C
Story
Fast Fourier Transforms Part 1: Cooley-Tukey
  • srean
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  • 4 hours ago
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At the root of the fast transform is the simple fact that

    ax + bx = (a+b)x
The right hand side has fewer arithmetic operations. It's about finding common factors and pushing parentheses in. Because of the inherent symmetry of the FT expression there are lots of opportunities for this optimization.

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.

How is belief propagation used for decoding LDPC codes related to FFT?
  • srean
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  • 34 minutes ago
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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).
Two extremely influential papers appeared back to back in transactions information theory. This is one of them.

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.

> 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.

  • srean
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  • 37 minutes ago
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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.
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.

  • ajross
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  • 11 minutes ago
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"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.

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.
Quite funny because now I can’t access the comment box at all.
  • f1shy
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  • 6 hours ago
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Same here. I think is intended as “feature” but extremely annoying.
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.
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  • 4 hours ago
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Fast Fourier transform was not invented by Cooley-Tukey, it was used by Gauss to compute trigonometric interpolation of orbits from observations.
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

  • srean
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  • 2 hours ago
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True. Before Fourier did Fourier.
Author here: thanks for sharing!
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