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 $f(t) = \sum_{j=1}^k a_j {\rm e}^{i\omega_j t} + \hat\nu$, $t \in{\mathbb Z\!}$; 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(klogklog(1/η)(logk + 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.