The Fast Fourier Transform (FFT): Most Ingenious Algorithm Ever?

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

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.

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.

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

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

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 .

Not a book, but I truly enjoy this guy’s videos: ▶ this video . I usually watch them while eating lunch. He does a truly fantastic job.

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.

Were you watching this video by any chance? ▶ this video ("The Fast Fourier Transform (FFT): Most Ingenious Algorithm Ever?" by Reducible)