Why it matters

FFT is one of most important algorithms. Understanding shapes broad application.

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The architecture

Bit-reverse input.

Iterative butterfly.

Roots of unity: exp(2*pi*i/n).

Cooley-Tukey FFTBit-reverseinput permutationButterflylog n stagesRoots of unitytwiddle factorsIterative implementation faster than recursive; better cache locality
FFT structure.
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How it works end to end

Iterative: outer loop over stages (1 to log n).

Inner: butterfly pairs.

Twiddle: precompute.

Inverse FFT: conjugate + divide by n.