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 $\mathbf{x}$ 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 $\mathcal{O}(m\log m\log (N/m))$ and requires $\mathcal{O}(m\log (N/m))$ 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 $\mathbf{x}$. The algorithm automatically recognizes and exploits a possible short support of the vector and falls back to a usual radix-2 FFT algorithm if $\mathbf{x}$ has (almost) full support. The numerical stability of the proposed algorithm ist shown by numerical examples.

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