What is the theory behind active noise cancellation?

Question

In a previous question, I had asked Why can't I simply negate the source time domain amplitude values to produce a destructive noise signal?

One of the posters said that while simply producing a inverses polarity (negated) signal will work in theory, in practice it is not possible

So I am asking, what is the fundamental approach (in a sort of semi technical way) to active noise cancellation?

Secondly, why are most literature on this topic in frequency domain?

Answer 1

It's rather simple.

By the time you send your inverted signal, the noise has already been heard.

You need to look at what frequencies are being generated, and then produce the appropriate inverted signals of those to cancel them out.

Answer 2

Noise cancellation is prediction. Your algorithm has to predict what the sound of the noise will be at some time in the future (that time given by the system and audio time latencies), and then predict what signal will produce the opposite sound at that same point in the future (which your system will distort and delay, so you have to figure in the opposite distortion and delay).

You might be able to use several successive FFTs to determine which frequencies in the noise are not changing, and assume or calculate some probability that they will continue for a short time into the future.

If you know the frequency response curve of the speaker, you might be able to figure out the frequency amplitudes of a signal needed to match some predicted noise spectrum. The phase angle of a sinusoid will change with time. If you know the time delay of your output signal, you might be able to calculate the phase of a sinusoid at some point in the future. If you have a predicted phase of a particular frequency of noise at some time and location, you can add π to that phase angle to estimate the noice-cancelling signal.

If you don't know the frequency response and delay of your system, then you won't know what frequencies, amplitudes or phases of signal to create for cancellation. You might well end up amplifying the noise instead of cancelling it.

Answer 3

It seems that what’s missing is the propagation delay required to intercept and negate a signal. The KISS rule will eventually prove this true. The FFT is a complex calculation and each N iteration will introduce resulting error due to the time required to process the signal. To cancel a sound wave it will need to be intercepted in advance, processed and inverted. Then the time constant of the transducer must. E considered. My experience is that a microphone near the source of “noise” connected by wire and amplification device and transducer near the location where It is to be cancelled.

Answer 4

edit: typo

The basic idea of ANC is to find repetitive sound and play the opposite of it. If the repetitive sound continue to play we'll be able to cancel it. That goes in direct contradiction to to the other answers, but I'll clarify.

Playing the opposite sound means playing it again with a precise power and delay, possibly inverting the waveform. The delay itself varies for each frequency. For example, for a 20Hz sound we have to replay the inverted sound on a precise multiple of 1/20 = 0.05s. For 23Hz, for example, the delay has to be a multiple of 1/23 ~= 0.04347s.

Since any waveform can be produced by sum of sinusoidal, one way of doing it would be to only worry about the N biggest sinusoids, measured in power (square of the amplitudes). For finding the sinusoidal's frequencies and power we use the Fourier Transform, typically with the FFT algorithm.

If we take, for example N=8, it means we are trying to eliminate the 8 most powerfull wave components. For each of them we store:

• wave's amplitude
• wave's offset, taking the computer's clock as a base.

than we constantly play 8 sinusoids, each on the correct power and with the correct delay. The hard part is what happens next. We need to keep listening to adapt, but now we are listening to the environment sound + our own sound. This algorithm is harder to implement, but conceptually is easier, and one could easily figure out how to do it by himself.

So, contrary to what the other answers say, managing the time delay is critical. Is not possible to create an ANC system without doing it. If you only care about the frequency domain, the only thing you could possibly do is filter those frequencies. On an ANC system this makes not sense.