Anybody know of a proof of the hard upper frequency limit of active noise cancellation using the ear canal volume as a boundary condition to solve for the upper frequency limit? I keep on encountering 1kHz as the upper limit in practice.
I don't have a mathematical proof for you, but here are 2 issues to think about.
The impulse response from the actuator (speaker) to the error measurement (microphone) is non-minimum phase. This is unavoidable because it takes a non-zero amount of time for an acoustic wave to travel from the speaker to the microphone. In a digital system there is additional latency introduced because you cant instantaneously sample, process, and output a signal. This latency limits your cancellation bandwidth.
The other issue is that even if you had a super low latency system and could get cancellation up to 10kHz it would still only be cancelling noise at the error microphone. At 10kHz your wavelength is about 3.4cm. The distance from the error microphone to the eardrum is a sizable portion of this wavelength, which means that whatever cancellation you thought you had at 10kHz is gone, and you might even be boosting now. So your perceived cancellation is limited by how accurately your microphone measurement approximates the sound field at the eardrum.