Signals (Oct 2021)

A Sparse Algorithm for Computing the DFT Using Its Real Eigenvectors

  • Rajesh Thomas,
  • Victor DeBrunner,
  • Linda S. DeBrunner

DOI
https://doi.org/10.3390/signals2040041
Journal volume & issue
Vol. 2, no. 4
pp. 688 – 705

Abstract

Read online

Direct computation of the discrete Fourier transform (DFT) and its FFT computational algorithms requires multiplication (and addition) of complex numbers. Complex number multiplication requires four real-valued multiplications and two real-valued additions, or three real-valued multiplications and five real-valued additions, as well as the requisite added memory for temporary storage. In this paper, we present a method for computing a DFT via a natively real-valued algorithm that is computationally equivalent to a N=2k-length DFT (where k is a positive integer), and is substantially more efficient for any other length, N. Our method uses the eigenstructure of the DFT, and the fact that sparse, real-valued, eigenvectors can be found and used to advantage. Computation using our method uses only vector dot products and vector-scalar products.

Keywords