Band limiting means that there's a maximum frequency. You can think of it as a limit on how fast the signal can change. There's only one possible way to draw a curve that goes through all the points without changing too quickly.

It's something like graphs of polynomials. There's only one possible line (first-degree polynomial) going through any two points. Three points completely determine a second-degree polynomial (parabola), etc. There are an infinite number of parabolas you can draw through two points, just as there are an infinite number of squiggles you could draw through the digital sample points. But you can't create a different solution without adding higher-order terms to the polynomial/higher frequencies to the signal. The number of points you have is sufficient to completely specify the curve.

Though there's only one theoretical solution, the converter can still "create it incorrectly".

This was incredibly helpful for my understanding. This is the one corner of this story that was always fuzzy for me. In the video he seemed to be making this point when he drew silly stuff all over the screen but it didn't quite click. Thank you.

Nyquist's theorem shows that you can reproduce a sine-wave, provided that you have strictly greater than two samples per cycle (and implicitly assume that you have a sine wave).

Any periodic signal (which, for audio, which is AC-coupled, is always) can be represented exactly by it's Fourier transform. The FT is simply a different representation of the same function, namely a sum of sine-waves. In general the FT of a signal has many terms (i.e., many spectral components or harmonics). However, in a band-limited system, only certain harmonics are allowed to pass. If your audio system claims to have, say, 20Hz to 20kHz bandwidth, then any Fourier component outside that range will be greatly attenuated (if the attenuation puts that component below the noise floor, we could say that the component has been completely eliminated). It is the act of attenuating those out-of-band components that causes the square wave to go squiggly.

This means that for any arbitrary signal which has been band-limited to half your sampling frequency, you can exactly reproduce the band-limited copy.

The sampling system usually (for consumer audio, always) has an anti-aliasing filter between the signal source and the sampler (not having an AA-filter is what allowed Tektronix sampling oscilloscopes to show multi-GHz signals in the 1960's, by effectively heterodyning the signal frequency down to something usable). That means that the digitizer only ever sees the band-limited signal. The bandwidth of the AA-filter is chosen so that the maximum frequency passed to the sampler is half (or less) than the sampling rate.

This also might help: http://dsp.stackexchange.com/questions/7879/how-does-subpixe...

> Isn't there also an argument that frequencies above 19khz can be heard by some people so need to be accurately represented?

That argument has been put forward, yes. "Back in the day" Bell Labs performed actual experiments with human test subjects to produce "equal loudness" curves. The upshot is that the power of an audio source must increase as the fundamental tone rises (starting from about 4kHz, or so). Then you must consider the effect of loud sounds on the human ear. Hearing damage is cumulative and the rate of damage increases with sound power, so it's hard to put an exact upper bound on the upper power limit of human hearing. However, at some point in the 18kHz to 22kHz range, the equal loudness curve crosses the damage curve. At that point you have to turn up the power of the audio source so high that you damage your hearing listening to it: So there is a definite upper frequency limit to human hearing. It may be slightly higher for some people than others, but 22.05kHz (the Nyquist limit for CD-rate sampling) is almost certainly greater than many standard deviations above the mean.

It's difficult to admit for a group of people (engineers and scientists) who are always striving to make things better, but we have come to a point where audio reproduction is "good enough." For a modest amount of money, the sampling, storage and reproduction is for all intents and purposes "perfect." Your living room is never going to sound like a concert hall with a live orchestra, but it's not because you don't have enough bits or samples or bandwidth.

The band limiting here means removing frequencies above the Nyquist limit. This allows for a unique solution. In the case these higher frequencies aren't removed, you get aliasing. One solution is the real one, and another solution folds back into the lower frequencies causing distortion.

You are probably already familiar with aliasing in the visual domain. This is the effect that causes tires to look like they are rotating backwards once they reach a fast enough forward spin.

"Isn't there also an argument that frequencies above 19khz can be heard by some people so need to be accurately represented?"

See http://people.xiph.org/~xiphmont/demo/neil-young.html for some details on this.

"Would be interesting to see a complex analogue signal being fed through the digital pipeline and then inverted against the original to see the differences."

The square wave is fairly complex and included in the video. While he doesn't invert the signal, he does show it fed through multiple times and then compensated for the delay.

Certainly the converter can be bad or malfunctioning... but there is only one bandlimited signal that passes through the sampling points.

The ideal behavior is perfect and described by simple (if surprising) math and we can measure any converter against the ideal. It turns out that _very_ good converters— ones with variation from the ideal are at the level of the thermal noise in the electronics— are commonly and cheaply available.

There was another chapter of the video that got dropped due to length, the desire to get something shipped, and because the tone didn't quite fit with the rest which compared the performance of a couple of consumer grade DAC/ADCs and basically showed they were all very good. Hopefully it should make its way out as 'bonus material' sometime soon.

I agree that was a little unclear. I think what he's saying is that since humans can't hear above ~20 KHz, frequencies above that are lost. The 'wobbles' in the square wave are what happens when you take a square wave (which is a summation of infinite sine saves with frequencies an integer multiple of the fundamental) and drop the frequencies above the ~20 KHz cutoff.

In other words, it has nothing to do with any analog/digital conversion. It's just what happens if you ignore frequencies above 20 KHz, which we can't hear anyways. We can't hear any difference between a 'perfect' square wave and the bandwidth limited one.

I also don't completely understand about their being only one possible solution. It makes sense if you know that the signal is a single sine wave, but since in general the signal is a summation of an unknown number of sine waves, it's not clear. I guess this is something he didn't have time to get into more depth for the quick overview :)

> Isn't there also an argument that frequencies above 19khz

The rule of thumb I've heard is 20 KHz. I also wondered if this is an issue. It'd be interesting to know the distribution of how many people can perceive frequencies how much higher than that. I think though that the 20 KHz is already fairly far along the long tail, and most people's cutoffs are actually lower.

I'll give this a stab. Assume you're connecting the dots (samples) with a pencil. You can see that the high frequency filter would not let the pencil go up or down several times between 2 adjacent dots because any movement at that rate would be too high a frequency right? Now extend that thinking: Any movement not along a specific path (the one possible solution) would produce frequencies above the limit of the filter. Even a teeny tiny little bend in that line would result in a tiny few decibels of sound at a frequency above the filter and hence are not allowed. Make sense? Experts: is that accurate?

Well. the converter add always small amounts of noise into the signal that could accumulate to 1 less significant bit or more.

With some special signals, mostly those made by man, like radar or sonar you could add more error(as signal accumulates so much in selected frequencies with exact multiples that have a strong bias) but most natural audio signals don't have much trouble.

"Isn't there also an argument that frequencies above 19khz can be heard by some people so need to be accurately represented?"

No, most kids could hear above 19khz, most adults can't, no argument about that, it is something you could test with a speaker and an electronic workbench using your kids or family members.

Nyquist proved that if you have a SAMPLED signal, with the FT you could reconstruct EXACTLY the same sampled signal, provided that you don't add quantization errors and so, that are small, you could even reconstruct square signals(with signal in all frequencies).

In the case you see those differences you will hear those transformation errors if they were significant(they are not), or you will have to turn on the volume and will hear a "sssssshhhhhhhhhh" sound.

If I can venture yet another explanation for this. Consider the projection

    π:signals --> signals,

> Can the converter itself not create the signal incorrectly? >

Let x the signal and π(x) be the band limited version of x. The video is talking about the general properties of projections, namely:

  π(π(x)) = π(x).

/\~---__/--\

If you raise that wave by five octaves it looks like this:

^-_^

You can play around with waves in my app OriginalSynth (GPL):

http://sourceforge.net/projects/originalsynth/

It's not a real demo but there's a drawing of it at 9:45 or so. The "quantization noise" is the noise introduced by digitization. And of course the rest of that whole section is the same thing, albeit with a simple 8-bit signal.

It was easier to use a real example than try to draw something by hand that was close to correct :-)