The Linear Algebra behind sound | Fourier Analysis #SoMePi

What is sound? How can we represent it mathematically & in a computer? And how can we manipulate sound by changing its frequency content using Fourier analysis? This videos gives an introduction into the linear algebra behind the Fourier theorem and motivates the underlying math by building a simple low-pass filter in Python (SageMath) to cut off high frequencies of a square wave signal. We will discuss it in a continuous setting first, then motivate the discrete setting (DFT) by analogy. This video is explicitly *not* about the Fast Fourier Transform (FFT).

✍ References
The theorems presented here are taken from the following amazing book:
"Linear algebra, signal processing, and wavelets. A unified approach". By Øyvind Rya. January 21, 2015 Python edition.

💡 Explore Fourier analysis yourself! Jupyter notebook

🌟 Acknowledgements
- Thank you Felix L. for the fruitful discussions about how to present the topics.
- Shoutout to the computer-assisted math seminar 2024 organized by Florent S. & Judith L.
- Thank you Luisa H. & Christian for reviewing the draft version of this video!

▶ Chapters
0:00 Intro
0:46 What is sound?
3:37 Fourier Series: Definitions
4:58 Fourier Series: Basis & best approximation
9:38 Implementation: Fourier coefficients
12:01 Discrete Fourier Series
15:10 Implementation: DFT
16:20 Low-pass filter
18:17 Outro & SageMath

[julioaurelio]

Another Roof sent me here. Awesome video.

[junemcontreras]

This video was amazing and cannot WAIT to check out the notebook!

[Filup]

Wow! You just covered close to a whole semester's overview of Fourier Analysis in under 20 minutes. I took a class on it last year and it was by far the most intense mathematics I have had to do so far in my degree. There was so much more towards the end of my semester that I just gave up on understanding in order to hopefully salvage my grades by focusing on less, so if you know more on this topic, I would absolutely love to see more!

[bernhardwerner]

Just a reminder: SoME4 will, presumably, take place in 2025. This is "only" SoMEπ. It is community-driven and there will be no price money and no 3Blue1Brown top 5 video this year.

[superuser]

Really great video. Also touches on the probability and real analysis 11:14 gem

[thegreengoblin]

You explained it very well, I understood most of the video although I don't even know one thing about Fourier stuff

[newbie]

This could have been a wonderful introduction to an undergraduate course in Speech Processing, but I did not research enough before opting for one such course being offered by the ECE dept at my uni, and we ended up brushing up Signals and Systems (DTFS, DTFT, DCT, FFT) for like 10weeks, jumping into the 'sound' component for only 5-6weeks...

The course instructor followed a book from the 1990's, the only enjoyable part we learnt was about Hidden Markov Models, which is a probabilistic model to predict the next word (or what I understood of it)

[forthrightgambitia]

Great video. I had no idea that old telephone sounds were made by the summation of two sine wats.