How do noise cancelling headphones *really* work?

[David Hess]

The reason is fairly obvious, as already pointed out in this thread:  the acoustic delay from a microphone mounted *on* the headphones, to the inside of your ear, is nowhere near long enough to do DSP, unless you gin up something really crazy with analog-based discrete-time processing.  The A/D and D/A delays alone would kill you.

If it can be done in the analog domain then it can be done in the digital domain.  The right converters can have a maximum latency of one sample and nothing prevents oversampling and decimation to reduce converter latency below that but diminishing returns are quickly reached because the latency of a single cycle converter starts out low.  Then it becomes a matter of clocking data through a finite impulse response filter which has the same group delay as the analog filter it replaces.

Unfortunately the only thing this improves is the ability to tune the filter digitally and it requires more power and is noisier when these are already problems in analog designs.

[InterestedTom]

It really is not that difficult.  Propagation through the mechanical structure of the headphone is slow compared to electronics so delay only becomes a problem at higher frequencies [...]

As I said earlier, that's one possibility. But I am not sure whether I quite believe it:

Let's say the distance between the outside microphone and the speaker is 30 mm in a headphone -- that only gives you 0.1ms of acoustical propagation delay to work with. The Infineon microphone alone which wraper linked to (reply #10 above) has 30° phase delay at 50 Hz; that's your electronic path lagging behind by 1.5ms. The speaker might add more lag; let alone digitization, if you apply digital filtering.

How do you make up for that without some predictive approach for your cancellation signal? Or what am I overlooking?

EDIT: Probably the group delay (along the electronic path for the cancellation signal) is what one should be looking at, rather than the phase delay. But that has a similar order of magnitude for the Infineon microohone: Extrapolating from the range shown in Fig. 10 of the application note, I would estimate 1.5ms group delay at 50 Hz.

The infineon microphone is a silicon microphone, I think you will find more conventional construction microphones will have a better phase response.