In reality noise cancelling headphones don't "just reverse the wave", since that's often not fast or accurate enough (too noisy). Instead it's more like:
mic -> fft -> secret suace analysis and filtering -> unfft -> speaker
A simple filtering would be a high pass filter, to avoid random high frequency sounds from constructively interfering producing more high frequency static.
Just a random tidbit, if you listen to rtings' examples or have a pair yourself I'd suggest playing some different colours of noise to see for yourself!
I don't know what audiophile circles you're in but when I started getting into headphones phone DACs were looked at with scrutiny. For most earbuds it's "good enough" whereas for higher end headphones it is both accepted and measurable that you're not experiencing all that your headphones have to offer.
In response there are many that purchase media players with DACs for this specific reason. When 3.5mm jacks started getting removed measurements also turned to the DACs within usb c/lightning to 3.5mm cables.
mic -> fft -> secret suace analysis and filtering -> unfft -> speaker
A simple filtering would be a high pass filter, to avoid random high frequency sounds from constructively interfering producing more high frequency static.
Just a random tidbit, if you listen to rtings' examples or have a pair yourself I'd suggest playing some different colours of noise to see for yourself!