These three sums correspond to portions of radix-2 (size N/2) and radix-4 (size N/4) Cooley–Tukey steps, respectively. (The underlying idea is that the even-index subtransform of radix-2 has no multiplicative factor in front of it, so it should be left as-is, while the odd-index subtransform of radix-2 benefits by combining a second recursive subdivision.) These smaller summations are now exactly DFTs of length N/2 and N/4, which can be performed recursively and then recombined. More specifically, let denote the result of the DFT of length N/2 (for ), and let and denote the results of the DFTs of length N/4 (for ). Then the output is simply: This, however, performs unnecessary calculations, since turn out to share many calculations with . In particular, if we add N/4 to k, the size-N/4 DFTs are not changed (because they are periodic in N/4), while the size-N/2 DFT is unchanged if we add N/2 to k. So, the only things that change are the and terms, known as twiddle factors. Here, we use the identities: to finally arrive at: which gives all of the outputs if we let range from to in the above four expressions. Notice that these expressions are arranged so that we need to combine the various DFT outputs by pairs of additions and subtractions, which are known as butterflies. In order to obtain the minimal operation count for this algorithm, one needs to take into account special cases for (where the twiddle factors are unity) and for (where the twiddle factors are and can be multiplied more quickly); see, e.g. Sorensen et al. (1986). Multiplications by and are ordinarily counted as free (all negations can be absorbed by converting additions into subtractions or vice versa). This decomposition is performed recursively when N is a power of two. The base cases of the recursion are N=1, where the DFT is just a copy , and N=2, where the DFT is an addition and a subtraction . These considerations result in a count: real additions and multiplications, for N>1 a power of two. This count assumes that, for odd powers of 2, the leftover factor of 2 (after all the split-radix steps, which divide N by 4) is handled directly by the DFT definition (4 real additions and multiplications), or equivalently by a radix-2 Cooley–Tukey FFT step. References R. Yavne, "An economical method for calculating the discrete Fourier transform," in Proc.