The Unique Evils of Digital Audio and How to Defeat Them

I disagree with much that was written at one level or another but skimmed it very quickly. At least they are trying! - Don

Benchmark products are highly thought of and used quite a lot in pro audio as well as consumer audio. Saying that you disagree with much that was written is a very broad brush statement. I am curious as to specifics of what you disagree with and what scientific data you would back up your claims with.
I'll have to write myself a note to tackle this one later after reading the article in more detail. I'm not saying anything about the products, as I have never heard them, just that some of the terms and definitions do not match what I am used to seeing or the way I use them. Some of this may be due to my focus on high-speed largely Nyquist converters vs. (relatively) low-speed audio converters and various flavors of delta-sigma modulators, so I need to think about it a bit. A couple of things caught me right off, but I haven't time to go back now (simulation finished -- back to work!)

For definitions, I use IEEE STD-1241/2000 (an update is forthcoming if not already out; my boss is on the committee):

IEEE Standard for Terminology and Test Methods for Analog-to-Digital Converters, IEEE Std 1241-2000, IEEE Press, 2001.

The definitions and much of the test info works with DACs, too, and the terms are frequency- and architecture-independent.
I use a Benchmark Product and it delivers a lot .. Well beyond decent. Makes you think twice about spending more .. Yet there were some of the point of the Article I, too, found somewhat overly simplistic like for example their defintion of "jItter"..

I will leave to those much knowledgeable to expand on this .. On this I have designate DonH50 ... Can I ? :)
Hey Frantz, you can try! I am typing whilst practicing -- gig tomorrow night -- so I doubt I follow up tonight (we'll see). I want to see if some of things I thought a bit "off" are my interpretation vs. truly nonstandard definitions (let alone errors).

For the record, there are plenty of companies I admire and respect, in many fields, that should really have the engineers review the marketing material, IMHO!!! :) Of course, sales could fall, so maybe that's why it rarely happens... - Don
A quick run-down, in the order I read them in the paper:

1. Analog or digital, there are analog elements that introduce nonlinearities, and these cause THD and IMD. I agree IMD is more objectionable to listeners because it generates non-harmonic tones that are more easily picked out. I see what they are trying to do in distinguishing analog from digital nonlinearities, but in truth thee are present in all systems. Digital converters also introduce their own "analog" nonlinearity through e.g. nonlinearities in the reference ladder or delta-sigma modulator's cells.

2. "Digital distortion" is a pretty ambiguous term; fortunately, they go on to define a few causes.

3. Jitter is actually explained well, imo. I do not equate it to phase modulation, nor does the standard, but I don't think there's anything wrong with that. They did it to relate it to other terms audiophiles might recognize. I can quibble about how accurate an analogy that is, but it's an analogy, no worries. Overall, the jitter write-up is pretty durn good.

4. The discussion on quantization noise is a little misleading though not inaccurate. Among things not said are that, as the signal fades away, you can't hear it anyway, whether it is above or below the quantization noise. And, we can actually pull signal out that is below the noise floor, a marvelous ability our ear/brain system has. And most DSP systems, for that matter (we and it can average out the noise to extract signal information below the noise peaks). And, those steps are pretty darn small, even for a 16-bit converter. Remember, the SNR is ~98 dB ideally, and most rooms have nowhere near that much dynamic range (typically maybe 40 - 50 dB up to perhaps 120 dB at "the chair", so realistic usable dynamic range is closer to 70 - 80 dB).

5. Dither does not "remove" distortion; it will decorrelate quantization noise and provide a noise floor that may sound a little less "harsh". This may be a bit of a semantic issue (similar to the discussion Amir and I had earlier): distortion to me is not the same as noise, and in my little world of ADC and DAC designers quantization noise and distortion are two different things, generated by different sources, with different spectral content, and different impact/implications on the system (audio, radio, radar, lidar, whatever). One of the largest benefits of dither is to suppress tones in delta sigma modulators. Those tones are not harmonically related to the signal, can be fairly large, and thus quite objectionable.

BTW, few converters (ADCs or DACs) over 16-bit resolution improve much in the way of noise floor or nonlinearity (distortion) in the real world, although the theory is there and some have certainly done so, though I am pretty sure the only true 24-bit ADCs I have seen have bandwidth far lower than audio. Also, dither reduces the SNR (it adds more noise, after all), but is generally worth the trade.

6. Delta-sigma modulators were around in the 30's, I think; Sony is being given a bit too much credit. What needed to happen, and has, is to develop better filters and practical digital processing to make them work in the real world. Sony was a major force on bringing them to market.

Doubling the sampling rate of a conventional converter adds 3 dB, not 6 dB in SNR. There are enough other "good" numbers that I suspect a typo.

I don't think noise shaping can improve the noise floor of a recording though noise shaping with filtering (interpolation) of the 16-bit source can help. "Perceived" may be the key word in that paragraph...

In my experience, 24-bit systems are used to provide increased dynamic range and headroom when multiple sources are recorded and combined through the mastering process. A good 16-bit recording and a good 24-bit recording are virtually indistinguishable in my experience once the mastering is done, but what happens along the way is easier with more bits to play with.

7. I was getting tired so just skimmed through the aliasing section. It appears correct and makes good points about the filtering problem Nyquist-rate converters all have (audio or microwave).

So, when all is said and done, after a bit harder look, there is actually little to quibble about and the overall article seems reasonable to me. I thiink a couple of things just stood out when I first read it. And, please do not take me for a pompous a** -- I was asked to express an opinion and did so. I am reasonably expert in my field, and admire the work and appreciate the effort (that article undoubtedly took hours and hours of work!) Benchmark put into that article. Two thumbs up with maybe a smudge here and there a little white-out might fix. :)

References -- many, many years of swearing at converters of various flavors, a decade or so of live and studio audio work and installing and repairing systems a couple of decades ago, the IEEE Standard cited earlier, a plethora of IEEE and AES articles, and perhaps:

Candy and Temes, Oversampling Delta-Sigma Data Converters, IEEE Press, 1991.
- an update of the delta-sigma book I did not bring home, sorry

Kinsler et. al., Fundamentals of Acoustics, 3rd ed., Wiley and Sons, 1982.

Everest, The Master handbook of Acoustics, 2nd ed, TAB, 1989.

Oppenheim and Schafer, Discrete-Time Signal Processing, Prentice-Hall, 1996 (I think; I only have the earlier version at home).

I could cite many, many more but am tired and lazy; please let me know if you want a specific topic reference.

HTH - Don (I was gonna' list credentials, but you have no way of verifying that I didn't just make them up, and virtually all my work has been behind the scenes and unpublished anyway, like that of many engineers; the fun's in the lab <and in the air, and even in space, in my case>, not the papers!)
5. Dither does not "remove" distortion; it will decorrelate quantization noise and provide a noise floor that may sound a little less "harsh". This may be a bit of a semantic issue

As you may know I'm not a math guy, favoring an empirical understanding. So maybe you can answer this:

A few weeks ago I was explaining dither to someone, and how to my ears it's needed mostly when going from 16-bit audio to only 8 bits. I told him of an experiment I did (16 bits to 8) where distortion when truncating went away with dither, though the noise seemed to be at the same level as the distortion. So the music sounded noisy rather than distorted. He asked me if the dither noise simply masks the distortion, or if there's more to it than that. I told him I didn't really know, but if there is a difference it's probably pretty slight.

So what's the real answer? :D Is it simply that dither noise fills in the rest of the spectrum, so you don't hear the individual distortion overtones?

Hey you caught one of my math mistakes (in the Fourier series)! Besides, I'm an engineer, which means I tolerate math because I have to use it all that time, not that I understand it. In this case I have a math reference but I think it's home (lunch hour for me now), and the math is pretty ugly (Bessel functions, statistic operators like expectation functions, other fun high-order stuff that turn an engineer's stomach). Anyway, I'll take a stab at this and hope I’m close. I also think there are really two things going on in your example...

When you truncate a signal, you throw away part of the lsb of the new samples. If you sampled with an 8-bit converter, it would catch samples at its thresholds, where the truncated version from 16 bits "floors" everything. Instead of 1.7487 (or whatever, a higher-precision number) becoming 2.0 when sampled, it becomes 1.0 when you truncate. In practice, this adds a little noise to the converter: instead of getting the 50 dB SNR of an ideal 8-bit, we get maybe 47 dB, around 3 dB lower SNR since the truncated sample is "right" about half the time (not the 6 dB we’d lose by dropping a bit). It does not add "distortion" in the way I think of it, but does raise the noise floor. This assumes no other nonlinearities and typical Gaussian noise (normal assumptions for quantization noise).

Now, quantization noise is comprised of little stair-step patterns that have sharp edges and tend to generate relatively more high-frequency energy than “analog” thermal or shot noise. That noise sounds harsher to our ears than typical analog noise. Think white vs. pink noise; the former sounds more "hissy" and "raspy" than the latter. Also, any nonlinearities in the sampling process also get reflected (aliased) into baseband, putting spurs in strange places rather than purely harmonic, and our ears pick up on that pretty well. For good converters these are pretty low spurs, but we're talking noise floor here, and now the noise is no longer truly random nor white (pink, brown – does anyone test with brown noise any more?)

Among the things dither does is to move the sampled values around, spreading the energy around more of the spectrum. This is a long-winded way of saying your last line is very close to the truth: noise decorrelation (dither) tends to spread the energy of all the samples. Over time, i.e. many cycles (which may take only ms of real time), the primary signals are repeated (auto-correlated) and will average to near their actual values (amplitude) as if there was no noise. Distortion terms such as HD and IMD will also average to near their "true" values; they are correlated to the signal, so dither does not significantly reduce that distortion, e.g. nonlinearity in the ADC or DAC (or anywhere else). Similarly, the quantization noise is related to the signal and the clock, so again can “build up” to create stronger tones in addition to those harsher sounds from the sharp edges. Dither is not correlated to the signal, so adding a little noise puts some “fuzz” on those sharp edges and moves the samples around a little bit. This spreads the quantization noise across the spectrum and does indeed give a more “analog” noise floor that does not sound as harsh. Adding dither does add noise, however, so the final SNR is lower than before you added dither; it just sounds better!

So, in your example, I think truncation itself did not really make the noise floor sound harsher; it was a side-effect of reduced resolution raising the quantization noise floor. The quantization noise got louder relative to the signal and thus your noise floor sounded harsher. Dither does indeed make that quantization noise “hash” sound better, thus it worked out as you expected.

HTH - Don
Interesting, thanks Don. I didn't think that truncation made the noise harsher, but that it made the harmonic distortion more prominent which itself might sound harsher than noise of any spectrum.

Also, are you sure that truncation always truncates downward? I had always ASSumed that saving a 24-bit file, or a 32-bit DAW's mix, as 16 bits would round the lowest bit up or down as appropriate. Of course I've never tested that. But wouldn't it be trivial for an audio editor program to do it that way?

Probably semantics, Ethan, and you are probably correct for real systems. I tend to be precise with terms and that trips me up in the real world now and then (forest, trees)... To me, "truncation" means to remove low-level bits, or for a decimal number to remove all digits after the decimal point (for fixed or floating-point numbers, or to whatever digit is defined as the limit). That is nearly the same as a "floor" function, which takes everything to the nearest integer below, while a "ceiling" function takes everything to the nearest integer above. Rounding is to the nearest integer above or below.

trunc(1.3) = trunc(1.7) = 1 = floor(1.3) or floor(1.7) assuming unit truncation
<aside: trunc1.3465e34 = 1e34>
round(1.3) = 1; round(1.7) = 2
ceil(1.3) = ceil(1.7) = 2

Some computers/programs implement these a little differently, but the main programs I have been using (Mathcad, Matlab) work in this way. As for what a specific audio system does, I cannot say. I have used systems that truncate (it is easiest -- just pick only the desired bits from a word, or word bus) and round (better results, but requires a digital comparison and an adder in the path).

You are correct that converting by rounding adds virtually no noise. For an audio editor program, I don't know. I suspect most of the new ones use floating-point math and even if not as you say rounding is pretty trivial. I have seen some that allow you to implement fairly sophisticated schemes, like predictive coding, to optimize the conversion. And, some that are fixed-point and just truncate. A long way of saying "I don't know" (I got blasted on another forum for "talking" too much; bad habit of mine, sorry!)

I have arrived to a quite decent Digital reproduction thru a dedicated iMac computer, i have gone all the way starting from the Wadia box to this recent setup which incorporates a license of Puremusic software. Digital has its evils, no doubt - just as analog or tubed amplification, the only difference I perceive is that the earliest is running FAST to resolve them.
Answering Ethan's questions differently :), dither does NOT work by masking the noise. If it did, it would not matter if you added the noise before or after conversion to 8-bits. But the math shows that it clearly makes a difference if you add it before conversion. That we call dither. If you add it after, it is just noise and doesn't help to nullify the harmonic distortion.

Here is another way to look at it. We can use noise-shaping to push dither noise to higher frequencies which are harder to hear. In that sense, we get rid of distortion in mid-band/audible frequencies without additional noise! Surely that would not work if the reason dither worked was due to noise covering up the distortion.

As to truncation, that is what people mean since it simply involves chopping off the extra bits in digital samples. Rounding requires work and if done right, can actually incorporate dither. So that is not usually what is meant by truncating.

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