If you model the noise canceling as a superposition of the input signal with the generated signal, where the generated signal is specifically designed to account for the processing delay and hence only targets lower-frequency noise, then wouldn't 2 things be true:
1) Any residual from imperfect cancellation would only be at the lower frequencies (because the generated signal itself contains no high frequencies, and we a linear superposition). Of course, the original high frequency content from the input that was never cancelled will still be present but this isn't any worse.
2) Even in the worst case that we completely mispredict, we will only double the sound intensity which is a 3db increase. This seems relatively safe?
That said, while it seems physically safe it's possible that ANC still wreaks havoc with the brain's audio processing. Maybe the brain relies on the existence of the low frequency content as a sort of gain control mechanism or something (there are anecdotal reports by some users of increased tinnitus with long-term ANC use, but it's also possible that it only increased their awareness of it). I've also read that some people are sensitive to this lack of low-frequency noise since the brain interprets it as a pressure differential.
If you take a sinusoids, invert it, slightly offset it, and combine them. You get a smaller signal always, unless you’re delay is larger then 90°.
I would recommend playing with some audio signals in an online simulator and see what you get, you realise that your 2D intuition does not apply well to analogue 1D signals. The strict digital nature of image processing done on a computer creates the possibility of results not easily possible when working on analogue signal. After all you can one pixel on a screen at max brightness, and it’s neighbour completely of, but it’s impossible to recreate a similar hard edge with an audio signal because it would require the speaker to be capable of infinite speed and acceleration.
Not that any of this is relevant to the original problem.
> the cancellation is not perfect and what I am actually getting into the ear is the residual high frequency noise, which may in fact be quite dangerous.
From GP:
> I can see quite clearly that the high frequency noise happens when you subtract two images, one of which is shifted by half-pixel for instance. What you are left with is the edges (high frequency) of the image.
Show me the part where the word “amplitude” appears in any parent post.
Additionally as mentioned in parent posts, noise cancelling headphone run a low pass filter over the input to the ANC system, specifically because achieving good alignment between your ANC signal and original signal is basically impossible at wave lengths shorter as you can’t know the exactly which direction the original signal came from (direct perpendicular to the head, or at a close tangential angle), and thus can’t compensate for the offset needed to ensure the two signals arrive at eardrum at the right time.
However, I don't agree that ANC always uses a low-pass filter, and it seems from kalal's followup that they are also talking about the using the full original signal. So the two of us were not talking about introducing high frequencies but about somehow enhancing the high frequencies already present, and that's what the figures I gave above are for. So we've been talking across each other. I apologise for my part in that.
("Amplitude" was a simple technical term to replace woolly terms that were being used, just as you are the first in the parent-chain to say "low pass filter".)
> What you are left with is the edges (high frequency)
So you subtract a slightly phase shifted high frequency signal, you're left with a high frequency signal that may be amplified at the edges depending on your phase shift. Nothing surprising here?
The question is can you create a high frequency residual by subtracting a lowpass filtered (gaussian blur?) image? I don't think so. You're just left with whatever high frequencies you had but you aren't creating any new ones.
https://www.howtogeek.com/423960/why-do-noise-canceling-head...
There are lots of other sources of pain, fatigue and discomfort, though. Essentially, since the noise cancellation isn't perfect, you can hear some subset of the sound, varying across the frequency spectrum. This can cause a feeling of being underwater, of having blocked ears or simply of having to listen more closely, all of which might cause ongoing muscle tension of different muscles.
On the good news side, a lot of people who find discomfort with one brand of noise-cancelling can find other brands fine, so it might be worth trying some other brands. (For Bose, the main similar-quality competitor is Sony.)
Finally, in the simplistic model, there does not have to be a delay. Sound only travels at 340 metres per second, so for every 10mm distance between the microphone and the emitter, one can have around 34 microseconds of processing time.
I've got a sony wh-1000xm2, some inear Jabra and the Bose 700, which came with my Pixel 6 purchase the other day.
It really is something the potential buyer should check out themselves in a store before buying, because they're really variable in quality.
According to mkbhd the Sony one's are the best at it, but they have very strong white noise background at least with my older model.
I actually had the same issue with that pressure on my ears, though it slowly went away and haven't noticed any displeasure in a long time now.
But the noise cancellation is subtly audible. I'm not a sound engineer, but it feels a little bit like sound compression being too high.
However, my Sennheiser over-ear ANC headphones (MB660) have a much better ANC effect compared to the AirPods but the strain on the ears is a lot less intense. I would expect, that the Bose QC-35 II are much closer to the MB660 than to the AirPods Pro.