Active noise cancelling headphones - How fast does a digital signal processor have to be to perfectly cancel out noise with an inverted signal?

Question

How fast does a digital signal processor have to be to perfectly cancel out noise with an inverted signal? Is it possible for a digital signal processor react so fast?

Answer 1

There's a "rule of thumb" for any phase-sensitive control loop that the processing phase lag has to be less than $\frac{\pi}{4}$ at the frequencies of interest to be stable.
For decent noise-cancelling, you'll want to do considerably better than that. I suspect you can find the actual time lag (not phase) for a given product by looking at the upper frequency limit and intuiting the time lag as something on the order of half the period of that limit. From that you can estimate the update rate of the digital processing algo.