All variants of the original Cooley-Tukey O(n log n) fast Fourier transform fundamentally exploit different ways to factor the discrete Fourier summation of length N.
For example, the split-radix FFT algorithm divides the Fourier summation of length N into three new Fourier summations: one of length N/2 and two of length N/4.
The prime factor FFT, divides the Fourier summation of length N, i...
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