7 HN users recommended this explainer8 months2021–2026score 10.5
Years with an HN recommendation, 2007–2026
HN repeatedly calls it a fantastic explanation that made FFT click, with some rewatching it periodically.
Reducible's visual walkthrough of the Fast Fourier Transform, motivated by polynomial multiplication. It shows how the FFT's key ideas can be discovered by asking the right questions.
“It derives the FFT algorithm from polynomial multiplication and it’s fantastic . I rewatch it every 6 months or so.” (HN, 2024)
As the author mentions, convolutions for signal processing and convolutions for probability are essentially equivalent things in two different contexts. It is interesting that the same concept is behind the multiplication of polynomials. In the discrete case we have the parallel of the vectors representing ... the (discretized) probability density <-> the (discrete) signal intensity <-> the coefficients of the polynomial ... in the respective context. If we look at the discrete case, since the operation of convolution is computationally expensive (O(n^3)), people often rely on the Fast Fourier Transform Algorithm (FFT), that runs in O(n log(n)). An excellent and very enjoyable video on the FFT applied to the polynomial multiplication is [this video] .
That video did the same for me. I also like the reducible video for Fast Fourier Transforms as well as the Veratasium piece that shares some fun history. Reducible - FFTs the most ingenious algorithm ever [this video] Veritasium - The Most Important Algorithm of All Time [link]
If you haven’t seen it yet, you should watch this video: [this video] It derives the FFT algorithm from polynomial multiplication and it’s fantastic . I rewatch it every 6 months or so.
I almost always prefer text over video, but this video on the same subject is fantastic: [this video] . I watch it at least once every year. Also, the title of the blog post (and by extension this HN post) is IMO not really correct: it's not about the DFT but specifically about the FFT.
And an FFT explanation by Reducible - [this video] It's in a similar style to 3blue1brown (uses the same visualisation library) but the video focuses specifically on the FFT algorithm, not on Fourier transforms in general. I'm pretty sure Grant himself recommended the Reducible video at some point.
FFT explainer in 3blue1brown style, focused on the algorithm
Related: Fast Fourier Transform (FFT) can be used for polynomial interpolation in O(nlogn) time (if you get to choose the points, and use “complex roots of unity”), where n is the degree of the polynomial, and is used as a step in multiplication of polynomials in O(nlogn) as well. This video covers it well: [this video] Also, chapter 30 of CLRS: [link]
Covers FFT for polynomial interpolation and multiplication
For anyone interested and with 30 minutes to spare, I would recommend the youtube video [1] by Reducible for an intuition on how FFT works. [1]: [this video]
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