Generation of all radix-2 fast Fourier transform algorithms using binary trees

In this work a systematic method to generate all possible fast Fourier transform (FFT) algorithms is proposed based on the relation to binary trees. The binary tree is used to represent the decomposition of a discrete Fourier transform (DFT) into sub-DFTs. The radix is adaptively changed according to compute sub-DFTs in proposed decomposition. In this work we determine the number of possible algorithms for 2ⁿ-point FFTs with radix-2 butterfly operation and propose a simple method to determine the twiddle factor indices for each algorithm based on the binary tree representation.