ar X iv : m at h / 05 02 35 7 v 1 [ m at h . N A ] 1 6 Fe b 20 05 A Sublinear Algorithm of Sparse Fourier Transform for Nonequispaced Data ∗

We present a sublinear randomized algorithm to compute a sparse Fourier transform for nonequispaced data. Suppose a signal S is known to consist of N equispaced samples, of which only L < N are available. If the ratio p = L/N is not close to 1, the available data are typically non-equispaced samples. Then our algorithm reconstructs a near-optimal B-term representation R with high probability 1−δ, in time and space poly(B, log(L), log p, log(1/δ), ǫ−1), such that ‖S − R‖2 ≤ (1 + ǫ)‖S − RB opt‖, where RB opt is the optimal B-term Fourier representation of signal S. The sublinear poly(logL) time is compared to the superlinear O(N logN + L) time requirement of the present best known Inverse Nonequispaced Fast Fourier Transform (INFFT) algorithms. Numerical experiments support the advantage in speed of our algorithm over other methods for sparse signals: it already outperforms INFFT for large but realistic size N and works well even in the situation of a large percentage of missing data and in the presence of noise.