The Riff
Published in

The Riff


The Mathematics of Active Noise Cancellation

The French physicist who made our deep work effortless and listening to low-volume music easier.

Joseph Fourier, French mathematician and physicist. CNRS News

Even if you’re not a music enthusiast, active noise cancellation (ANC) can do wonders for your focus by eliminating a barrier you didn’t know had so much impact.

My purchase of Airpods Pro was not impulsive. I was waiting for an excuse to get it, and an accident did me a favour.

From then on, I couldn’t live in a noisy environment without my silencer buddies.

The outdated not-so-effective noise cancellation.

Passive noise cancellation is when physical accessories become a barrier to external noise. They are heavy on ears and are not friendly for extended use.

For example, the headphones that cover your ear completely. You can listen to music clearly, but ear-covering headphones don’t go light on your ears when it comes to meditation or preparing a ppt that might take hours.

The weight becomes another burden. I have tried once, and that was my last time.

Enter active noise cancellation (ANC).

Since I’m an Apple fanboy, my first option was to try Airpods Pro first and see if the ANC benefits me. One year is over, and I can’t imagine my deep work without it.

Active noise cancellation works by nullifying the external noise without weighing down your ears. Your ears — and head — are free to move for as long as you want, which is not the case with noise cancellation using physical barriers.

The mathematics of ANC

Without complex maths, understanding noise cancellation lies in basic maths. For example, if you have the number 14, how will you make it zero? Yeah, baby, by adding -14 to it.

ANC works the same way. But here, we don’t have simple integers.

We have sounds of varying loudness. We represent them as waves.

The technique for ANC lies in a formula created by a French mathematician, Joseph Fourier.

The principle of Fourier Transform for ANC works like this:

“Any sound wave can be decomposed into a combination of sine-wave.”

For a visual explanation, here is how noise reaches us and the mechanism of ANC earphones.

Source: Techabal

By producing a wave which is precisely the opposite of the noisy wave, the loudness of noise entering your ear is undetectable. The loudness is so low that it fades into the background.

I wish I could explain comprehensively what Fourier Transform is and how it works for all signals. But since the feature of ANC was the aim of this article, I’ll leave the simple explanation here.

Do yourself a favour and try ANC earphones.

Your head will feel light, you’ll be able to ace your deep work sessions, and you’ll have Joseph Fourier to thank for it.

If you want to receive more stories like this, my lifelong learning newsletter is for you.

Sanjeev is a mentor, writer, and fitness enthusiast from India. He writes about lifelong learning, personal growth, and positive psychology. When he’s not engaging with students in solving their doubts or busy with writing, he’s sweating either in a workout, PC gaming or playing 8-ball pool. You can also find him on Twitter.



Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store