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I'm trying to understand the principle of active noise control, a technique to reduce noise by using interference. Wikipedia says:

A noise-cancellation speaker emits a sound wave with the same amplitude but with inverted phase (also known as antiphase) to the original sound. The waves combine to form a new wave, in a process called interference, and effectively cancel each other out – an effect which is called destructive interference.

This simple explantion makes perfect sense to me except one thing — what exactly is "antiphase"?

If we are talking about pure sinusoidal sound waves, then it is pretty simple. Two waves with a phase difference of 180˚ are in antiphase, and if their amplitudes are the same, they will interfere with each other and cancel each other.

My question arises when talking about non-sinusoidal periodic waves. In order to cancel such a wave F(t), we need an inverted wave with the same amplitude, i.e. -F(t). But now, because it is not a sinusoidal curve, this inverted wave -F(t) is not identical to the phase-shifted original wave by 180˚ F(t+1/2*L) where L is the cycle length of F(t). Can we still call this inverted wave -F(t) as "antiphase"? Isn't it literally an inverted in-phase wave (because we haven't changed the phase at all)?

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“Phase” isn’t particularly well defined for non-sinusoidal waveforms.

But we still use the language because we’re used to thinking of complicated waveforms as being made up of multiple sinusoidal components of different frequencies.

In that case, when you apply anti phase to each of those components separately, you get the inverse you’re looking for.

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  • $\begingroup$ > “Phase” isn’t particularly well defined for non-sinusoidal waveforms. That's so true. So there's Fourier transform-like view behind this usage of antiphase. Thanks. $\endgroup$ – Kouichi C. Nakamura Nov 19 '19 at 4:25

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