WEBVTT
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Hi, I'm Monty Montgomery from Red Hat and Xiph.Org.
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A few months ago, I wrote an article on digital audio and why 24bit/192kHz music downloads don't make sense.
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In the article, I mentioned--almost in passing--that a digital waveform is not a stairstep,
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and you certainly don't get a stairstep when you convert from digital back to analog.
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Of everything in the entire article, that was the number one thing people wrote about.
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In fact, more than half the mail I got was questions and comments
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about basic digital signal behavior.
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Since there's interest, let's take a little time to play with some simple digital signals.
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Pretend for a moment
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that we have no idea how digital signals really behave.
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In that case it doesn't make sense for us
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to use digital test equipment either.
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Fortunately for this exercise, there's still
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plenty of working analog lab equipment out there.
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First up, we need a signal generator
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to provide us with analog input signals--
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in this case, an HP3325 from 1978.
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It's still a pretty good generator,
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so if you don't mind the size,
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the weight,
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the power consumption,
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and the noisy fan,
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you can find them on eBay.
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Occasionally for only slightly more than you'll pay for shipping.
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Next, we'll observe our analog waveforms on analog oscilloscopes,
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like this Tektronix 2246 from the mid-90s,
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one of the last and very best analog scopes ever made.
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Every home lab should have one.
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And finally inspect the frequency spectrum of our signals
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using an analog spectrum analyzer.
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This HP3585 from the same product line as the signal generator.
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Like the other equipment here it has a rudimentary
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and hilariously large microcontroller,
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but the signal path from input to what you see on the screen
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is completely analog.
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All of this equipment is vintage,
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but aside from its raw tonnage,
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the specs are still quite good.
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At the moment, we have our signal generator
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set to output a nice 1kHz sine wave at one volt RMS,
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we see the sine wave on the oscilloscope,
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can verify that it is indeed 1kHz at one volt RMS,
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which is 2.8V peak-to-peak,
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and that matches the measurement on the spectrum analyzer as well.
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The analyzer also shows some low-level white noise
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and just a bit of harmonic distortion,
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with the highest peak about 70dB or so below the fundamental.
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Now, this doesn't matter at all in our demos,
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but I wanted to point it out now
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just in case you didn't notice it until later.
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Now, we drop digital sampling in the middle.
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For the conversion, we'll use a boring,
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consumer-grade, eMagic USB1 audio device.
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It's also more than ten years old at this point,
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and it's getting obsolete.
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A recent converter can easily have an order of magnitude better specs.
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Flatness, linearity, jitter, noise behavior, everything...
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you may not have noticed.
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Just because we can measure an improvement
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doesn't mean we can hear it,
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and even these old consumer boxes were already
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at the edge of ideal transparency.
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The eMagic connects to my ThinkPad,
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which displays a digital waveform and spectrum for comparison,
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then the ThinkPad sends the digital signal right back out
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to the eMagic for re-conversion to analog
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and observation on the output scopes.
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Input to output, left to right.
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OK, it's go time.
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We begin by converting an analog signal to digital
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and then right back to analog again with no other steps.
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The signal generator is set to produce
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a 1kHz sine wave just like before.
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We can see our analog sine wave on our input-side oscilloscope.
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We digitize our signal to 16 bit PCM at 44.1kHz,
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same as on a CD.
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The spectrum of the digitized signal matches what we saw earlier. and...
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what we see now on the analog spectrum analyzer,
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aside from its high-impedance input being just a smidge noisier.
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For now
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the waveform display shows our digitized sine wave
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as a stairstep pattern, one step for each sample.
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And when we look at the output signal
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that's been converted from digital back to analog, we see...
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It's exactly like the original sine wave.
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No stairsteps.
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OK, 1kHz is still a fairly low frequency,
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maybe the stairsteps are just
hard to see or they're being smoothed away.
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Fair enough. Let's choose
a higher frequency, something close to Nyquist, say 15kHz.
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Now the sine wave is represented by less than three samples per cycle, and...
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the digital waveform looks pretty awful.
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Well, looks can be deceiving. The analog output...
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is still a perfect sine wave, exactly like the original.
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Let's keep going up.
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16kHz....
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17kHz...
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18kHz...
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19kHz...
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20kHz.
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Welcome to the upper limits of human hearing.
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The output waveform is still perfect.
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No jagged edges, no dropoff, no stairsteps.
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So where'd the stairsteps go?
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Don't answer, it's a trick question.
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They were never there.
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Drawing a digital waveform as a stairstep
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was wrong to begin with.
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Why?
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A stairstep is a continuous-time function.
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It's jagged, and it's piecewise,
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but it has a defined value at every point in time.
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A sampled signal is entirely different.
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It's discrete-time;
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it's only got a value right at each instantaneous sample point
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and it's undefined, there is no value at all, everywhere between.
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A discrete-time signal is properly drawn as a lollipop graph.
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The continuous, analog counterpart of a digital signal
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passes smoothly through each sample point,
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and that's just as true for high frequencies as it is for low.
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Now, the interesting and not at all obvious bit is:
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there's only one bandlimited signal that passes
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exactly through each sample point.
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It's a unique solution.
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So if you sample a bandlimited signal
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and then convert it back,
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the original input is also the only possible output.
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And before you say,
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"Oh, I can draw a different signal that passes through those points."
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Well, yes you can, but...
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if it differs even minutely from the original,
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it contains frequency content at or beyond Nyquist,
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breaks the bandlimiting requirement
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and isn't a valid solution.
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So how did everyone get confused
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and start thinking of digital signals as stairsteps?
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I can think of two good reasons.
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First: It's easy enough to convert a sampled signal
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to a true stairstep.
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Just extend each sample value forward until the next sample period.
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This is called a zero-order hold,
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and it's an important part of how some digital-to-analog converters work,
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especially the simplest ones.
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So, anyone who looks up digital-to-analog conversion
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is probably going to see a diagram of a stairstep waveform somewhere,
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but that's not a finished conversion,
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and it's not the signal that comes out.
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Second,
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and this is probably the more likely reason,
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engineers who supposedly know better,
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like me,
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draw stairsteps even though they're technically wrong.
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It's a sort of like a one-dimensional version of
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fat bits in an image editor.
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Pixels aren't squares either,
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they're samples of a 2-dimensional function space and so they're also,
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conceptually, infinitely small points.
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Practically, it's a real pain in the ass to see
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or manipulate infinitely small anything.
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So big squares it is.
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Digital stairstep drawings are exactly the same thing.
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It's just a convenient drawing.
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The stairsteps aren't really there.
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When we convert a digital signal back to analog,
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the result is also smooth regardless of the bit depth.
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24 bits or 16 bits...
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or 8 bits...
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it doesn't matter.
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So does that mean that the digital bit depth
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makes no difference at all?
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Of course not.
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Channel 2 here is the same sine wave input,
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but we quantize with dither down to eight bits.
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On the scope, we still see a nice
smooth sine wave on channel 2.
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Look very close, and you'll also see a
bit more noise.
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That's a clue.
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If we look at the spectrum of the signal...
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aha!
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Our sine wave is still there unaffected,
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but the noise level of the eight-bit signal
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on the second channel is much higher!
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And that's the difference the number of bits makes.
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That's it!
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When we digitize a signal, first we sample it.
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The sampling step is perfect; it loses nothing.
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But then we quantize it,
and quantization adds noise.
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The number of bits determines how much noise
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and so the level of the
noise floor.
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What does this dithered quantization noise sound like?
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Let's listen to our eight-bit sine wave.
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That may have been hard to hear anything but the tone.
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Let's listen to just the noise after we notch out the sine wave
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and then bring the gain up a bit because the noise is quiet.
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Those of you who have used analog recording equipment
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may have just thought to yourselves,
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"My goodness! That sounds like tape hiss!"
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Well, it doesn't just sound like tape hiss,
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it acts like it too,
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and if we use a gaussian dither
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then it's mathematically equivalent in every way.
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It is tape hiss.
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Intuitively, that means that we can measure tape hiss
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and thus the noise floor of magnetic audio tape
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in bits instead of decibels,
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in order to put things in a digital perspective.
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Compact cassettes...
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for those of you who are old enough to remember them,
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could reach as
deep as nine bits in perfect conditions,
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though five to six bits was more typical,
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especially if it was a recording made on a tape deck.
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That's right... your mix tapes were only about six bits
deep... if you were lucky!
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The very best professional open reel tape
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used in studios could barely hit...
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any guesses?...
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13 bits
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with advanced noise reduction.
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And that's why seeing 'DDD' on a Compact Disc
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used to be such a big, high-end deal.
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I keep saying that I'm quantizing with dither,
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so what is dither exactly?
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More importantly, what does it do?
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The simple way to quantize a signal is to choose
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the digital amplitude value closest
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to the original analog amplitude.
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Obvious, right?
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Unfortunately, the exact noise you get
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from this simple quantization scheme
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depends somewhat on the input signal,
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so we may get noise that's inconsistent,
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or causes distortion,
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or is undesirable in some other way.
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Dither is specially-constructed noise that
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substitutes for the noise produced by simple quantization.
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Dither doesn't drown out or mask quantization noise,
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it actually replaces it
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with noise characteristics of our choosing
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that aren't influenced by the input.
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Let's watch what dither does.
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The signal generator has too much noise for this test
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so we'll produce a mathematically
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perfect sine wave with the ThinkPad
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and quantize it to eight bits with dithering.
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We see a nice sine wave on the waveform display
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and output scope
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and...
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once the analog spectrum analyzer catches up...
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a clean frequency peak with a uniform noise floor
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on both spectral displays
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just like before
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Again, this is with dither.
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Now I turn dithering off.
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The quantization noise, that dither had spread out
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into a nice, flat noise floor,
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piles up into harmonic distortion peaks.
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The noise floor is lower, but the level of distortion becomes nonzero,
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and the distortion peaks sit higher than the dithering noise did.
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At eight bits this effect is exaggerated.
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At sixteen bits,
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even without dither,
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harmonic distortion is going to be so low
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as to be completely inaudible.
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Still, we can use dither to eliminate it completely
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if we so choose.
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Turning the dither off again for a moment,
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you'll notice that the absolute level of distortion
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from undithered quantization stays approximately constant
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regardless of the input amplitude.
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But when the signal level drops below a half a bit,
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everything quantizes to zero.
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In a sense,
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everything quantizing to zero is just 100% distortion!
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Dither eliminates this distortion too.
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We reenable dither and...
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there's our signal back at 1/4 bit,
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with our nice flat noise floor.
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The noise floor doesn't have to be flat.
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Dither is noise of our choosing,
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so let's choose a noise as inoffensive
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and difficult to notice as possible.
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Our hearing is most sensitive in the midrange from 2kHz to 4kHz,
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so that's where background noise is going to be the most obvious.
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We can shape dithering noise away from sensitive frequencies
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to where hearing is less sensitive,
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usually the highest frequencies.
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16-bit dithering noise is normally much too quiet to hear at all,
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but let's listen to our noise shaping example,
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again with the gain brought way up...
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Lastly, dithered quantization noise is higher power overall
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than undithered quantization noise even when it sounds quieter.
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You can see that on a VU meter during passages of near-silence.
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But dither isn't only an on or off choice.
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We can reduce the dither's power to balance less noise against
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a bit of distortion to minimize the overall effect.
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We'll also modulate the input signal like this:
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...to show how a varying input affects the quantization noise.
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At full dithering power, the noise is uniform, constant,
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and featureless just like we expect:
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As we reduce the dither's power,
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the input increasingly affects the amplitude and the character
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of the quantization noise:
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Shaped dither behaves similarly,
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but noise shaping lends one more nice advantage.
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To make a long story short, it can use
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a somewhat lower dither power before the input
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has as much effect on the output.
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Despite all the time I just spent on dither,
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we're talking about differences
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that start 100 decibels below full scale.
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Maybe if the CD had been 14 bits as originally designed,
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dither might be more important.
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Maybe.
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At 16 bits, really, it's mostly a wash.
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You can think of dither as an insurance policy
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that gives several extra decibels of dynamic range
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just in case.
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The simple fact is, though,
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no one ever ruined a great recording
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by not dithering the final master.
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We've been using sine waves.
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They're the obvious choice when what we want to see
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is a system's behavior at a given isolated frequency.
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00:17:32.212 --> 00:17:34.217
Now let's look at something a bit more complex.
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What should we expect to happen
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when I change the input to a square wave...
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The input scope confirms our 1kHz square wave.
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The output scope shows..
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Exactly what it should.
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What is a square wave really?
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Well, we can say it's a waveform that's some positive value
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for half a cycle and then transitions instantaneously
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to a negative value for the other half.
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But that doesn't really tell us anything useful
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about how this input
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becomes this output.
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Then we remember that any waveform
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is also the sum of discrete frequencies,
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and a square wave is a particularly simple sum
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a fundamental and
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an infinite series of odd harmonics.
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Sum them all up, you get a square wave.
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At first glance,
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that doesn't seem very useful either.
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You have to sum up an infinite number of harmonics
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to get the answer.
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00:18:33.108 --> 00:18:35.977
Ah, but we don't have an infinite number of harmonics.
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We're using a quite sharp anti-aliasing filter
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that cuts off right above 20kHz,
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so our signal is band-limited,
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which means we get this:
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..and that's exactly what we see on the output scope.
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The rippling you see around sharp edges in a bandlimited signal
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is called the Gibbs effect.
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00:19:00.926 --> 00:19:04.137
It happens whenever you slice off part of the frequency domain
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in the middle of nonzero energy.
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00:19:07.006 --> 00:19:09.854
The usual rule of thumb you'll hear is the sharper the cutoff,
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the stronger the rippling,
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which is approximately true,
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but we have to be careful how we think about it.
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For example...
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what would you expect our quite sharp anti-aliasing filter
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00:19:19.529 --> 00:19:23.181
to do if I run our signal through it a second time?
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00:19:34.136 --> 00:19:37.588
Aside from adding a few fractional cycles of delay,
362
00:19:37.588 --> 00:19:39.348
the answer is...
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00:19:39.348 --> 00:19:40.857
nothing at all.
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The signal is already bandlimited.
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00:19:43.656 --> 00:19:46.590
Bandlimiting it again doesn't do anything.
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00:19:46.590 --> 00:19:50.686
A second pass can't remove frequencies that we already removed.
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And that's important.
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People tend to think of the ripples as a kind of artifact
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00:19:56.233 --> 00:19:59.945
that's added by anti-aliasing and anti-imaging filters,
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00:19:59.945 --> 00:20:01.737
implying that the ripples get worse
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00:20:01.737 --> 00:20:03.913
each time the signal passes through.
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00:20:03.913 --> 00:20:05.950
We can see that in this case that didn't happen.
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00:20:05.950 --> 00:20:09.492
So was it really the filter that added the ripples the first time through?
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00:20:09.492 --> 00:20:10.537
No, not really.
375
00:20:10.537 --> 00:20:12.126
It's a subtle distinction,
376
00:20:12.126 --> 00:20:15.252
but Gibbs effect ripples aren't added by filters,
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00:20:15.252 --> 00:20:18.836
they're just part of what a bandlimited signal is.
378
00:20:18.836 --> 00:20:20.798
Even if we synthetically construct
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00:20:20.798 --> 00:20:23.508
what looks like a perfect digital square wave,
380
00:20:23.508 --> 00:20:26.206
it's still limited to the channel bandwidth.
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00:20:26.206 --> 00:20:29.140
Remember the stairstep representation is misleading.
382
00:20:29.140 --> 00:20:32.222
What we really have here are instantaneous sample points,
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00:20:32.222 --> 00:20:36.148
and only one bandlimited signal fits those points.
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00:20:36.148 --> 00:20:39.614
All we did when we drew our apparently perfect square wave
385
00:20:39.614 --> 00:20:43.198
was line up the sample points just right so it appeared
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00:20:43.198 --> 00:20:47.785
that there were no ripples if we played connect-the-dots.
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00:20:47.785 --> 00:20:49.449
But the original bandlimited signal,
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00:20:49.449 --> 00:20:52.742
complete with ripples, was still there.
389
00:20:54.004 --> 00:20:56.542
And that leads us to one more important point.
390
00:20:56.542 --> 00:20:59.550
You've probably heard that the timing precision of a digital signal
391
00:20:59.550 --> 00:21:02.409
is limited by its sample rate; put another way,
392
00:21:02.409 --> 00:21:05.140
that digital signals can't represent anything
393
00:21:05.140 --> 00:21:08.041
that falls between the samples...
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00:21:08.041 --> 00:21:11.422
implying that impulses or fast attacks have to align
395
00:21:11.422 --> 00:21:14.473
exactly with a sample, or the timing gets mangled...
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00:21:14.473 --> 00:21:16.219
or they just disappear.
397
00:21:16.711 --> 00:21:20.820
At this point, we can easily see why that's wrong.
398
00:21:20.820 --> 00:21:23.742
Again, our input signals are bandlimited.
399
00:21:23.742 --> 00:21:26.036
And digital signals are samples,
400
00:21:26.036 --> 00:21:29.340
not stairsteps, not 'connect-the-dots'.
401
00:21:31.572 --> 00:21:34.592
We most certainly can, for example,
402
00:21:36.777 --> 00:21:39.337
put the rising edge of our bandlimited square wave
403
00:21:39.337 --> 00:21:42.004
anywhere we want between samples.
404
00:21:42.004 --> 00:21:44.354
It's represented perfectly
405
00:21:47.508 --> 00:21:50.218
and it's reconstructed perfectly.
406
00:22:04.620 --> 00:22:06.526
Just like in the previous episode,
407
00:22:06.526 --> 00:22:08.393
we've covered a broad range of topics,
408
00:22:08.393 --> 00:22:10.868
and yet barely scratched the surface of each one.
409
00:22:10.868 --> 00:22:13.620
If anything, my sins of omission are greater this time around...
410
00:22:13.620 --> 00:22:16.286
but this is a good stopping point.
411
00:22:16.286 --> 00:22:17.833
Or maybe, a good starting point.
412
00:22:17.833 --> 00:22:18.708
Dig deeper.
413
00:22:18.708 --> 00:22:19.710
Experiment.
414
00:22:19.710 --> 00:22:21.374
I chose my demos very carefully
415
00:22:21.374 --> 00:22:23.668
to be simple and give clear results.
416
00:22:23.668 --> 00:22:26.217
You can reproduce every one of them on your own if you like.
417
00:22:26.217 --> 00:22:28.766
But let's face it, sometimes we learn the most
418
00:22:28.766 --> 00:22:30.516
about a spiffy toy by breaking it open
419
00:22:30.516 --> 00:22:32.553
and studying all the pieces that fall out.
420
00:22:32.553 --> 00:22:35.230
That's OK, we're engineers.
421
00:22:35.230 --> 00:22:36.350
Play with the demo parameters,
422
00:22:36.350 --> 00:22:37.972
hack up the code,
423
00:22:37.972 --> 00:22:39.774
set up alternate experiments.
424
00:22:39.774 --> 00:22:40.692
The source code for everything,
425
00:22:40.692 --> 00:22:42.398
including the little pushbutton demo application,
426
00:22:42.398 --> 00:22:44.361
is up at Xiph.Org.
427
00:22:44.361 --> 00:22:45.940
In the course of experimentation,
428
00:22:45.940 --> 00:22:47.401
you're likely to run into something
429
00:22:47.401 --> 00:22:49.950
that you didn't expect and can't explain.
430
00:22:49.950 --> 00:22:51.198
Don't worry!
431
00:22:51.198 --> 00:22:54.537
My earlier snark aside, Wikipedia is fantastic for
432
00:22:54.537 --> 00:22:56.788
exactly this kind of casual research.
433
00:22:56.788 --> 00:22:59.956
If you're really serious about understanding signals,
434
00:22:59.956 --> 00:23:03.337
several universities have advanced materials online,
435
00:23:03.337 --> 00:23:07.380
such as the 6.003 and 6.007 Signals and Systems modules
436
00:23:07.380 --> 00:23:08.798
at MIT OpenCourseWare.
437
00:23:08.798 --> 00:23:11.593
And of course, there's always the community here at Xiph.Org.
438
00:23:12.792 --> 00:23:13.929
Digging deeper or not,
439
00:23:13.929 --> 00:23:14.974
I am out of coffee,
440
00:23:14.974 --> 00:23:16.436
so, until next time,
441
00:23:16.436 --> 00:23:19.316
happy hacking!