Average Running Time of the Fast Fourier Transform

Abstract We compare several algorithms for computing the discrete Fourier transform of n numbers. The number of “operations” of the original Cooley-Tukey algorithm is approximately 2n A(n), where A(n) is the sum of the prime divisors of n. We show that the average number of operations satisfies 1 x )∑ n≤x 2n A(n) ∼ ( π 2 9 )( x 2 log x ) . The average is not a good indication of the number of operations. For example, it is shown that for about half of the integers n less than x, the number of “operations” is less than n1.61. A similar analysis is given for Good's algorithm and for two algorithms that compute the discrete Fourier transform in O(n log n) operations: the chirp-z transform and the mixed-radix algorithm that computes the transform of a series of prime length p in O(p log p) operations.