Const-me36 minutes ago
I have recently needed a decently performing FFT. Instead of doing Cooley-Tukey, I have realized the bruteforce version essentially computes two vector×matrix products, so I have interleaved and reshaped the matrices for sequential full-vector loads, and did bruteforce version with AVX1 and FMA3 intrinsics. Good enough for my use case of moderately sized FFT where matrices fit in L2 cache.
HarHarVeryFunny16 minutes ago
I'm curious why you wouldn't just use a library like FFTW or Intel's IPP (or NVidia's cuFFT if applicable) ?
srean6 hours ago
At the root of the fast transform is the simple fact that
ax + bx = (a+b)x
Efficient decoding of LDPC codes also use the same idea. LDPCs were quite a revolution (pun intended) in coding/information theory.
On the other hand, something completely random, few days ago I found out that Tukey (then a Prof) and Feynman (then a student) along with other students were so enamored and intrigued by flexagons that they had set up an informal committee to understand them. Unfortunately their technical report never got published because the war intervened.
Strangely, it does not find a mention in Surely You're Joking.
rigtorp4 hours ago
How is belief propagation used for decoding LDPC codes related to FFT?
srean2 hours ago
At the core both derive their optimization from the distributive property. If the expression graph has symmetry, you get more optimization out of it.
https://www.cs.ubc.ca/~murphyk/Teaching/Papers/GDL.pdf
Check out the first paragraph
THE humble distributive
law, in its simplest form
states that...this leads
to a large family of fast
algorithms, including
Viterbi’s algorithm and
the fast Fourier
transform (FFT).
The other is
https://vision.unipv.it/IA2/Factor graphs and the sum-product algorithm.pdf
Both are absolute gems of papers. The editor made sure that both appear in the same volume.
kqbx18 minutes ago
The second link is broken on HN because it contains a space. Here's a clickable version: https://vision.unipv.it/IA2/Factor%20graphs%20and%20the%20su...
ajross3 hours ago
> At the root of the fast transform is the simple fact that
Actually... no? That's a constant factor optimization; the second expression has 75% the operations of the first. The FFT is algorithmically faster. It's O(N·log2(N)) in the number of samples instead of O(N²).
That property doesn't come from factorization per se, but from the fact that the factorization can be applied recursively by creatively ordering the terms.
srean2 hours ago
It's the symmetry that gives recursive opportunities to apply the optimization. It's the same optimization folded over and over again. Butterfly diagrams are great for understanding this. https://news.ycombinator.com/item?id=45291978 has pointers to more in depth exploration of the idea.
emil-lp2 hours ago
Well, actually ... Summation is linear time, multiplication is superlinear (eg n log n in number of digits).
Meaning that this takes k summations and one multiplication rather than k multiplications and k summations.
... Where k is the number of terms.
ajross2 hours ago
"Digits" are constant in an FFT (or rather ignored, really, precision is out of scope of the algorithm definition).
Obviously in practice these are implemented as (pairs of, for a complex FFT, though real-valued DCTs are much more common) machine words in practice, and modern multipliers and adders pipeline at one per cycle.
terabytest8 hours ago
This website appears broken in a very unique way on my iOS device. Whenever I swipe to scroll, the page gets zoomed out and it zooms back in when I stop swiping, but half of the content is cut off.
bonefolder7 hours ago
Quite funny because now I can’t access the comment box at all.
f1shy8 hours ago
Same here. I think is intended as “feature” but extremely annoying.
sunrunner6 hours ago
I'm struggling to imagine what the feature is intended to be. Being able to see a larger portion of the page while scrolling? This...doesn't help at all, sadly.
[deleted]6 hours agocollapsed
Mikhail_K4 hours ago
Fast Fourier transform was not invented by Cooley-Tukey, it was used by Gauss to compute trigonometric interpolation of orbits from observations.
ajross4 hours ago
The factorization trick was reinvented several times. The algorithm that uses it to do a frequency decomposition was presented just once by named authors. This happens all the time. Freaking out about naming and attribution isn't really very informative.
Edit: as always, Wikipedia is a better source than comment pedantry: https://en.wikipedia.org/wiki/Fast_Fourier_transform#History
srean4 hours ago
True. Before Fourier did Fourier.
connorboyle3 hours ago
Author here: thanks for sharing!
curtisszmania2 hours ago
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