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-delta, in time and space poly(B,log(L),log p, log(1/delta), epsilon^{-1}, such that ||S-R||^2 < (1+epsilon) ||S-R_{opt}^B||^2, where R_{opt}^B is the optimal B-term Fourier representation of signal S. The sublinear poly(logL) time is compared to the superlinear O(Nlog N+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.