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This is such an awesome video. I really hope there's a follow-up at some point, because as someone with a decent amount of experience with computer graphics, bu

This is such an awesome video. I really hope there's a follow-up at some point, because as someone with a decent amount of experience with computer graphics, but only a fairly general understanding of sampling theorem, the stuff that would naturally come next is what most interests me, namely: - What happens when you start combining waves together to create more complex signals? This is pretty important, since any real instrument produces hundreds of harmonics with complex attack and decay properties, so decomposition of the fundamental frequencies won't be as accurate as with toy examples. - Following on from that: the effects of aliasing. It definitely exists and I have a very good understanding of aliasing with respect to computer graphics, but what effect does aliasing really have on an audio signal? In CG it's something that's talked about all the time and there are tons of papers about it, but it seems (from the outside at least) that audio guys only ever talk about aliasing in very hand-wavy terms. - Although we can't hear sounds (much) above 20KHz, we can detect artefacts such as "beats" produced when harmonics are slightly mismatched. Is is possible to show that such information either isn't lost, or that what is lost is either below the noise floor or outside the audible frequency range? This particular one is a fairly common complaint made by audiophiles about 44KHz/Nyquist, so it would be nice to see it addressed head-on. FWIW, I'm a bit of an audiophile myself, but not one of those people who thinks they can hear a difference between 192kbps MP3 and uncompressed, let alone 16/44 vs 24/192. Generally the noise floor in the original recording is too high to tell the difference, even if you could hear a difference in artificially constructed pathological cases. But I am interested in really understanding what information is lost, what isn't, and why that may or may not make any difference. In other words, what are those pathological cases? This video is a really good start and clears up a lot, but in some ways it only scratches the surface.

rspeer 12y ago
I can point you to a follow-up on one of those: An example on Wikipedia [1] lets you hear the effect of aliasing on a sawtooth wave. When audio is aliased, the high harmonics will wrap around the Nyquist frequency and come back as audible, inharmonic tones. Aliased audio really sounds quite bad, in a way that would make any listener say "oh god make that horrible noise stop". This is probably why it's easier to recognize the effects of aliasing in video than in audio -- you're not hearing much aliased audio. Aliased video is fairly common and seen as acceptable in some contexts, while anyone who produces aliased audio is going to fix their mistake before inflicting it on listeners. [1] https://en.wikipedia.org/wiki/Aliasing#Online_audio_example https://en.wikipedia.org/wiki/Aliasing#Online_audio_example
- robert_tweed 12y ago
  Presumably though the effects of aliasing will also also produce distortions at a smaller scale, which might not be as obvious. That would result in a loss of information, but would not be quite so easy to spot and filter out. The pathological case is really just a good illustration of what happens and why, whereas what's trickier is understanding the true impact it has on real-world data. For instance, you won't notice Moiré patterns in photos very often, but it's easy to construct a test image that demonstrates the problem. The underlying issue of sampling error hasn't gone away in the photo: it just gets lost in the background noise or among smooth-shaded surfaces without hard edges, which makes it harder to see. But maybe now and then you'll have a sharp edge in a real-world image where the effect is noticeable. Depending on how close it is to the pathological case, it might not jump out and make itself immediately obvious. I mention Moiré patterns because these are (I believe) precisely the same effect as the given audio aliasing example. Moiré patterns aren't a form of sampling error per se, but they can be caused by uniform sampling (which is how digital quantisation typically works for both audio and images).
  - aidenn0 12y ago
    Moiré patterns are the same thing, but if you put your video signal through an analog low-pass filter before sampling, then you wouldn't get Moiré patterns. Audio is typically low-passed before sampling for this reason.
  - gaze 12y ago
    "Presumably though the effects of aliasing will also also produce distortions at a smaller scale, which might not be as obvious" Why? What do you mean? What scale? What do you mean by distortion? The only only only thing sampling does is move frequencies beyond the nyquest frequency into the sampled signal by taking the signal at f, and sending it to f mod f_n where f_n is the nyquist frequency.
- keenerd 12y ago
  Aliasing on a triangle wave can sound quite beautiful: http://kmkeen.com/awk-music/ http://kmkeen.com/awk-music/
fnordfnordfnord 12y ago
There is an Episode 1. http://xiph.org/video/ http://xiph.org/video/ See also http://xiph.org/~xiphmont/demo/neil-young.html http://xiph.org/~xiphmont/demo/neil-young.html which was posted here a week ago and received 441 comments. https://news.ycombinator.com/item?id=8689231 https://news.ycombinator.com/item?id=8689231
jeffwass 12y ago
Not entirely related to your question on audio aliasing, but many moons ago I was an EE on a team using electro-optics to design a much better Analog-to Digital converter than you could buy commercially. The signals we sampled were in hundreds of MHz but the concepts are exactly the same as in audio frequencies. What we effectively did was use a pulsed fiber laser, which produces _very_ short pulse widths of light. Connecting this to an electro-optical modulator, we can effectively capture a very fast snapshot of the electrical signal. Use some analog circuitry to hold that level constant for the slower analog-to-digital conversion, and we had an AD converter that while sampling relatively slowly, see's only a tiny portion of the signal for it's actually sample. What was amazing about this was that the laser pulses were so short, and the 'timing jitter' of the pulses so low that we could cleanly see the expected aliased sampling frequency when saming a sine wave signal at large multiples of our sample frequency (iirc up to the 40's). Eg, inputting a sine wave around 10 GHZ and we'd see only a single sine wave in the sampled signal at it's row sampled frequency with minimal artifacts. So if you are wondering how aliasing affects audio samping, draw a sine wave and see what happens if you sample it at a much lower frequency. You'll get a lower frequency sine wave. Without filtering the input signal, it's impossible to tell if this is a real low-frequency sine wave or something above Nyquist that has been down sampled. N.B. If you're wondering why our AD converter was better than a commercial one, we also added optical demultiexing to get an extra order of magnitude improvement in sample rate. This is possible using our electro optical setup due to the laser's low jitter and the ability to hold the signal steady, a purely electronic version will have difficulty with the timing to see much actual improvement in SNR.
dmicah 12y ago
With respect to aliasing, an analog to digital converter contains an analog low-pass filter which reduces the amplitude of the signal content above the Nyquist frequency prior to digitization. This effectively eliminates alsiasing in the digitized signal. In terms of more complex signals, or frequency "beats", the same principles apply. If two high frequency (above-20kHz) signals interact in such a way to create an audible (below-20kHz) beating, the beating will be accurately recorded by the ADC.
gaze 12y ago
> What happens when you start combining waves together to create more complex signals Aliasing is a linear effect. If L is an operator that aliases at some frequency, L(Aa(t) + Bb(t))=AL(a(t))+BL(b(t)). Aliasing is in fact multiplicative! Any time you wish to understand aliasing, you really want write the signal multiplied by a dirac comb (a train of equally spaced infinitely narrow pulses). If you wish to understand nonfinite sampling widths, it's effectively convolution with the shape of sampling pulse... which looks like low-pass filtering. Ya dig? This is the frequency on your scope that's stupidly above the sampling rate. Why's my scope have a bazillion ghz bandwidth when it only samples at 4 samples oer week? It's because it's real good at getting you that signal, aliased down into the sampled signal. > what effect does aliasing really have on an audio signal PRECISELY the same effect it does on video, except in one dimension, and you haven't understood aliasing until you've understood it in one dimension. > Is is possible to show that such information either isn't lost It's lost. Completely... if done properly. First you need to understand Whittaker-shannon interpolation. The result of which is that a CONTINUOUS BAND LIMITED signal is PERFECTLY represented by a train of discrete samples. This is WEIRD. Really weird. Prove it to yourself. It's VERY important in sampling. Say you then perfectly sample a signal with harmonic content ALL over the place. EVERYTHING gets mirrored down into your view, if the dirac comb has infinitely narrow pulses. So at the least you lose frequency information! You only know the original frequencies modulo the nyquist frequency! If you sample CORRECTLY, you band pass filter your incoming signal so there's NOTHING above the nyquist frequency, then you have a perfect digital representation of your original signal. If you wanna understand signal processing, you really can't use your intuition. Practice understanding things in the frequency domain and time domain, and generally there's only one proper basis to understand a phenomenon. Filtering is best understood in the frequency domain, for instance. If you try to understand filtering in the time domain, you might begin to think that it bounds the speed at which the signal can vary but that's not a useful intuition. Aliasing is best understood in the frequency domain, since it moves stuff down in frequency mod the nyquist frequency. If you try to build an intuition in the time domain, you'll think that there's some signal lost when you multiply by zero between pulses... and you'll guess at what it was and think "distortion" or whatever but what's lost is just WHERE each frequency is... so be very careful about what basis you choose!
ArkyBeagle 12y ago
Bit depth and sample rate are completely separable. If you are recording something important, it may well be worth the higher SR just in case some technology comes along that can exploit it. But the 20-20kHz bandlimit is a pretty well-entrenched thing. People are working in it. It's tough sledding. So the rule of thumb is that each bit gives -6dB more noise floor. This gives us a rough digital noise floor of -96dB for 16 bit or -144dB for 24 bit. But you can add dither to randomize the digital noise, and turn it into "white noise" or the like. Aliasing is taken care of by the "reconstruction filter" ( named for the "reconstruction" part of the Shannon-Nyquist theorem). This has been a solved problem for some time now. There's a super-high speed digital filter in the DAC phase that can be measured to eliminate all aliasing products. Implementation quality varies, but even cheap stuff now does a good job of it. Yes; it's unusual to have a room for recording that gets to -70dB much less -96dB. -40 to -50 is much more typical for, say acoustic guitar. Noise doesn't add like you'd think; the dominant noise wins, basically. Combined waveforms make for another waveform. They add brilliantly and this isn't lossy at all. There are pathological cases, but they're not likely found in nature. Transducers - headphones, speakers, microphones - all impart much more error than the electronics in the middle.

rspeer 12y ago

I can point you to a follow-up on one of those: An example on Wikipedia [1] lets you hear the effect of aliasing on a sawtooth wave. When audio is aliased, the high harmonics will wrap around the Nyquist frequency and come back as audible, inharmonic tones. Aliased audio really sounds quite bad, in a way that would make any listener say "oh god make that horrible noise stop". This is probably why it's easier to recognize the effects of aliasing in video than in audio -- you're not hearing much aliased audio. Aliased video is fairly common and seen as acceptable in some contexts, while anyone who produces aliased audio is going to fix their mistake before inflicting it on listeners. [1] https://en.wikipedia.org/wiki/Aliasing#Online_audio_example https://en.wikipedia.org/wiki/Aliasing#Online_audio_example

robert_tweed 12y ago
Presumably though the effects of aliasing will also also produce distortions at a smaller scale, which might not be as obvious. That would result in a loss of information, but would not be quite so easy to spot and filter out. The pathological case is really just a good illustration of what happens and why, whereas what's trickier is understanding the true impact it has on real-world data. For instance, you won't notice Moiré patterns in photos very often, but it's easy to construct a test image that demonstrates the problem. The underlying issue of sampling error hasn't gone away in the photo: it just gets lost in the background noise or among smooth-shaded surfaces without hard edges, which makes it harder to see. But maybe now and then you'll have a sharp edge in a real-world image where the effect is noticeable. Depending on how close it is to the pathological case, it might not jump out and make itself immediately obvious. I mention Moiré patterns because these are (I believe) precisely the same effect as the given audio aliasing example. Moiré patterns aren't a form of sampling error per se, but they can be caused by uniform sampling (which is how digital quantisation typically works for both audio and images).
- aidenn0 12y ago
  Moiré patterns are the same thing, but if you put your video signal through an analog low-pass filter before sampling, then you wouldn't get Moiré patterns. Audio is typically low-passed before sampling for this reason.
- gaze 12y ago
  "Presumably though the effects of aliasing will also also produce distortions at a smaller scale, which might not be as obvious" Why? What do you mean? What scale? What do you mean by distortion? The only only only thing sampling does is move frequencies beyond the nyquest frequency into the sampled signal by taking the signal at f, and sending it to f mod f_n where f_n is the nyquist frequency.
keenerd 12y ago
Aliasing on a triangle wave can sound quite beautiful: http://kmkeen.com/awk-music/ http://kmkeen.com/awk-music/

fnordfnordfnord 12y ago

There is an Episode 1. http://xiph.org/video/ http://xiph.org/video/ See also http://xiph.org/~xiphmont/demo/neil-young.html http://xiph.org/~xiphmont/demo/neil-young.html which was posted here a week ago and received 441 comments. https://news.ycombinator.com/item?id=8689231 https://news.ycombinator.com/item?id=8689231

jeffwass 12y ago

Not entirely related to your question on audio aliasing, but many moons ago I was an EE on a team using electro-optics to design a much better Analog-to Digital converter than you could buy commercially. The signals we sampled were in hundreds of MHz but the concepts are exactly the same as in audio frequencies. What we effectively did was use a pulsed fiber laser, which produces _very_ short pulse widths of light. Connecting this to an electro-optical modulator, we can effectively capture a very fast snapshot of the electrical signal. Use some analog circuitry to hold that level constant for the slower analog-to-digital conversion, and we had an AD converter that while sampling relatively slowly, see's only a tiny portion of the signal for it's actually sample. What was amazing about this was that the laser pulses were so short, and the 'timing jitter' of the pulses so low that we could cleanly see the expected aliased sampling frequency when saming a sine wave signal at large multiples of our sample frequency (iirc up to the 40's). Eg, inputting a sine wave around 10 GHZ and we'd see only a single sine wave in the sampled signal at it's row sampled frequency with minimal artifacts. So if you are wondering how aliasing affects audio samping, draw a sine wave and see what happens if you sample it at a much lower frequency. You'll get a lower frequency sine wave. Without filtering the input signal, it's impossible to tell if this is a real low-frequency sine wave or something above Nyquist that has been down sampled. N.B. If you're wondering why our AD converter was better than a commercial one, we also added optical demultiexing to get an extra order of magnitude improvement in sample rate. This is possible using our electro optical setup due to the laser's low jitter and the ability to hold the signal steady, a purely electronic version will have difficulty with the timing to see much actual improvement in SNR.

dmicah 12y ago

With respect to aliasing, an analog to digital converter contains an analog low-pass filter which reduces the amplitude of the signal content above the Nyquist frequency prior to digitization. This effectively eliminates alsiasing in the digitized signal. In terms of more complex signals, or frequency "beats", the same principles apply. If two high frequency (above-20kHz) signals interact in such a way to create an audible (below-20kHz) beating, the beating will be accurately recorded by the ADC.

gaze 12y ago

> What happens when you start combining waves together to create more complex signals Aliasing is a linear effect. If L is an operator that aliases at some frequency, L(Aa(t) + Bb(t))=AL(a(t))+BL(b(t)). Aliasing is in fact multiplicative! Any time you wish to understand aliasing, you really want write the signal multiplied by a dirac comb (a train of equally spaced infinitely narrow pulses). If you wish to understand nonfinite sampling widths, it's effectively convolution with the shape of sampling pulse... which looks like low-pass filtering. Ya dig? This is the frequency on your scope that's stupidly above the sampling rate. Why's my scope have a bazillion ghz bandwidth when it only samples at 4 samples oer week? It's because it's real good at getting you that signal, aliased down into the sampled signal. > what effect does aliasing really have on an audio signal PRECISELY the same effect it does on video, except in one dimension, and you haven't understood aliasing until you've understood it in one dimension. > Is is possible to show that such information either isn't lost It's lost. Completely... if done properly. First you need to understand Whittaker-shannon interpolation. The result of which is that a CONTINUOUS BAND LIMITED signal is PERFECTLY represented by a train of discrete samples. This is WEIRD. Really weird. Prove it to yourself. It's VERY important in sampling. Say you then perfectly sample a signal with harmonic content ALL over the place. EVERYTHING gets mirrored down into your view, if the dirac comb has infinitely narrow pulses. So at the least you lose frequency information! You only know the original frequencies modulo the nyquist frequency! If you sample CORRECTLY, you band pass filter your incoming signal so there's NOTHING above the nyquist frequency, then you have a perfect digital representation of your original signal. If you wanna understand signal processing, you really can't use your intuition. Practice understanding things in the frequency domain and time domain, and generally there's only one proper basis to understand a phenomenon. Filtering is best understood in the frequency domain, for instance. If you try to understand filtering in the time domain, you might begin to think that it bounds the speed at which the signal can vary but that's not a useful intuition. Aliasing is best understood in the frequency domain, since it moves stuff down in frequency mod the nyquist frequency. If you try to build an intuition in the time domain, you'll think that there's some signal lost when you multiply by zero between pulses... and you'll guess at what it was and think "distortion" or whatever but what's lost is just WHERE each frequency is... so be very careful about what basis you choose!

ArkyBeagle 12y ago

Bit depth and sample rate are completely separable. If you are recording something important, it may well be worth the higher SR just in case some technology comes along that can exploit it. But the 20-20kHz bandlimit is a pretty well-entrenched thing. People are working in it. It's tough sledding. So the rule of thumb is that each bit gives -6dB more noise floor. This gives us a rough digital noise floor of -96dB for 16 bit or -144dB for 24 bit. But you can add dither to randomize the digital noise, and turn it into "white noise" or the like. Aliasing is taken care of by the "reconstruction filter" ( named for the "reconstruction" part of the Shannon-Nyquist theorem). This has been a solved problem for some time now. There's a super-high speed digital filter in the DAC phase that can be measured to eliminate all aliasing products. Implementation quality varies, but even cheap stuff now does a good job of it. Yes; it's unusual to have a room for recording that gets to -70dB much less -96dB. -40 to -50 is much more typical for, say acoustic guitar. Noise doesn't add like you'd think; the dominant noise wins, basically. Combined waveforms make for another waveform. They add brilliantly and this isn't lossy at all. There are pathological cases, but they're not likely found in nature. Transducers - headphones, speakers, microphones - all impart much more error than the electronics in the middle.