A deterministic sparse FFT algorithm for vectors with small support

In this paper we consider the special case where a signal x∈ℂN${\in }\,\mathbb {C}^{N}$ 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 x̂∈ℂN$\widehat {\mathbf x}\,{\in }\,\mathbb {C}^{N}$. In case of exact Fourier measurements we require only O${\mathcal O}$(mlog$\log $m) arithmetical operations. For noisy measurements, we propose a stable O${\mathcal O}$(mlog$\log $N) algorithm.