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In this paper we consider the special case where a signal x ∈ C N is known to vanish outside a support interval of length m < N . If the support length m of x or a good bound of it is a-priori known we derive a sublinear deterministic algorithm to compute x from its discrete Fourier transform x ∈ C N . In case of exact Fourier measurements we require only O(m log m) arithmetical operations. For noisy measurements, we propose a stable O(m log N ) algorithm.

In this paper we propose a new fast Fourier transform to recover a real nonnegative signal x ∈ R N + from its discrete Fourier transform x = F N x ∈ C N . 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(m log m log(N/m)) and requires 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 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 ist shown by numerical examples.

In this paper we consider sparse signals ${\bf x}\in {\bf C}^N$ which are known to vanish outside a support interval of length bounded by m < N. For the case that m is known, we propose a deterministic algorithm of complexity ${\cal O}(m\log m)$ for reconstruction of x from its discrete Fourier transform $\widehat{\bf x}\in {\bf C}^N.$ (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)