The Mathematics behind Active Noise Cancellation

This paper aims to decipher the mathematics which makes Active Noise Cancellation (ANC) possible. We consider the general equation of a sound wave and the method used by headphones with ANC to effectively cancel out unwanted sound waves, providing both a physical explanation and a mathematical formulation using the Fast Fourier Transform (FFT), applied to a representative example signal. Additionally, we investigate a method by which ambient noise may be predicted ahead of time in order to improve cancellation performance.