Sadly, both of you are quite wrong. ANC can easily cancel wideband sudden sounds. It may have problems at very high frequencies due to frame sizes and required group delay precision. Computing FFT or even more complex transforms with very low latency is possible, as is combining time domain and frequency domain processing. You cannot do analogue ANC at high frequencies, as the internal reflections in the headphone/ear system ruin it, while a digital system can measure these and respond well enough while not damaging the sound. Plus it is very hard to represent psychoacoustics without digital processing and those help preserve sound quality immensely.