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.
Link: https://www.uio.no/studier/emner/matn...

💡 Explore Fourier analysis yourself! Jupyter notebook
Felix L. and me built a Jupyter notebook for you that contains all explanations from the video alongside the code. Feel free to explore it here: https://github.com/Splines/fourier-an...

🌟 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!
Background image showing some synthesizers by TStudio_lv on Unsplash: https://unsplash.com/de/fotos/ein-hau...

▶ 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