Difference between revisions of "Videos/Digital Show and Tell"
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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:
<center><div style="background-color:#;border-color:#;border-style:solid;width:80%;padding:0 1em 1em 1em;text-align:left;">
* make the pipe fifos for the applications to communicate (only needs to be done once)
* make the pipe fifos for the applications to communicate (only needs to be done once)
mkfifo pipe0 pipe1
mkfifo pipe0 pipe1
Revision as of 03:10, 26 February 2013
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!
Veritas ex machina
All of this equipment is vintage, but aside from its raw tonnage, the specs are still quite good.
At the moment, we have our signal generator set to output a nice 1 kHz sine wave at one Volt RMS. We see the sine wave on the oscilloscope, can verify that it is indeed 1 kHz at 1 Volt RMS, which is 2.8 Volts peak-to-peak, and that matches the measurement on the spectrum analyzer as well.
The analyzer also shows some low-level white noise and just a bit of harmonic distortion, with the highest peak about 70dB or so below the fundamental. Now, this doesn't matter at all in our demos, but I wanted to point it out now just in case you didn't notice it until later.
Now, we drop digital sampling in the middle.
Input to output, left to right.
“OK, it's go time. We begin by converting an analog signal to digital and then right back to analog again with no other steps.”
“The signal generator is set to produce a 1kHz sine wave just like before.”
“We can see our analog sine wave on our input-side oscilloscope.”
“We digitize our signal to 16 bit PCM at 44.1kHz, same as on a CD.”
“The spectrum of the digitized signal 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 a stairstep pattern, one step for each sample.”
“And when we look at the output signal that's been converted from digital back to analog, we see...”
“It's exactly like the original sine wave. No stairsteps.”
“OK, 1 kHz is still a fairly low frequency, maybe the stairsteps are just hard to see or they're being smoothed away. Fair enough. Let's choose a higher frequency, something close to Nyquist, say 15kHz.”
“Now the sine wave is represented by less than three samples per cycle, and... the digital waveform looks pretty awful. Well, looks can be deceiving. The analog output... is still a perfect sine wave, exactly like the original.”
“Let's keep going up.”
“16kHz.... 17kHz... 18kHz... 19kHz...”
“20kHz. Welcome to the upper limits of human hearing. The output waveform is still perfect. No jagged edges, no dropoff, no stairsteps.”
“So where'd the stairsteps go? Don't answer, it's a trick question. They were never there.”
“Drawing a digital waveform as a stairstep was wrong to begin with.”
“Why? 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's undefined, 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 high frequencies as it is for low.”
“Now, the interesting and not at all obvious bit is: there's only one bandlimited signal that passes exactly through each sample point. It's a unique solution. So if you sample a bandlimited signal and then convert it back, the original input is also the only possible output.”
“And before you say, "oh, I can draw a different signal that passes through those points", well, yes you can, but if it differs even minutely from the original, it 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 enough to convert a sampled signal to a true stairstep. Just extend each sample value forward until the next sample period. This is called a zero-order hold, and it's an important part of how some digital-to-analog converters work, especially the simplest ones.”
“So, anyone who looks up digital-to-analog converter or digital-to-analog conversion is probably going to see a diagram of a stairstep 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 who supposedly know better, like me, draw stairsteps even though they're technically wrong. It's a sort of like a one-dimensional version of fat bits in an image editor.”
“Pixels aren't squares either, they're samples of a 2-dimensional function space and so they're also, conceptually, infinitely small points. Practically, it's a real pain in the ass to see or manipulate infinitely 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.”
When we convert a digital signal back to analog, the result is also smooth regardless of the bit depth. 24 bits or 16 bits... or 8 bits... it doesn't matter.
So does that mean that the digital bit depth makes no difference at all? Of course not.
Channel 2 here is the same sine wave input, but we quantize with dither down to 8 bits.
If we look at the spectrum of the signal... aha! Our sine wave is still there unaffected, but the noise level of the 8-bit signal on the second channel is much higher!
And that's the difference the number of bits makes. That's it!
The number of bits determines how much noise and so the level of the noise floor.
What does this dithered quantization noise sound like? Let's listen to our 8-bit sine wave.
Those of you who have used analog recording equipment may have just thought to yourselves, "My goodness! That sounds like tape hiss!" Well, it doesn't just sound like tape hiss, it acts like it too, and if we use a gaussian dither then it's mathematically equivalent in every way. It is tape hiss.
Intuitively, that means that we can measure tape hiss and thus the noise floor of magnetic audio tape in bits instead of decibels, in order to put things in a digital perspective. Compact cassettes (for those of you who are old enough to remember them) could reach as deep as 9 bits in perfect conditions, though 5 to 6 bits was more typical, especially if it was a recording made on a tape deck. That's right... your mix tapes were only about 6 bits deep... if you were lucky!
The very best professional open reel tape used in studios could barely hit... any guesses? 13 bits with advanced noise reduction. And that's why seeing 'D D D' on a Compact Disc used to be such a big, high-end deal.
so we may get noise that's inconsistent, or causes distortion, or is
undesirable in some other way.
Dither is specially-constructed noise that substitutes for the noise produced by simple quantization. Dither doesn't drown out or mask quantization noise, it actually replaces it with noise characteristics of our choosing that aren't influenced by the input.
Let's watch what dither does. The signal generator has too much noise for this test so we'll produce a mathematically perfect sine wave with the ThinkPad and quantize it to 8 bits with dithering.
We see a nice sine wave on the waveform display and output scope and, once the analog spectrum analyzer catches up... a clean frequency peak with a uniform noise floor on both spectral displays just like before. Again, this is with dither.
Now I turn dithering off.
At 8 bits this effect is exaggerated. At 16 bits, even without dither, harmonic distortion is going to be so low as to be 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 the absolute level of distortion from undithered quantization stays approximately constant regardless of the input amplitude. But when the signal level drops below a half a bit, everything quantizes to zero.
In a sense, everything quantizing to zero is just 100% distortion! Dither eliminates this distortion too. We reenable dither and ... there's our signal back at 1/4 bit, with our nice flat noise floor.
Our hearing is most sensitive in the midrange from 2kHz to 4kHz, so that's where background noise is going to be the most obvious. We can shape dithering noise away from sensitive frequencies to where hearing is less sensitive, usually the highest frequencies.
Lastly, dithered quantization noise is higher power overall than undithered quantization noise even when it sounds quieter, and you can see that on a VU meter during passages of near-silence. But dither isn't only an on or off choice. We can reduce the dither's power to balance less noise against a bit of distortion to minimize the overall effect.
We'll also modulate the input signal like this to show how a varying input affects the quantization noise. At full dithering power, the noise is uniform, constant, and featureless just like we expect:
As we reduce the dither's power, the input increasingly affects the amplitude and the character of the quantization noise. Shaped dither behaves similarly, but noise shaping lends one more nice advantage. To make a long story short, it can use a somewhat lower dither power before the input has as much effect on the output.
Despite all the time I just spent on dither, we're talking about differences that start 100 decibels and more below full scale. Maybe if the CD had been 14 bits as originally designed, dither might be more important. Maybe. At 16 bits, really, it's mostly a wash. You can think of dither as an insurance policy that gives several extra decibels of dynamic range just in case. The simple fact is, though, no one ever ruined a great recording by not dithering the final master.
Bandlimitation and timing
The input scope confirms our 1kHz square wave. The output scope shows..
Exactly what it should. ... What is a square wave really?
Well, 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.
Then 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. Sum them all up, you get a square wave.
At first glance, that doesn't seem very useful either. You have to sum up an infinite number of harmonics to get the answer. Ah, but we don't have an infinite number of harmonics.
..and that's exactly what we see on the output scope.
The usual rule of thumb you'll hear is "the sharper the cutoff, the stronger the rippling", which is approximately true, but we have to be careful how we think about it. For example... what would you expect our quite sharp anti-aliasing filter to do if I run our signal through it a second time?
And that's important. People tend to think of the ripples as a kind of artifact that's added by anti-aliasing and anti-imaging filters, implying that the ripples get worse each time the signal passes through. We can see that in this case that didn't happen. So was it really the filter that added the ripples the first time through? No, not really. It's a subtle distinction, but Gibbs effect ripples aren't added by filters, they're just part of what a bandlimited signal is.
Even if we synthetically construct what looks like a perfect digital square wave,
it's still limited to the channel bandwidth. Remember, the stairstep representation is misleading.
What we really have here are instantaneous sample points,
But the original bandlimited signal, complete with ripples, was still there.
At this point, we can easily see why that's wrong.
It's represented perfectly and it's reconstructed perfectly.
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. And, 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.
Digging deeper or not, I am out of coffee, so, until next time, happy hacking!
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
This Video Was Produced Entirely With Free and Open Source Software:
All trademarks are the property of their respective owners.
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:
Spectrum and Waveform both expect an input stream on the command line, either as raw data or as a WAV file.
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.
Gtk-bounce consists of eleven pushbutton panels (numbered zero through ten) that can be selected by scrolling up and dwon with the arrow buttons on the right side. Each panel is intended for a specific demo or part of a demo.
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...