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Sound BlasterX G6 Hi-Res 130dB 32bit/384kHz Gaming DAC, External USB Sound Card with Xamp Headphone Amp, Dolby Digital, 7.1 Virtual Surround Sound, Sidetone/Speaker Control for PS4, Xbox One, Nintendo.

Continuing in the 'firehose' tradition of Episode 01, Xiph.Org's second video on digital media explores multiple facets of digital audio signals and how they really behave in the real world.

Demonstrations of sampling, quantization, bit-depth, and dither explore digital audio behavior on real audio equipment using both modern digital analysis and vintage analog bench equipment, just in case we can't trust those newfangled digital gizmos. You can download the source code for each demo and try it all for yourself!

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• Jan 08, 2021 The first thing to look for in a sound card is the quality of the Digital-to-Analog-Converter. This module is the main component responsible for playback; the better the quality of DAC is, the more noise-free your audio will be. So how do you tell the quality of the DAC? Usually by its SNR, or Signal to Noise Ratio.

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“Hi, I'm Monty Montgomery from Red Hat and Xiph.Org.

“A few months ago, I wrotean article on digital audio and why 24bit/192kHz music downloads don't make sense.In the article, Imentioned--almost in passing--that a digital waveform isnot a stairstep,and you certainly don't get a stairstep when youconvert from digital back to analog.

“Of everything in the entire article, that was the number one thingpeople wrote about. In fact, more than half the mail I got was questions andcomments about basic digital signal behavior. Since there's interest, let'stake a little time to play with some simple digital signals. ”

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Veritas ex machina

If we pretend for a moment that we have no idea how digital signals reallybehave, then it doesn't make sense for us to use digital testequipment. Fortunately for this exercise, there's still plentyof working analog lab equipment out there.

We need a signal generator to provide us with analog inputsignals--in this case, anHP3325from 1978.

We'll observe our analog waveforms on analog oscilloscopes,like this Tektronix 2246 from the mid-90s, one of the last and best analog scopes made.

Finally, we'll inspect the frequency spectrum of our signals using ananalog spectrum analyzer, thisHP3585from the same product line asthe signal generator. Like the other equipment here it hasa rudimentary and hilariously large microcontroller,but the signal pathfrom input to what you see on the screen is completely analog.

All of this equipment is vintage, but the specs are still quite good.We start with the signal generator set to output a 1 kHzsine wave at one VoltRMS.We see the sine wave on the oscilloscope, can verify that it is indeed1 kHz at 1 Volt RMS, which is 2.8 Voltspeak-to-peak,and that matches themeasurement on the spectrum analyzer as well.

The analyzer also shows some low-level white noiseand just a bit of harmonic distortion,with the highest peak about 70dB or so belowthe fundamental.This doesn't matter to the demos, but it's good to take notice of it now to avoid confusion later.

For digital conversion, we use a boring, consumer-grade, eMagic USB1audio device. It's more than ten years old at this point, and it'sgetting obsolete.

A recent converter can easily have an order of magnitude better specs.Flatness,linearity,jitter,noise behavior,everything...You may nothave noticed. Just because we can measure an improvement doesn'tmean we can hear it, and even these old consumer boxes were already atthe edge of ideal transparency.

The eMagic connects to my ThinkPad, which displays a digitalwaveform and spectrum for comparison, then the ThinkPadsends the digital signal right back out to the eMagic forre-conversion to analog and observation on the output scopes.

Stairsteps

First demo: We begin by converting an analog signal to digital andthen right back to analog again with no other steps.

The signal generator is set to produce a 1kHz sine wave just likebefore and we can see the analog sine wave on the input-side oscilloscope. The eMagic digitizes our signal to16 bit PCM at 44.1kHz,same as on a CD. The spectrum of the digitized signal on the Thinkpad matches what we saw earlier and what we see now on the analog spectrum analyzer, aside from its high-impedance input being just a smidge noisier. For now, the waveform display shows our digitized sine wave as astairstep pattern, one step for each sample.

When we look at the output signal that's been convertedfrom digital back to analog, we see that it's exactly like the original sine wave. No stairsteps.

1 kHz is still a fairly low frequency, so perhaps the stairsteps are justhard to see or they're being smoothed away. Next, set the signal generator to 15kHz, which is much closer to Nyquist.Now the sine wave is represented by less than three samples per cycle, and the digital waveform appears rather poor! Yet the analog output is still a perfect sine wave, exactly like the original.As we keep increasing frequency, all the way to 20kHz, the output waveform is still perfect. No jagged edges, no dropoff, no stairsteps.

So where'd the stairsteps go? It's a trick question; they were never there. Drawing a digital waveform as a stairstep was wrong to begin with.

A stairstep is a continuous-time function. It's jagged, and it's piecewise, but it has a defined value at every point in time.A sampled signal is entirely different. It's discrete-time; it's only got a value right at each instantaneous sample point and it'sundefined, there is no value at all, everywhere between. A discrete-time signal is properly drawn as a lollipop graph.The continuous, analog counterpart of a digital signal passes smoothly through each sample point, and that's just as true for highfrequencies as it is for low.

The interesting and non-obvious bit is that there's only onebandlimited signal that passes exactly through each sample point; it's a unique solution. If you sample a bandlimited signal and then convert it back, the original input is also the only possible output.A signal that differs even minutely from the original includes frequency content at or beyond Nyquist, breaks the bandlimiting requirement and isn't a valid solution.

So how did everyone get confused and start thinking of digital signals as stairsteps? I can think of two good reasons.

First: it's easy to convert a sampled signal to a true stairstep. Justextend each sample value forward until the next sample period. This iscalled a zero-order hold, and it's an important part of how somedigital-to-analog converters work, especially the simplest ones.As a result, anyone who looks up digital-to-analog converter ordigital-to-analog conversion is probably going to see a diagram of astairstep waveform somewhere, but that's not a finished conversion,and it's not the signal that comes out.

Second, and this is probably the more likely reason, engineers whosupposedly know better (yes, even I) draw stairsteps even though they'retechnically wrong. It's a sort of one-dimensional version offat bits in an image editor.Pixels aren't squares either, they're samples of a 2-dimensionalfunction space and so they're also, conceptually, infinitely smallpoints. Practically, it's a real pain in the ass to see or manipulateinfinitely small anything, so big squares it is.

Digital stairstep drawings are exactly the same thing. It's just a convenient drawing. The stairsteps aren't really there.

Bit-depth

When we convert a digital signal back to analog, the result isalso smooth regardless of the bit depth. It doesn't matter if it's 24 bits or 16 bits or 8 bits.So does that mean that the digital bit depth makes no difference atall? Of course not.

Channel 2 is the same sine wave input, but we quantize it withdither down to 8 bits.On the scope, we still see a nicesmooth sine wave on channel 2. Look very close, and you'll also see abit more noise. That's a clue.

If we look at the spectrum of the signal, our sine wave isstill there unaffected, but the noise level of the 8-bit signal onthe second channel is much higher. And that's the difference, the only difference, the number of bits makes.

When we digitize a signal, first we sample it. Thesampling step is perfect; it loses nothing. But then we quantize it,and quantization adds noise.The number of bits determines how much noise and so the level of thenoise floor.

What does this dithered quantization noise sound like?Those of you who have used analog recording equipment might think to yourselves, 'My goodness! That sounds like tape hiss!'Well, it doesn't just sound like tape hiss, it acts like it too, andif we use a gaussian dither then it'smathematically equivalent in every way. It is tape hiss.

Intuitively, that means that we can measure tape hiss and thus the noise floorof magnetic audio tapein bits instead of decibels, in order to put things in adigital perspective. Compact cassettes, for those of you who are old enough to remember them, could reach asdeep as 9 bits in perfect conditions. 5 to 6 bits wasmore typical, especially if it was a recording made on atape deck. That's right; your old mix tapes were only about 6 bitsdeep if you were lucky!

The very best professional open reel tape used in studios could barelyhit 13 bits withadvanced noise reduction. That's why seeing 'D D D' on a Compact Disc used to be such a big,high-end deal.

Dither

We've been quantizing with dither. What is ditherexactly and, more importantly, what does it do?

The simplest way to quantize a signal is to choose the digitalamplitude value closest to the original analog amplitude.Unfortunately, the exact noise that results from this simplequantization scheme depends somewhat on the input signal.It may be inconsistent, cause distortion, or beundesirable in some other way.

Going deeper…

• Cameron Nicklaus Christou's thesis Optimal Dither and Noise Shaping in Image Processing provides an excellent explanation of dither and noise shaping.

Dither is specially-constructed noise that substitutes for the noiseproduced by simple quantization. Dither doesn't drown out or maskquantization noise, it replaces it with noise characteristicsof our choosing that aren't influenced by the input.

The signal generator has too much noise for this test so we produce a mathematically perfect sine wave with the ThinkPad and quantize it to 8 bits with dithering.We see the sine wave on waveform display and output scope, and a clean frequency peak with a uniform noise floor on both spectral displaysjust like before. Again, this is with dither.

Now I turn dithering off.

The quantization noise that dither had spread out into a nice, flat noisefloor, piles up into harmonic distortion peaks. The noise floor islower, but the level of distortion becomes nonzero, and the distortionpeaks sit higher than the dithering noise did.

At 8 bits this effect is exaggerated. At 16 bits,even without dither, harmonic distortion is going to be so low as tobe completely inaudible. Still, we can use dither to eliminate it completely if we so choose.

Turning the dither off again for a moment, you'll notice that theabsolute level of distortion from undithered quantization staysapproximately constant regardless of the input amplitude.But when the signal level drops below a half a bit, everythingquantizes to zero.

In a sense, everything quantizing to zero is just 100% distortion!Dither eliminates this distortion too. When we reenable dither, we clearly see our signal at 1/4 bit with a nice flat noise floor.

The noise floor doesn't have to be flat. Dither is noise of ourchoosing, so it makes sense to choose a noise as inoffensive anddifficult to noticeas possible.

Human hearing is most sensitive in the midrange from 2kHz to 4kHz; that's where background noise is going to be the most obvious.We can shape dithering noise away from sensitive frequencies to wherehearing is less sensitive, usually the highest frequencies.

Lastly, dithered quantization noise is higher power overallthan undithered quantization noise, even though it often sounds quieter, andyou can see that on a VU meter during passages of near-silence. However,dither isn't only an on or off choice. We can reduce the dither'spower to balance less noise against a bit of distortion to minimizethe overall effect.

For the next test, we also modulate the input signal like this to show how a varying input affects the quantization noise. Atfull dithering power, the noise is uniform, constant, and featurelessjust like we expect.

As we reduce the dither's power, the input increasinglyaffects the amplitude and the character of the quantization noise.Shaped dither behaves similarly, but noise shaping lends one more niceadvantage; it can use a somewhat lowerdither power before the input has as much effect on the output.

Despite all this text spent on dither, the differences exist 100 decibels or more below full scale. If the CD had been14 bits as originally designed,perhaps dither might bemore important. At 16 bits it's mostly a wash. It's reasonable to treatdither as an insurance policy that gives several extradecibels of dynamic range just in case. That said noone ever ruined a great recording by not dithering the final master.

Bandlimitation and timing

We've been using sine waves. They're the obvious choice when what wewant to see is a system's behavior at a given isolated frequency. Now let's look at something a bit more complex. What should we expect to happen when I change the input to a square wave?

The input scope confirms a 1kHz square wave. The output scope shows... exactly what it should.

What is a square wave really?

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We can say it's a waveform that's some positive value for half a cycle and then transitions instantaneously to a negative value for the other half.

But that doesn't really tell us anything useful about how that input becomes this output.

We remember that any waveform is also the sum of discrete frequencies,and a square wave is particularly simple sum: a fundamental and an infinite series of odd harmonics.

${\displaystyle \begin{array}{r}squarewave(t)=\frac{4}{\pi }\sin (\omega t)+\frac{4}{3\pi }\sin (3\omega t)+\frac{4}{5\pi }\sin (5\omega t)+\\ \frac{4}{7\pi }\sin (7\omega t)+\frac{4}{9\pi }\sin (9\omega t)+\frac{4}{11\pi }\sin (11\omega t)+\\ \frac{4}{13\pi }\sin (13\omega t)+\frac{4}{15\pi }\sin (15\omega t)+\frac{4}{17\pi }\sin (17\omega t)+\\ \frac{4}{19\pi }\sin (19\omega t)+\frac{4}{21\pi }\sin (21\omega t)+\frac{4}{23\pi }\sin (23\omega t)+\\ \frac{4}{25\pi }\sin (25\omega t)+\frac{4}{27\pi }\sin (27\omega t)+\frac{4}{29\pi }\sin (29\omega t)+\\ \frac{4}{31\pi }\sin (31\omega t)+\frac{4}{33\pi }\sin (33\omega t)+\cdots \end{array}}$

At first glance, that doesn't seem very useful either; you'd have to sum an infinite number of harmonics to get the answer! However, we don't have an infinite number of harmonics.

We're using a quite sharp anti-aliasing filter that cuts off rightabove 20kHz, so our signal is bandlimited. Only the first ten terms make it through, and that's exactly what we see on the output scope.

Going deeper…

• In modern web browsers you can program audio synthesizers directly in javascript. Use the two square wave formulas to get a square wave out of this page. (Note: The scope is not very accurate/useful.)

The rippling you see around sharp edges in a bandlimited signal is called the Gibbs effect. It happens whenever you slice off part of the frequency domain in the middle of nonzero energy.

The usual rule of thumb you'll hear is 'the sharper the cutoff, thestronger the rippling', which is approximately true, but we have to becareful how we think about it. For example, what would you expect our quite sharp anti-aliasing filterto do if I run our signal through it a second time?

Aside from adding a few fractional cycles of delay, the answer is:Nothing at all. The signal is already bandlimited. Bandlimiting itagain doesn't do anything. A second pass can't remove frequenciesthat we already removed.

That's important. People tend to think of the ripples asa kind of artifact that's added by anti-aliasing and anti-imagingfilters, implying that the ripples get worse each time the signalpasses through. We see that in this case that didn't happen, soit wasn't really the filter that added the ripples the first timethrough. It's a subtle distinction, but Gibbs effectripples aren't added by filters, they're just part of what abandlimited signal is.

Even if we synthetically construct what looks like a perfect digitalsquare wave it's still limited to the channel bandwidth. Remember thatthe stairstep representation is misleading. What we really have here are instantaneous sample pointsand only one bandlimited signal fits those points. All we did when wedrew our apparently perfect square wave was line up the sample pointsjust right so it appeared that there were no ripples if we playedconnect-the-dots. The original bandlimited signal, complete with ripples, wasstill there.

That leads us to one more important point. You've probably heardthat the timing precision of a digital signal is limited by its samplerate; put another way, that digital signals can't represent anything that falls between thesamples.. implying that impulses orfast attacks have to align exactlywith a sample, or the timing gets mangled or they just disappear.At this point, we can easily see why that's wrong.

Again, our input signals are bandlimited. And digital signals aresamples, not stairsteps, not 'connect-the-dots'. We most certainlycan, for example, put the rising edge of our bandlimited square waveanywhere we want between samples.

It's represented perfectly and it's reconstructed perfectly.

Epilogue

Like in A Digital Media Primer for Geeks, we've covered a broad range of topics, and yet barely scratched the surface of each one. If anything, my sins of omission are greater this time around.

Thus I encourage you to dig deeper and experiment. I chose my demos carefully to be simple and give clear results. You can reproduce every one of them on your own if you like, but let's face it: sometimes we learn the most about a spiffy toy by breaking it open and studying all the pieces that fall out. Play with the demo parameters, hack up the code, set up alternate experiments. The source code for everything, including the little push-button demo application, is at the end of this transcript.

In the course of experimentation, you're likely to run into something that you didn't expect and can't explain. Don't worry! My earlier snark aside, Wikipedia is fantastic for exactly this kind of casual research. If you're really serious about understanding signals, several universities have advanced materials online, such as the 6.003 and RES.6-007 Signals and Systems modules at MIT OpenCourseWare. And, of course, there's always the community here at Xiph.Org.

Credits

Written by: Christopher (Monty) Montgomery and the Xiph.Org Community

Special thanks to:

• Heidi Baumgartner, for the second Tektronix oscilloscope
• Gregory Maxwell and Dr. Timothy Terriberry, for additional technical review

Intro, title and credits music:
'Andy Warhol Is Gone', by Lousy Robot
Used by permission of Lousy Robot.
Original source track All Rights Reserved.
www.lousyrobot.com

This Video Was Produced Entirely With Free and Open Source Software:

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• GNU
  
• Linux
  
• Fedora
  
• Cinelerra
  
• The Gimp
  
• Audacity
  
• Postfish
  
• Gstreamer
  

All trademarks are the property of their respective owners.

• Complete videoCC-BY-SA
  
• Text transcript and Wiki editionCC-BY-SA
  

A Co-Production of Xiph.Org and Red Hat, Inc.
(C) 2012-2013, Some Rights Reserved

Use The Source Luke

As stated in the Epilogue, everything that appears in the video demos is driven by open source software, which means the source is both available for inspection and freely usable by the community. The Thinkpad that appears in the video was running Fedora 17 and Gnome Shell (Gnome 3). The demonstration software does not require Fedora specifically, but it does require Gnu/Linux to run in its current form. In all, the video involved just under 50,000 lines of new and custom-purpose code (including contributions to non-Xiph projects such as Cinelerra and Gromit).

The Spectrum and Waveform Viewer

The realtime software spectrum analyzer application that appears in the video was a preexisting application that was dusted off and updated for use in the video. The waveform viewer (effectively a simple software oscilloscope) was written from scratch making use of some of the internals from the spectrum analyzer application. Both are available from Xiph.Org svn:

• Source for the Spectrum and Waveform applications is found at:

• The source can be checked out of svn using the following command line:

svn co https://svn.xiph.org/trunk/spectrum

• Trac is a convenient way to browse the source without checking out a copy:

Spectrum and Waveform both expect an input stream on the command line, either as raw data or as a WAV file.

GTK-Bounce

Computer Sound Cards

The touch-controlled application used in the video is named 'gtk-bounce' and was custom-written for the sole purpose of the in-video demonstrations. It is so named because, for the most part, all it does is read the input from an audio device, and then immediately write the same data back out for playback. It also forwards a copy of this data to up to two external monitoring applications, and in several demos, applies simple filters or generates simple waveforms. It includes several demos not included in the video.

• Source for gtk-bounce is found at:

• The source can be checked out of svn using the following command line:

svn co https://svn.xiph.org/trunk/Xiph-episode-II/bounce/

• Trac is a convenient way to browse the source without checking out a copy:

Starting Gtk-bounce

The application is somewhat hardwired for specific demo parameters, but most of the hardwired settings can be found at the top of each source file. As found in SVN, the application expects an ALSA hardware audio device at hw:1, and if none if found, it will wait for one to appear. Once a sound device is successfully initialized, it expects to find and open two pipes named pipe0 and pipe1 for output in the current directory. In the video, the waveform and spectrum applications are started to take input from pipe0 and pipe1 respectively. The output sent to the two pipes is identical, and in most demos matches the output data sent to the hardware device for conversion to analog. The only exception is the tenth demo panel (which does not appear in the video) where gtk-bounce can be set to monitor the hardware inputs instead while the outputs are used to produce test waveforms.

Assuming gtk-bounce, spectrum and waveform have been checked out and built, the configuration seen in the video can be started using the following commands:

• make the pipe fifos for the applications to communicate (only needs to be done once)

mkfifo pipe0 pipe1

• start all three applications

waveform pipe0 & spectrum pipe1 & gtk-bounce &

Using Gtk-bounce

Gtk-bounce consists of eleven pushbutton panels (numbered zero through ten) that can be selected by scrolling up and down with the arrow buttons on the right side. Each panel is intended for a specific demo or part of a demo.

• Panel 0: This panel presents buttons that allow the sound card to be configured in several sampling rates and bit depths. Samples read from the audio inputs are sent to the output pipes and audio outputs for playback without modification.

• Panel 1: Both channels are forwarded to the outputs, however the user may select the bit depth of each channel independently. When the sound card is running in 16 bit mode and 16-bit depth is selected, the data is untouched. Requantization to a lower bit depth is performed with a flat triangle dither.

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• Panel 2: Both channels are re-quantized to the selected bit depth. Requantization to a lower bit depth is performed with a flat triangle dither.

• Panel 3: 'generate sine wave' discards the audio inputs and instead internally generates a sine wave at 32 bit precision, which is then quantized to the selected bit depth, optionally with dither. The resulting signal is then forwarded to the output.

• Panel 4: gtk-bounce generates a 16-bit sine wave of the selected amplitude, optionally with dither, and forwards the resulting signal to the outputs. The audio input from the audio device is discarded. Note that the slider sets the peak amplitude, not the peak-to-peak amplitude.

• Panel 5: generates a 16-bit sine wave, optionally quantized using dither. The user may additionally select a flat or a shaped dither. The 'notch and gain' button applies a notch filter to the resulting signal, and boosts the gain of the remaining noise so that it's easily audible. The audio input from the audio device is discarded.

• Panel 6: allows the user to play with the power of the dithering noise applied before quantizing the sine wave. Shaped or flat dither are available. The sine wave may also be modulated with a varying amplitude to highlight correlations between the input and the resulting quantization noise. The 'notch and gain' button applies a notch filter to the resulting signal, and boosts the gain of the remaining noise so that it's easily audible. The audio input from the audio device is discarded.

• Panel 7: applies a sharper antialiasing (lowpass) filter than is likely to be built into the sound-card hardware (as there's generally no reason to use a filter quite this sharp in practice). The very sharp filter allows us to bandpass the demonstration square wave without any harmonics landing in the transition band. The input is read from the audio device, passed through this sharper filter, and then forwarded to the outputs.

• Panel 8: when selected, generate a synthetic 'square wave' (this is not quite equivalent to a bandlimited analog square wave; the harmonic amplitudes are a bit different) that when aligned with the sampling phase just right gives the appearance of having infinite rise and fall time. The slider allows us to shift the waveform sample alignment back and forth by +/- one sample to reveal that the underlying signal is still band-limited.

• Panel 9: as in panel 8, generate a 'perfect' synthetic 'square wave'. However, the slider now allows us to shift the sample alignment of the second channel with respect to the first, instead of shifting both channels. This allows us the trigger/lock the scope timing to the channel 1 waveform so we can see the fractional sample movement and alignment of the waveform on channel 2. The audio input from the audio device is discarded.

• Panel 10: not used in the video; The audio device is configured to 24-bit input/output. The user may produce one of a range of test signals that are output to both the external applications and the audio device on the first channel. The input on the second channel is passed-through to the applications and audio device outputs unchanged. The first channel input is unused unless 'two input mode' is selected. When two input mode is selected, both input channels are read and the data sent to the external applications. Generated test signals are sent only to the audio hardware (on the first channel). This combination of test signals and input modes allows self-references frequency response, phase, noise, distortion and crosstalk testing of a given audio device.

Cairo Animations

Sound Cards 7.1

The animations featured throughout the Episode 2 video were rapid-development spaghetti hack-jobs coded by hand in raw Cairo. Each module generated a series of PNG stills that were then stitched into an animation with Cinelerra or mplayer. In the interest of pointing and laughing at what really bad code looks like...

• Source for the Cairo animations is found at:

• The source can be checked out of svn using the following command line:

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svn co https://svn.xiph.org/trunk/Xiph-episode-II/cairo/

• Trac is a convenient way to browse the source without checking out a copy: