A sparse fast Fourier algorithm for real non-negative vectors

Abstract

In this paper we propose a new fast Fourier transform to recover a real non-negative signal x∈R+N from its discrete Fourier transform x̂=FNx∈CN. If the signal 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 O(mlogmlog(N/m)) and requires O(mlog(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 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 x has (almost) full support. The numerical stability of the proposed algorithm is shown by numerical examples.