Fast Discrete Fourier Transform algorithms requiring less than 0(NlogN) multiplications

Abstract

In the paper it is shown that there exist infinite classes of fast DFT algorithms having multiplicative complexity lower than O(NlogN), i.e. smaller than their arithmetical complexity. The derivation starts with nesting of Discrete Fourier Transform (DFT) of size N = q_1 q_2 ... q_r, where q_i are powers of prime numbers: DFT is mapped into multidimensional one, Rader convolutions of q_i-point DFTs extracted, and combined into multidimensional convolutions processing data in parallel. Crucial to further optimization is the observation that multiplicative complexity of such algorithm is upper bounded by 0(Nlog M_max), where M_max is the size of the greatest structure containing multiplications. Then the size of the structures is diminished: Firstly, computation of a circular convolution can be done as in Rader-Winograd algorithms. Secondly, multidimensional convolutions can be computed using polynomial transforms. It is shown that careful choice of q_i values leads to important reduction of M_max value: Multiplicative complexity of the new DFT algorithms is O(Nlog^c log N) for c\le 1, while for more addition-orietnted ones it is O(Nlog^{1/m} N), m is a natural number denoting class of q_i values. Smaller values of c, 1/m are obtained for algorithms requiring more additions, part of algorithms for c = 1, m=2 have arithmetical complexity smaller than that for the radix-2 FFT for any comparable DFT size, and even lower than that of split-radix FFT for N\le 65520. The approach can be used for finding theoretical lower limit on the DFT multiplicative complexity.