# Maximum delay for effective active noise cancelling?

Active noise cancelling reduces unwanted sound by sending the inverted phase of the original phase:

(Source: Wikipedia)

Theoretically, this seems logical to me. However, in real life, the anti-noise must be created by some hardware or software system (like active noise cancelling headphones), which takes time. So I assume that the anti-noise is always delayed to the original sound:

My questions:

• How much (in milliseconds or whatever) is the maximum delay which is "allowed" for active noise cancelling so that the hearer of the noise+antinoise still notices the effect?
• Does the "allowed" delay depend on which noise has to be cancelled (e.g. a car driving, people speaking, music)?
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@Downvoter Could you explain the downvote, please? If something is wrong with the question I would like to have the chance to improve it. –  Uooo Apr 10 at 5:34
+1 It's an interesting physical effect @w4rumy You know that the delay depends on the surroundings and the sound source. You can't work out the reason for believing there is an actual delay? Is this reason for the delay also part of the question? Ah ... welcome to Physics SE! So be more specific about thought of the effect instead software delay for effect creation :) –  Stefan Bischof Apr 10 at 5:47
@StefanBischof My question is rather from a developer's point of view: "If I want to build an Active Noise Cancelling system, which maximum delay should I aim for?". Reading articles about this topic seem that the "optimum" would be 0ms, but I assume that's not possible because the original sound signal has to be processed. Maybe Physics SE is not the right place to ask this? Would there be any better SE site to ask? However it seems that understanding the physics behind ACN help answering this question. –  Uooo Apr 10 at 6:02
The problem is a control system design, so my clue for you is to google it that way, in automation, delays are a common issue. mathworks.com/products/control/examples.html?file=/products/… –  HDE Apr 12 at 16:03
To get people thinking in the right direction, I propose thinking about this as "given an amplitude (or square amplitude?) tolerance, what is the allowable phase offset (from 180 degrees) one can have when adding two sinusoids of equal amplitude and frequency to get cancellation within tolerance" Throw in the frequency range of interest to convert phases to times. As a bonus, someone should think about whether and how the specifics of the spectrum of sound matters. –  Chris White Apr 12 at 16:37
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The key to this is the physical principle that the quantity you're asking about (delay between noise and noise cancelling) carries dimensional information (i.e. it's a time) and therefore it has to depend on the specific situation.

The simplest case is trying to cancel out a pure note, with a sinusoidal waveform, then the delay can be as long as you want: you just wait a whole number of periods and it doesn't change anything. (The delay precision, though, has to be quite high! The absolute delay can be as large as you want, but its precision must be much smaller than the pure note's period (1/its frequency) for the noise cancelling to work.)

A pure note, however, is not really a physical thing. All sounds have a finite duration, and therefore a more physical model is a finite waveform such as

Here it's clear that you can't wait forever, or the "noise cancelling" will just be an echo. The relevant timescale your delay must act on is that in which the pulse is changing. This is given by the input's bandwidth: you can represent your noise as a (Fourier) superposition of pure notes drawn over some interval of frequencies $[0,\nu_\text{max}]$; the shortest timescales over which the waveform can change are of the order of $1/\nu_\text{max}$. This is the maximum delay for the noise-cancelling system to be effective.

In practice, it works the other way. Such systems have a fixed delay, which determines which noise bandwidths they are effective against. Designers try to make their headphones effective against most usual sources of noise, but this can't always be managed. Some systems (including the headphones I'm currently wearing) include a choice of bandwidths - mine says "low" and "wide", best for trains and airplanes or viceversa - to adapt to different noise sources.

The bottom line, then, is an emphatic yes to your second question. The delay should be shorter than 1/(highest noise frequency) for whichever source you're dealing with.

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