A sparse Fast Fourier Algorithm for Real Nonnegative Vectors

Abstract

In this paper we propose a new fast Fourier transform to recover a real nonnegative signal ${\bf x}$ from its discrete Fourier transform. If the signal ${\mathbf x}$ appears to have a short support, i.e., vanishes outside a support interval of length $m < N$, then the algorithm has an arithmetical complexity of only ${\cal O}(m \log m \log (N/m)) $ and requires ${\cal O}(m \log (N/m))$ Fourier samples for this computation. In contrast to other approaches there is no a priori knowledge needed about sparsity or support bounds for the vector ${\bf x}$. The algorithm automatically recognizes and exploits a possible short support of the vector and falls back to a usual radix-2 FFT algorithm if ${\bf x}$ has (almost) full support. The numerical stability of the proposed algorithm ist shown by numerical examples.