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(N log N ), i.e. smaller than their arithmetical complexity. The derivation starts with nesting of Discrete Fourier Transform (DFT) of size N = q1 • q2 • . . . qr, where qi are powers of prime numbers: DFT is mapped into multidimensional one, Rader convolutions of qi-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(N log Mmax), where Mmax 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 qi values leads to important reduction of Mmax value: Multiplicative complexity of the new DFT algorithms is O(N log c log N ) for c ≤ 1, while for more addition-orietnted ones it is O(N log 1/m N ), m is a natural number denoting class of qi 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 ≤ 65520. The approach can be used for finding theoretical lower limit on the DFT multiplicative complexity.

Refined convolution‐based measures for real‐time harmonic distortions estimation in power system dominated by inverter‐based resources

“…The DFT-based method requires N data points and involves N 2 complex multiplications and N (N − 1) complex additions [19]. In addition, identifying the harmonic components requires further computations.…”