Why it matters

FFT is one of the most important algorithms in computing. Understanding matters for signal processing and algorithm design.

Advertisement

The architecture

Polynomial evaluation at n points × pointwise mult + interpolation.

DFT: O(n²). FFT: divide-and-conquer using roots of unity, O(n log n).

FFT structureDivideeven + odd coefsConquerrecursivelyCombineusing roots of unityComplex arithmetic; NTT for integer-only via modular arithmetic
FFT divide-and-conquer.
Advertisement

How it works end to end

Cooley-Tukey: standard radix-2 FFT. O(n log n).

Inverse FFT: convert back. Similar algorithm with conjugate roots.

Applications: polynomial mult, big integer multiplication (via poly), convolution, signal analysis.

Number Theoretic Transform (NTT): integer-only variant using modular arithmetic. Avoids floating-point errors.