 This result, however, applies only to the unnormalized Fourier transform (which is a scaling of a unitary matrix by a factor of formula_19), and does not explain why the Fourier matrix is harder to compute than any other unitary matrix (including the identity matrix) under the same scaling.Pan (1986) proved an Ω("N" log "N") lower bound assuming a bound on a measure of the FFT algorithm's "asynchronicity", but the generality of this assumption is unclear.For the case of power-of-two "N", Papadimitriou (1979) argued that the number formula_20 of complex-number additions achieved by Cooley–Tukey algorithms is "optimal" under certain assumptions on the graph of the algorithm (his assumptions imply, among other things, that no additive identities in the roots of unity are exploited).