A deterministic sparse FFT algorithm for vectors with small support

Abstract

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