OK - this may be due to lack of sleep or I'm just being incredibly stupid and missing the obvious - but I can't get my brain around where the noise comes from!
View attachment 364662
So this is an image from the famous Monty video - and basically everything he says makes perfect sense.
I appreciate the red line is the difference between the original and the quantized signal - but surely the DAC has no knowledge of the original signal?
In my head, the DAC is reading t1=4, t2=5, t3=6, t4=5, t5=2, t6=-1, ... and constructs the only continuous path to join the dots and creates the the yellow line.
So the analogue signal out the DAC is the yellow line. It doesn't know the green line at all - so where is the red line in the output!? In my head we just have 2 waveforms, the input and the output which are just very slightly different!
I think to think I'm pretty intelligent and have a pure maths degree, so I'm wondering why I'm finding this so hard to understand!!!
I think this is a really good question, and a fascinating one at that!
As I see it, you've asked three main questions and two of them have been pretty thoroughly answered by others' helpful and astute comments:
1. How do we know what the red line is - in other words, how can we accurately measure/map the quantization noise? The easiest way to think about it is that the quantization noise is not usually directly observed or heard. It's derived by subtracting the yellow line's values from the green line's values (or vice-versa - doesn't really matter for purposes of determining the average magnitude of the noise).
2. Where is the noise in the yellow line? isn't the yellow line just a signal rather than being the green signal plus noise? When we're talking about properly generated output using 16-bit or 24-bit formats, the answer is that for practical purposes Yes, the yellow line is just a signal and the noise is "nowhere" in the sense that we don't hear it. Of course it's possible to hear the noise: take a 16-bit source, put on headphones, and during the silence between songs, crank up the volume all the way - you might hear it (although you still might not if the source was produced with noise-shaping dither). But in the overwhelming majority of situations we don't hear quantization noise.
3. How do we ever know what the green line is exactly? Audio isn't like that photo where we can see the flaws. This, I think, is the one question to which you haven't necessarily gotten a complete answer.
One way of addressing this is to say, as has been noted already, that we can generate simple signals where the voltages can be measured fairly easily and we can show definitively that the voltage values cannot be perfectly encoded in a digital sampling system with finite bit depth. (This can also be proven mathematically, but this would be a practical example where it could also be directly measured.)
Another way of addressing this is to go back to Monty's video where you got your original screenshot. The beauty of digital sampling is that we don't need to know or "see" the green line - the original signal - because as Monty says, digital sampling can reproduce the original signal perfectly, except with quantization noise. He demonstrates this by reducing the bit depth to 8 bits (and if memory serves maybe even to 6 or 4, I don't recall at the moment) - and you can still hear the original signal, just with more noise.
So it doesn't matter what the green line is, as long as our yellow line contains the full range of audible frequencies that were in the green line, and as long as the yellow line doesn't have audible noise. With a 16-bit, 44.1kHz sample rate system with noise-shaping dither and a well-chosen reconstruction filter, that's the case. In other words, we don't need to care what the green line looks like - we only need to have a system that we know can create a yellow line with sufficient frequency response capability, sufficiently low noise floor, and good rejection of out of band frequencies with minimal aliasing.
