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I did not build the full system, but I did build an analog circuit that when fed an input would generate the complement of the signal and then sum the signals (which if it works sum to 0). In addition, the summer had a third input for an audio input. This is the basic idea of how they work.
Noise cancelling headsets are commonly used in small aircraft cockpits.
Just two microphones, one to pick up noise, the other the voice. Noise signal is inverted and then mixed with the voice signal to cancel the noise.
It seems to work