Digital Audio Demystified

[NTK]

Amir has a thread on the fundamentals of sigma-delta DAC. As Don has alluded to, the highest performance audio DACs these days are sigma-delta.

Delta-Sigma Digital Audio Converters (DAC)

Note: this article was originally written by our DAC/technical expert, @DonH56. The previous thread on DAC fundamentals provided an overview of conventional (Nyquist) DACs. This thread will introduce oversampling and the delta-sigma architecture that dominates the DACs used in consumer audio...

www.audiosciencereview.com

However, I have in my plan to write a post on DACs that use sample-and-hold, which is the most intuitive method and therefore how DACs were first implemented. This produces the stair-step waveform (see post #2) as part of the digital to analog conversion process. I just started preparing the materials and hopefully will have a post by this weekend.

 

[DVDdoug]

Monty's video showed the post-DAC signal as a perfect sine wave with no stair stepping. I think he even said that the smoothness isn't even due to filtering a stair step. He made it sound like the DAC chip can literally somehow fit the correct curve to the sample points and output it smoothly.

I don't know how they work but interpolation would be "smoothing" although possibly not as smooth as analog filtering.

Filtering/smoothing isn't as important as you might think. One time I was doing some "weekend experiments" with a soundcard and an oscilloscope at work. I don't even remember what I was doing but I was shocked to see rectangular unfiltered signals coming out of the (cheap) soundcard!!! I was listening fairly regularly (on "cheap computer speakers". A little 2.1 setup) and I never noticed anything wrong. I never tried it on a better system so I'm not sure if I would hear anything or not. Then when I started thinking about it, the harmonics are ultrasonic so I can't hear them, the speakers will provide "mechanical" filtering, and the amplifier might not pass ultrasonic signals either.

 

[txbdan]

Yeh, I'm a EE and at one time passed, although barely, the Signals and Systems course. hah. My understanding was that a DAC always had some small sample and hold induced stair step type feature. I don't work much with DACS or analog circuits so haven't thought much bout it. Monty's video where he's adamant about the output NOT being a stair step is what got my head spinning. Even a Sigma-Delta which seems sort of like a LPF'ed PWM signal may be plenty smooth enough, but isn't quite what Monty was implying I think.

 

[LTig]

It's the analog reconstruction filter which removes the high frequency components (the "step" part) of the signal coming out of the DAC chip.

 

[voodooless]

It's the analog reconstruction filter which removes the high frequency components (the "step" part) of the signal coming out of the DAC chip.

… if you’re lucky.

Here is some info on why sample-and-hold is fundamentally wrong:

CHORD M-Scaler Review (Upsampler)

People have compared filter-less NOS dacs with delta-sigma dacs, reported audible differences, and even stated some advantages. How I imagine one might reply to this on this site is by saying that they tricked themselves or that their brain did, that the distortion gave to a sound effect or that...

www.audiosciencereview.com

Best to first dig into on how the sampling theorem works as well.

 

[DonH56]

But if each bit is being mapped to a discrete voltage, isn't that in fact a stairstep signal?

Monty's video showed the post-DAC signal as a perfect sine wave with no stair stepping. I think he even said that the smoothness isn't even due to filtering a stair step. He made it sound like the DAC chip can literally somehow fit the correct curve to the sample points and output it smoothly.

Each bit is mapped to a specific voltage no matter the DAC architecture. The output will be a stairstep at the output of the DAC, but in this context the DAC is the actual converter circuit. To an audiophile, the DAC is a box that includes the input buffer, clock recovery and digital pre-filters, DAC itself (often a highly-integrated chip), and output anti-imaging filter (often along with a bunch of other stuff). The output anti-image filter is required to remove all frequencies above Nyquist, 1/2 the sampling rate. That gets rid of the steps and the images at the DAC's output. A delta-sigma, or any oversampled design, can use a much smoother (lower-order) roll off above the audio band but the final effect will be the same: smoothed analog output with no stair steps.

There are a number of articles on ASR about ADC/DAC background and such; see the list in my signature as well as this thread and hopefully others from @NTK. My articles cover the basics of sampling and DAC reconstruction but I did not include the output filter in my basic model (at least I don't think I did, didn't go back and look, lazy). Filter design is a big subject and my stuff was just to introduce the basics.

HTH - Don

 

[RayDunzl]

Monty's video showed the post-DAC signal as a perfect sine wave with no stair stepping

16 bit samples gives up to 32,767 positive and 32768 negative "steps".

24 bit samples gives up to 8,388,607 positive and 8,388,608 negative "steps".

Is the resolution of your display able to discern those variations?

My monitor has only 1600 pixels (steps) vertically, giving at most (full screen) +/- 800 steps, so, I suppose there should be some jaggies present merely due to the display resolution, but I rarely notice even those crude steps.

 

[antcollinet]

16 bit samples gives up to 32,767 positive and 32768 negative "steps".

24 bit samples gives up to 8,388,607 positive and 8,388,608 negative "steps".

Is the resolution of your display able to discern those variations?

My monitor has only 1600 pixels (steps) vertically, giving at most (full screen) +/- 800 steps, so, I suppose there should be some jaggies present merely due to the display resolution, but I rarely notice even those crude steps.

But again, the output of a properly designed dac has NO steps whatsoever - regardless of bit depth. The reconstruction filter can perfectly restore the original smooth signal.

All increased bit depth gives you is a lower noise floor.

 

[fpitas]

But if each bit is being mapped to a discrete voltage, isn't that in fact a stairstep signal?

Monty's video showed the post-DAC signal as a perfect sine wave with no stair stepping. I think he even said that the smoothness isn't even due to filtering a stair step. He made it sound like the DAC chip can literally somehow fit the correct curve to the sample points and output it smoothly.

No DAC chip can output a smooth signal. But after low-pass filtering, it can come very close. I suppose one could consider the filter as part of a correctly designed DAC.

 

[fpitas]

For those curious what's under the hood, so to speak:

https://www.analog.com/media/en/training-seminars/design-handbooks/Basic-Linear-Design/Chapter6.pdf

 

[txbdan]

NTK's first post in this thread describes sync interpolation. It showings that storing and even processing audio information in discrete time works very well. But this applies to digital processors and DSPs and has nothing to do with how DACs work. (and that is the intent of his post which is fine)

What got me thinking is Monty's video here:

. It comes right out of the gate saying that post-DAC digital audio is not stair-stepped. Is his point simply that its stair-stepped but then filtered to be smooth? If so, wouldn't that be much simpler to just say that? But even using a 3bit sample rate example, he shows the output as totally smooth:

He then goes onto explain how given two sample points per period (Nyquist) there is only one sine wave solution that fits the samples. This is of course true and explains how digital signal processing works accurately, but my question is how the heck does the DAC do this? A DAC isn't "smart" enough to curve fit sine wave to data points or event to do sync interpolation as far as I know.

Ok, maybe the DAC doesn't and it does produce stair steps which have a lot of high frequency content. If we LPF the output, the steps soften and soften until things are mostly smooth. But wouldn't we need an extremely tight bandpass filter centered at the fundamental to produce a perfect sine wave? But how would that filter work for broadband music?

If the answer is simply that if the resolution is high enough, then the steps are small enough, and we can smooth most of it out with filters, and that distortion is well below human hearing.. I get it. But if that's the case, why doesn't the video say that? I feel like there is some point in the video that I'm not getting that allows this to work even better than that.

 

[voodooless]

It comes right out of the gate saying that post-DAC digital audio is not stair-stepped.

Correct, at least current delta sigma DAC’s

Is his point simply that its stair-stepped but then filtered to be smooth? If so, wouldn't that be much simpler to just say that?

It would be simpler, but wrong

He then goes onto explain how given two sample points per period (Nyquist) there is only one sine wave solution that fits the samples. This is of course true and explains how digital signal processing works accurately, but my question is how the heck does the DAC do this? A DAC isn't "smart" enough to curve fit sine wave to data points or event to do sync interpolation as far as I know.

It oversampling the audio to a much higher rate, like 8x to 128x. This is done by adding zero samples in between the real ones. All that is then needed is a low pass to “fill-in” the zero samples. This will then reconstruct you pretty sine wave. The quality of the reconstruction depends on the properties of the filter.

Next one can put this data into a (multi-bit) delta-sigma modulator. These do not generate stair steps and need little analog filtering.

An R2R DAC will generate steps though. These have to be filtered in the analog domain.

Ok, maybe the ADC doesn't and it does produce stair steps which have a lot of high frequency content.

An ADC does not. The sample values they sample are only valid for the sample moment, not the whole same period.

Best to forget steps altogether.. they are a wrong representation of digital audio.

 

[fpitas]

NTK's first post in this thread describes sync interpolation. It showings that storing and even processing audio information in discrete time works very well. But this applies to digital processors and DSPs and has nothing to do with how DACs work. (and that is the intent of his post which is fine)

What got me thinking is Monty's video here:

. It comes right out of the gate saying that post-DAC digital audio is not stair-stepped. Is his point simply that its stair-stepped but then filtered to be smooth? If so, wouldn't that be much simpler to just say that? But even using a 3bit sample rate example, he shows the output as totally smooth:

He then goes onto explain how given two sample points per period (Nyquist) there is only one sine wave solution that fits the samples. This is of course true and explains how digital signal processing works accurately, but my question is how the heck does the DAC do this? A DAC isn't "smart" enough to curve fit sine wave to data points or event to do sync interpolation as far as I know.

Ok, maybe the ADC doesn't and it does produce stair steps which have a lot of high frequency content. If we LPF the output, the steps soften and soften until things are mostly smooth. But wouldn't we need an extremely tight bandpass filter centered at the fundamental to produce a perfect sine wave? But how would that filter work for broadband music?

If the answer is simply that if the resolution is high enough, then the steps are small enough, and we can smooth most of it out with filters, and that distortion is well below human hearing.. I get it. But if that's the case, why doesn't the video say that? I feel like there is some point in the video that I'm not getting that allows this to work even better than that.

When any smoothed stairsteps are in the noise floor, that's pretty much perfection.

 

[antcollinet]

Correct, at least current delta sigma DAC’s

It would be simpler, but wrong

It oversampling the audio to a much higher rate, like 8x to 128x. This is done by adding zero samples in between the real ones. All that is then needed is a low pass to “fill-in” the zero samples. This will then reconstruct you pretty sine wave. The quality of the reconstruction depends on the properties of the filter.

Next one can put this data into a (multi-bit) delta-sigma modulator. These do not generate stair steps and need little analog filtering.

An R2R DAC will generate steps though. These have to be filtered in the analog domain.

An ADC does not. The sample values they sample are only valid for the sample moment, not the whole same period.

Best to forget steps altogether.. they are a wrong representation of digital audio.

OK - so my understanding is faulty also.

So the oversampling and low pass filtering is done in the digital domain - correct?

But then you still have a sampled signal - (say 5.6448 Msamples/s for 44.1K 128x oversampled). Which must be output from the DAC Chip before going into an analogue reconstruction (low pass) filter? Correct?

So what is that waveform - a series of voltage "spikes" At each sample time? Or a 'held' voltage from one sample to the next (stair stepped)?

 

[voodooless]

So the oversampling and low pass filtering is done in the digital domain - correct?

Yes

But then you still have a sampled signal - (say 5.6448 Msamples/s for 44.1K 128x oversampled). Which must be output from the DAC Chip before going into an analogue reconstruction (low pass) filter? Correct?

Yes

So what is that waveform - a series of voltage "spikes" At each sample time? Or a 'held' voltage from one sample to the next (stair stepped)?

That depends on whatever does that job. An R2R DAC will output a stepped waveform, a delta sigma DAC will output a 1 bit PWM signal.

 

[antcollinet]

Yes

Yes

That depends on whatever does that job. An R2R DAC will output a stepped waveform, a delta sigma DAC will output a 1 bit PWM signal.

Thanks

*wanders off to read up more on delta sigma converters*

 

[voodooless]

Thanks

*wanders off to read up more on delta sigma converters*

For completeness here are some RAW waveforms from these DACs:

Yeah.. it looks awful, but in the end it all works out

 

[DonH56]

So the oversampling and low pass filtering is done in the digital domain - correct?

For a delta-sigma DAC oversampling is part of the design and yes is digital. The very steep low-pass filtering is also done digitally. There is still an output anti-imaging filter on the analog output but, since the sampling rate is well above the audio band, it can be fairly simple (low order).

You can also oversample an R2R (etc. "conventional") DAC and add digital filters before the conversion to analog, followed by a low-order anti-imaging filter. I have not seen this done very often, but again audio is not my day job.

I realize my earlier answer was about conventional DACs. Delta-sigma designs output a fast (oversampled) pulse stream so you will not see conventional stair steps at its output. It still passes through discrete (fixed) steps, however, just harder to actually see them since a lot of filtering happens prior to the first analog output so initial smoothing is done "inside the box".

But then you still have a sampled signal - (say 5.6448 Msamples/s for 44.1K 128x oversampled). Which must be output from the DAC Chip before going into an analogue reconstruction (low pass) filter? Correct?

Yes. But what the signal looks like depends upon the DAC's architecture.

So what is that waveform - a series of voltage "spikes" At each sample time? Or a 'held' voltage from one sample to the next (stair stepped)?

There are special DACs that create quasi-impulse outputs, but usually (vast majority IME) a conventional DAC will create steps (like stair steps) at the output before the filter. A delta-sigma creates a pulse train (typically PCM signal) so toggles between "1" and "0" with varying pulse widths.

The classic intro for me was an IEEE book (tutorial and collection of papers) edited by one of my early teachers: Oversampling Delta-Sigma Data Converters by James C. Candy and Gabor C. Temes. There was what many considered a follow-on text years later with more information and practical background: Delta-SIGMA Data Converters: Theory, Design, and Simulation, by Norsworthy, Schreier, and Temes -- I no longer have that one (borrowed and never returned, arrrgh!!!) There are many others, natch, since these were early 1990's books, but they have the basics.

These articles show conventional DACs: https://www.audiosciencereview.com/forum/index.php?threads/digital-audio-sampling-101.1919/ and https://www.audiosciencereview.com/...ital-audio-converters-dacs-fundamentals.1927/

Delta-sigma DACs here: https://www.audiosciencereview.com/...ital-audio-converters-dacs-fundamentals.1927/ -- I did not include time-domain pictures, looking back a major oversight, but @JohnYang1997 added some plots that make it clear.

Hard to believe some of those early posts are more than 10 years old, and original articles are from papers I gave over 20 years ago. Blah.

HTH - Don

Edit: @voodooless already answered, I'm late to the party again, oh well...

 

[antcollinet]

For a delta-sigma DAC oversampling is part of the design and yes is digital. The very steep low-pass filtering is also done digitally. There is still an output anti-imaging filter on the analog output but, since the sampling rate is well above the audio band, it can be fairly simple (low order).

You can also oversample an R2R (etc. "conventional") DAC and add digital filters before the conversion to analog, followed by a low-order anti-imaging filter. I have not seen this done very often, but again audio is not my day job.

I realize my earlier answer was about conventional DACs. Delta-sigma designs output a fast (oversampled) pulse stream so you will not see conventional stair steps at its output. It still passes through discrete (fixed) steps, however, just harder to actually see them since a lot of filtering happens prior to the first analog output so initial smoothing is done "inside the box".


Yes. But what the signal looks like depends upon the DAC's architecture.


There are special DACs that create quasi-impulse outputs, but usually (vast majority IME) a conventional DAC will create steps (like stair steps) at the output before the filter. A delta-sigma creates a pulse train (typically PCM signal) so toggles between "1" and "0" with varying pulse widths.

The classic intro for me was an IEEE book (tutorial and collection of papers) edited by one of my early teachers: Oversampling Delta-Sigma Data Converters by James C. Candy and Gabor C. Temes. There was what many considered a follow-on text years later with more information and practical background: Delta-SIGMA Data Converters: Theory, Design, and Simulation, by Norsworthy, Schreier, and Temes -- I no longer have that one (borrowed and never returned, arrrgh!!!) There are many others, natch, since these were early 1990's books, but they have the basics.

These articles show conventional DACs: https://www.audiosciencereview.com/forum/index.php?threads/digital-audio-sampling-101.1919/ and https://www.audiosciencereview.com/...ital-audio-converters-dacs-fundamentals.1927/

Delta-sigma DACs here: https://www.audiosciencereview.com/...ital-audio-converters-dacs-fundamentals.1927/ -- I did not include time-domain pictures, looking back a major oversight, but @JohnYang1997 added some plots that make it clear.

Hard to believe some of those early posts are more than 10 years old, and original articles are from papers I gave over 20 years ago. Blah.

HTH - Don

Edit: @voodooless already answered, I'm late to the party again, oh well...

No probs - the more times I read it, and the more words used, the better chance I have of some of it sinking in

 

[LTig]

Just as a reminder: It's not the output of the DAC chip which matters but the output of the whole device. Searching for stair steps at the output of the chip is futile and tells us nothing about what comes out of the device at its analog output.

DAC devices implement a reconstruction filter behind the output of the DAC chip which is required to output the same signal as was digitized earlier. This is what Monty shows in his great video.

Devices without a reconstruction filter I consider as broken per design, and the designer has not understood how digital audio works.