The Discrete Fourier Transform, But With Triangles
Here’s something I’ve been wondering about lately: what happens if you replace all the sine waves in the fourier transform with triangle waves? If you are like me and would like to play around with such a thing, this post has a method to get a representation of a signal as scaled-and-shifted band-limited triangle waves in O(nlogn)O(n\log n)O(nlogn) time. This is fast enough that we can compute first and ask questions about what it’s good for later (I’ve yet to find anything). Actually, the stuff in this post will work for any periodic function, not just triangle waves, but this was the idea that motivated me to figure this out.