What's the frequency, Kenneth?: Sublinear fourier sampling off the grid

Abstract

We design a sublinear Fourier sampling algorithm for a case of sparse off-grid frequency recovery. These are signals with the form f(t) = Σ j=1k a je iwjt + v̂, t ∈ ZZ; i.e., exponential polynomials with a noise term. The frequencies {ω j} satisfy ω j ∈ [η,2π-η] and min i≠j |ω i - ω j | ≥ η for some η > 0. We design a sublinear time randomized algorithm, which takes O(k log k log(1/η)(log k + log(∥a∥ 1/ ∥ν∥ 1)) samples of f(t) and runs in time proportional to number of samples, recovering {ω j} and {a j} such that, with probability Ω(1), the approximation error satisfies |ω′ j - ω j| ≤ η/k and |a j -∈a′ j|≤∥ν∥ 1/k for all j with |a j| ≥∥ν∥ 1/k. © 2012 Springer-Verlag.