMagic output filters. Read this with what Keith wrote, above. Together that may help.

And yes, the gentle analogue filter of my now ANCIENT Magnevox top loader which oversampled at 176.4 is a VERY nice solution. If I could find a laser for that player, I'd be tempted to get it fixed.
I think it was the 2nd player on sale in the US, back in about '83 and was sold in Europe as the Philips FD-1000. Find the Stereophile test, and prepare to share their amazement at how good it sounded.
For the time.

[KeithL]

I am very curious to hear Schiit's new DAC (but it seems to be taking a bit longer than originally planned).

The short answer to Garbulky's question is that it seems that we humans are simply wired to detect some things very accurately and not others. For example, most of we humans can distinguish minute differences in color (put two tiles of slightly different colors next to each other), but we are pathetically unable to identify different colors when they aren't next to each other. (All of that "white balance" stuff with your camera isn't because the camera isn't accurate; it's because your brain will cheerfully delude itself into believing that a ping pong ball in sunlight and that same ping pong ball under a fluorescent lamp are both "white" - even though the actual color of the light reaching your eye from each isn't even close to being the same.)

The short explanation is that the filters used in oversampling DACs (which are absolutely necessary) tend to produce "time errors". You feed in what looks like a perfect little vertical line, and it comes back out with ringing before and after the line. In the physical world it seems perfectly obvious that a glass will ring for a few seconds after you tap it, but would seem absurd that it might start ringing before you touched it. In the mathematical and digital world, the opposite is true, nothing can start moving instantly, and so, in order "for the numbers to work out right", the glass must start ringing at least a tiny while before the primary impulse. Reality is actually somewhere in between; there is really no such thing a "perfectly sharp drumbeat"; even if you hit a drum with a sledge hammer, there is a time interval while the hammer transfers energy to the drum head, which takes a measurable time. However, all impulses acquire some ringing on their way through a modern DAC (luckily all musical instruments also exhibit this anyway, being mechanical, so it's mixed in with a lot of the same).

The reality is that we humans aren't wired to be especially sensitive to harmonic distortion. Since most instruments produce a mix of primary tones and harmonics to begin with, all your amplifier is doing if it has harmonic distortion is altering the mix of harmonics by adding more of certain ones. If I were to play a perfect 440 Hz sine wave, with no harmonic content, it would sound like a clean sort of whistle; if I were to then add some harmonic distortion, it would sound quite different. However, if I were to start with a piano note at 440 Hz, which already contains lots of harmonics, adding some extra harmonic content would make much less difference (because, instead of adding something that wasn't there before, all I'm doing is slightly altering the ratio of a bunch of harmonics that were there already anyway).

Back to transients and such....

In most natural sounds, when you have a transient, it is followed by a gradual decay. Think tapping a glass; think a cymbal; think hitting a cowbell. You get a relatively sharp transient, followed by a slow decay; you do not hear the cymbal ringing before the stick hits it. Now, we play that same signal through our nice new digital filter and, instead of only having the ringing after the "hit" (which was there to begin with), our digital filter has added a tiny bit of ringing BEFORE the hit. Remember that, if we add up the energy involved, this energy belongs there, so it doesn't make our frequency response not flat; since it's part of the original content, it probably won't even show up as distortion; but what it is is some energy showing up at the wrong time.

Now, since most natural transients show a sharp hit followed by a gradual decay anyway, adding a tiny bit more energy during that decay isn't going to be audible at all. However, the tiny bit of ringing that gets added BEFORE the hit is "out there for all to hear" - it isn't being covered up by the sound of the hit, and it isn't hidden by the naturally occurring decay; it's purely a very faint ringing that shouldn't be there. And many of us seem quite prone to noticing it.

OK, now here's what those various choices of digital filter usually offer:

One of then is "normal" - that's the filter that provides the flattest frequency response, and has the right amount of energy at all frequencies, but has some ringing before and some after the transient. This is usually a sharp filter (sometimes named something like "sharp knee").

Another one is excessively damped - it uses some mathematical tricks to reduce the amount of ringing altogether. The result is that the signal sounds very clean; unfortunately, a side effect of filters that avoid almost all ringing is that they almost always also roll off the high frequency response (so you get transients with almost no ringing, but your CD is down to -3 dB at 20 kHz). Unfortunately, since the frequency response is so far from flat, it's difficult to tell whether you're hearing the lack of ringing or the -3 dB at 20 kHz.

The newest favorite trick is what is often referred to as an "apodizing filter" (a cool term for "changes stuff"). What that does is to use some very cool math tricks to "trade" on the ringing. It eliminates all or most of the ringing before transients, but at the cost of adding more ringing AFTER the transient. As it turns out, since most natural sounds have after-ringing anyway, and since we are accustomed to hearing it a lot of the time anyway, but we are not at all used to pre-ringing, we find this "post ringing" a lot less annoying than even a similar amount of pre-ringing. (Unfortunately, these filters also always seem to roll off the high frequency response.)

The problem is that each of us experiences music differently; you may listen to mostly vocals and be especially sensitive to pre-ringing, so you may prefer that apodizing filter.
Whereas I love vocals, and I really like my cymbals to sound right - which unfortunately rules out the filter that's -3 dB at 20 kHz.

I've vastly oversimplified because in reality there are many different filter parameters, and a virtually unlimited number of combinations and minor variations.
What you get on many DACs is an assortment of different "preselected options" to choose from.

Now, to throw my experience into the arena here about how much all this matters......

First off, every digital recording ever made has been passed through at least one band-limiting filter, so the transients have already been altered (and you don't know the details of what filter was used).
(And, just to be clear, the bandwidth limitations and nonlinearities of ANALOG recordings virtually ensure that they will mess up transients even worse than any modern digital filter.)
Second, these differences are subtle, and it's virtually impossible to justify calling one or the other "more accurate overall with all content".
In general it comes down to one or the other filter sounding a tiny bit better with a particular recording - but generally you end up with one or two which usually sound best and one or two that only sound best with one or two particular recordings.
On all the DACs I've owned that offered multiple filter choices, I've always ended up picking one and just using it - and being quite satisfied doing so.
(Honestly, other than the frequency response issues with some of them, these tend to be very minute differences in sound with no clear "correct" or "winner".)

But, to get back to the original comment, you can take two DACs, both of which have ridiculously low THD and absurdly flat frequency response, and you may well see obvious differences in the output you get from each when you input a transient (even a properly band-limited transient as you would find on a real digital recording). Unfortunately, as they say, a picture is worth a lot of words, and it's not easy to quantify all the differences into a single specification about "which one is closer to right".

Incidentally, most DAC manufacturers include one or more default filters in the DAC chip itself (we use the one internal filter Analog Devices included for the DC-1 because we believe they did a good job designing it and it works well). Some DAC chip makers allow you to choose between several internal filters (the Wolfson WM8741 gives you 20 or so filter combinations to pick from). Many makers of very high-priced DACs design their own digital filters; unfortunately not all of them are better than the one included inside the DAC already.

(Also incidentally, what Schiit is attempting to do with Yggdrasil is to use a new filter design that avoids some of the flaws in current digital filter implementations.)

To answer your question about Yggdrasil.... I personally don't find Delta-Sigma DACs to be "horribly flawed", or to sound especially different than other types of DACs (at most I've seen some technical arguments that Delta-Sigma types are more sensitive to jitter; which obviously only matters if you haven't included some form of jitter removal in your design). However, I am quite sure that the guys over at Schiit know what they're doing, and that it will sound very good. (But I'm not convinced that will be because it isn't a Delta-Sigma design.)

Oct 12, 2014 17:09:20 GMT -5 garbulky said:

Thank you Keith. This must be the reason I hear differences in DAC's when all of them have insanely low THD values and frequency repsonses 20-20khz +/- 0.1 db. I don't have golden ears nor am I able to tell between 1% THD and 0.0001 %THD. So why do they all sound different? This is a good start. I would like it expanded a bit more
Have you heard of Schiit's Yggdrasil? They are trying to go away from delta sigma and the approximations. Do you have any comments as to the idea?
www.head-fi.org/t/701900/schiit-happened-the-story-of-the-worlds-most-improbable-start-up/3045#post_10944447

[vcautokid]

Oct 12, 2014 17:09:20 GMT -5 garbulky said:

Thank you Keith. This must be the reason I hear differences in DAC's when all of them have insanely low THD values and frequency repsonses 20-20khz +/- 0.1 db. I don't have golden ears nor am I able to tell between 1% THD and 0.0001 %THD. So why do they all sound different? This is a good start. I would like it expanded a bit more
Have you heard of Schiit's Yggdrasil? They are trying to go away from delta sigma and the approximations. Do you have any comments as to the idea?
www.head-fi.org/t/701900/schiit-happened-the-story-of-the-worlds-most-improbable-start-up/3045#post_10944447

Yggdrasil is not yet a reality, and Delta Sigma is the sauce in most DACs. I want to hear how it would sound that Yggdrasil.

[garbulky]

@keithl : I appreciate that nice informative post You have any thoughts on multi bit dacs vs the one that's not multi bit? I've been comparing the an asus xonar which has a Burr brown PCM 1792 which appears to also be a delta sigma dac but of interest - it is a multi bit. I had no clue that all dacs weren't multi bit. What does that mean?

[ericl]

Here is a picture & non-mathematical description of 1 bit versus multi-bit DAC's:

www.mother-of-tone.com/conversion.htm

Read an interesting usb DAC shoot-out with several folks doing blind listening tests on about 10 different usb DAC's.  The winner was the most expensive $3K +.  Second place (a very close second) was a $600 unit.  The most expensive refused to state which DAC IC they used.  Second place used ESS Sabre mult-bit.
From the standpoint of those that manufacture DAC IC's and those like Emotiva that uses various DAC IC's in products, the main difference is in the cost & complexity of producing an analog output that is of high quality from the standpoint of having low noise/distortion.
I think the cost difference between a best DAC's IC's  & so-so DAC's IC's is about 2 trips to Starbucks.  

Best regards, Eric
.

[KeithL]

Interesting article... however, be warned that, while most of the technical details are pretty much correct, the author... shall we say... deliberately states certain things in a misleading manner to support his claims.

The terms "R2R DAC" and "multi-bit" and "single-bit" refer to HOW THE DAC CHIP DOES WHAT IT DOES INTERNALLY. While there may be technical reasons why one or the other is superior in terms of absolute performance or manufacturing cost, all three methods are theoretically equal in the sense that they accept the same 16 bit or 24 bit numbers as input and, assuming perfect operation, GIVE YOU THE SAME EQUALLY ACCURATE ANALOG AUDIO AS OUTPUT. The Delta-Sigma DAC does NOT give you less resolution or less accurate output than the R2R DAC! It's just another way to achieve what is basically the same result - although you'd have to spend an awful lot of money to find an R2R DAC that performs on a level with even our XDA-2. (If that's all you were wondering, then you can stop reading here .... )

Beyond that, I have a few bones to pick with the article...

First off, by conspicuously using terms with negative connotations (like "el cheapo") he makes it plain that he considers saving money to automatically be a bad thing. Oddly enough, most of the makers of Delta-Sigma DACs tend to promote their being able to deliver better performance for a lot lower cost to be a benefit. And second, or more generally, he is clearly guilty of saying misleading things like "don't be afraid of hearing what you eyes tell you to expect to hear". Now, while this is all well and good, and I certainly agree that you shouldn't be surprised to hear what the specifications or graphs suggest, I would also remind you not to be afraid of hearing something DIFFERENT than what an uninformed look at a specific graph or diagram might lead you to expect to hear. There's really not a lot of point in looking at "the scary pictures" unless you do understand what the math behind them means. (I personally find that a glass of Coca Cola tastes nothing like what its chemical formula looks like.)

Once you dig into it, though, he makes several statements that are quite misleading. For example, I don't think that any competent digital engineer thinks "that if only the switch was toggled fast enough, one could also achieve many different voltages". The fact is that, starting with a single voltage, if you vary the frequency of a fixed width pulse, or vary the duty cycle of a pulse train, then feed it through an integrating filter, you ACTUALLY CAN achieve as many different analog voltages as you want - with very high precision. And he's also technically correct that the method used by Delta-Sigma DACs results in the generation of a lot of very high frequency noise; but he neglects to mention the fact that this high frequency noise, because it's far from the audio band, is quite simple to filter out - using relatively simple filters that don't cause much disturbance in the audio band. (Unlike those "harmless looking" little steps generated by a non-oversampling ladder DAC which, because they fall very close to the top of the audio band, are extremely difficult to filter out without interfering with the frequency response inside the audio band).

Both Delta-Sigma DACs and R2R DACs have drawbacks.... but, quite simply, Delta-Sigma DACs have FEWER drawbacks, and deliver equivalent or better performance at a much lower cost.

I've finally figured out an analogy that works to explain this.....

Imagine you had a piece of wire that was about fifteen feet long and you wanted to measure it very accurately.
(You stretched it along something that you need to measure very precisely and cut it to the same exact length.)

You can use:

A one-foot ruler that's accurate to 1/100 inch - and can be gotten for a reasonable price.
Or a twenty foot tape measure that's accurate to 1/10 of an inch - and is also pretty cheap.
Or a twenty foot metal tape that's accurate to 1/100 of an inch - but is very expensive.

So, we could measure it using the economical twenty foot tape and get a reading that's accurate to 1/10 inch (this would be the 16 bit R2R DAC).
Or we could buy the expensive twenty foot metal tape and use that (that would be a 24 bit R2R DAC).
Or, since we only need to know the total length, we could cut it into short pieces, measure each very quickly and accurately using our accurate one-foot ruler, than add the numbers together (that would be our Delta-Sigma DAC).

Where the R2R DAC simply converts the input value to a voltage "all at once", a Delta-Sigma DAC splits the number up into smaller numbers (in a mathematically precise way), converts those smaller numbers individually, then combines them into a single very accurate total value. While this sounds like an awful long way around, it actually turns out to be much easier and more efficient to do it this way. Instead of doing 44,000 conversions a second, each one 24 bits, you may end up doing 4 million conversions a second, but each one only being 4 bits of 5 bits instead of 24 bits. It turns out that it's a lot easier to do very fast 5 bit conversions, the parts you need to do very accurate 5 bit conversions are a lot cheaper than the ones you need to do accurate 24 bit conversions, and putting it all back together afterwards isn't complicated either. On the digital side, there's a lot more math involved, but processing power is cheap, so that doesn't really matter. On the analog side, since we've ended up with a lot more values per second, with smaller time steps between them, the noise those steps represent is at a much higher frequency. This makes it much easier to design a filter that will eliminate all of that noise without altering the frequency response of the analog audio we want. (You have more noise, but it's a lot easier to filter it without messing up anything else, because it's at a much higher frequency. In contrast, it's actually much harder to filter out the lower-frequency noise produced by the R2R DAC without messing up the audio, even though there's less of it, because it's so close to the audio band.)

Going  back to my wire analogy, by using the Delta-Sigma approach, you've used some fancy math to arrange things so you can measure your fifteen foot wire, accurate to 1/100 inch, but at a cost equal to that of the cheap tape measure. Yes, there's a lot more stuff going on inside the Delta-Sigma chip, but the end result is excellent performance at a very reasonable price. (Most of those R2R non-oversampling DACs you hear about are only 16 bits; 24 bit R2R DACs are quite expensive, and I'm not even sure if you can buy a 32 bit R2R DAC in a single chip - if you can it would cost an awful lot - and building one out of loose parts would cost a fortune.)

There are two real drawbacks to the Delta-Sigma approach - neither of which was mentioned in that article. First, the digital filters required as part of the process do slightly alter the transient response; and, if they're done very badly, this can lead to what the author complained about - where instruments sort of blur together, and a cymbal sounds sort of like a steam leak. However, to put it bluntly, well designed digital filters don't sound that way. Their transient response may not be absolutely perfect when you look at it on an oscilloscope, but it's probably better than any microphone or speaker out there. This is oen of the reasons why some DACs do indeed sound better than others. The second drawback is that Delta-Sigma DACs are more sensitive to jitter. This means that, if there is excessive jitter on your incoming digital signal, it will probably affect the way a Delta-Sigma DAC sounds more than it would affect an R2R DAC. Since jitter is never good anyway, the obvious answer there is to not have jitter. (Our XDA-2 and our DC-1 have ASRC jitter reduction on all their digital inputs; our XDA-2, our DC-1, and pretty well every other good USB DAC out there, also have asynch USB inputs, which are basically totally immune to jitter anyway.)

Apr 6, 2015 12:22:22 GMT -5 ericl said:

Here is a picture & non-mathematical description of 1 bit versus multi-bit DAC's:

www.mother-of-tone.com/conversion.htm

Read an interesting usb DAC shoot-out with several folks doing blind listening tests on about 10 different usb DAC's.  The winner was the most expensive $3K +.  Second place (a very close second) was a $600 unit.  The most expensive refused to state which DAC IC they used.  Second place used ESS Sabre mult-bit.
From the standpoint of those that manufacture DAC IC's and those like Emotiva that uses various DAC IC's in products, the main difference is in the cost & complexity of producing an analog output that is of high quality from the standpoint of having low noise/distortion.
I think the cost difference between a best DAC's IC's  & so-so DAC's IC's is about 2 trips to Starbucks.  

Best regards, Eric
.

[audiosyndrome]

Very nice writeup Keith.

Russ

[ericl]

Hi Keith.  My only reason for choosing the article was that it had some pictures versus the wiki articles written by PhD's for PhD's (which I am not).  I'd have to confess to not reading the article; my apologies.
Iv'e got a technical background but I am a user of the technology & highly doubt I could listen to 25 different DAC's/Processors & tell which one's had 1 bit versus multi bit DAC IC's.

Best regards, Eric

[KeithL]

No apology necessary - and it's not even that it's a bad article. It just follows a common trend where articles about very complicated material provide decent explanations of the actual technical details, but then do a sort of segue into conclusions or opinions of the author 's that aren't exactly supported by the data. Unfortunately, digital audio is one of those subjects that sort of varies in complexity (and it's ability to be confusing) depending on how deep you get into it. The author of that article clearly doesn't like certain types of digital audio; he likes multi-bit DACs and hates DSD. Another author of another article thinks that DSD is clearly superior to multi-bit PCM; and, other than his conclusions, he bases that opinion on pretty much the same data. (The pictures in that article are actually quite useful as part of the explanations of how the different systems work; but they can be misleading otherwise.)

The reality is that neither argument against either type of DAC is compelling. R2R DACs reproduce transients pretty well, but they tend to have trouble with achieving a good frequency response without sacrificing other things, and it tends to be very expensive to make a high-quality R2R DAC. Delta-Sigma DACs usually deliver really excellent frequency response and distortion figures, but also do tend to alter that shape of transients a little bit. However, compared to R2R DACs, Delta-Sigma DACs tend to offer a lot more "bang for the buck" and really are quite capable of sounding just as good - or even better. Vinyl lovers will quote both of these reasons for why they think analog audio should sound better; however, if you've ever seen a 'scope trace of a phono cartridge trying to reproduce a clean high-frequency square wave, you'll realize that the analog system does a worse job of butchering transients than either type of DAC. And, at this point, we should also remember that neither microphones nor speakers can reproduce transients very accurately either.

The fact is that ALL digital audio reproduction systems generate large amounts of nasty high-frequency noise. However, since that noise is a necessary part of the conversion process, and is removed along the way anyway, comparing whose noise is nastier is really pretty much a waste of time... but it does make for great propaganda.... 

I've definitely heard Delta-Sigma DACs that make wire brush cymbals sound like a leaky steam valve (which is why the author of that article says they're no good); however, I've also heard Delta-Sigma DACs that DON'T sound that way - and that, in fact, sounded very clean. At the moment I own a non-oversampling DAC which has a reputation for sounding remarkably clean and smooth on vocals; unfortunately, it also rolls off high frequencies rather badly, and sounds downright odd with some things other than vocals (although it sounds very nice on plucked guitars). Oddly though, our DC-1 (which cost half as much), does just as good a job on vocals, but manages to avoid sounding odd on the other stuff.  However, it's easy to see how someone who doesn't like Delta-Sigma DACs, and expects them all to "make all instruments sound the same", can find plenty of individual models to support his claim (and his expectations).

I've listened to a lot of DACs, and I still own way too many, and I wish I could finally deliver a "theory of everything" about DACs - but I can't (at least not yet). I think I can claim a correlation between jitter and that "homogenized sound" that the author of that article was complaining about. I have definitely found some DACs which sounded as he described, and which sounded much better when I used an external jitter-removal device on the signal I was sending to them. And there is some legitimate scientific theory to suggest that, when excess jitter is present, it affects Delta-Sigma DACs worse than R2R DACs. (Our XDA-2 and DC-1 have the jitter removal circuitry built in, which removes jitter from all their inputs, and makes that consideration moot; most modern USB DACs, including ours, operate in asynchronous mode, which means that the DAC clocks the signal, which should also render them quite insensitive to jitter anyway.)

Unfortunately, this sort of throws a monkey wrench into using the architecture of a DAC to decide whether you should expect it to sound good or not.....

So I guess we're stuck listening to them   

Apr 7, 2015 12:01:36 GMT -5 ericl said:

Hi Keith.  My only reason for choosing the article was that it had some pictures versus the wiki articles written by PhD's for PhD's (which I am not).  I'd have to confess to not reading the article; my apologies.
Iv'e got a technical background but I am a user of the technology & highly doubt I could listen to 25 different DAC's/Processors & tell which one's had 1 bit versus multi bit DAC IC's.

Best regards, Eric

[ericl]

Thanks Keith.  I appreciate & respect what you have to say here & on the other threads.

Could you speak to the DAC section on the XMC-1? My current DAC is a peachtree Nova - love it but it is limited to 48K/24 bit.

I tried a demo of DIRAC & I have become obsessed with having DIRAC.  From an architectural standpoint (I am a computer consultant/architect) running DIRAC  on the stage right before the analog amp(s) is superior to the other alternatives of running Dirac on a computer or minidsp in that anything that is an input to the XMC-1 (except native DSD) can be diractisized (OK DIRAC DSP engine sounds more eloquent but whatever).

Anyway, guess I miss vinyl about as much as you do .  When all the dust on the needle causes it to jump off-track, just replace the nickel on the tone arm with a quarter.

Best regards, Eric

[leonski]

IMPLIMENTATION is key.
Good trumps bad, whether 1 bit or 16 bit or whatever.

I DO remember back when '1 bit' came out, seeing a test of several players. The INFERIOR implimentation of the 1 bit stuff clearly gave the Philips player the edge. Even statistically, when many listeners were involved, they identified and preferred the philips player (TD1541S?) over any of the 1 bit.

And Eric, IS 48k/24bit REALLY a 'limit' or just propaganda? I'll bet a best-case 44.1/16 bit 'redbook' would better other systems when running inferior software.

[yves]

Let me stir the pot just a little.

www.aes.org/e-lib/browse.cfm?elib=17497