What’s the Frequency, Kenneth?: Sublinear Fourier Sampling Off the Grid

We design a sublinear Fourier sampling algorithm for a case of sparse off-grid frequency recovery. These are signals with the form $$f(t) = \sum _{j=1}^k a_j \mathrm{e}^{i\omega _j t} + \int \nu (\omega )\mathrm{e}^{i\omega t}d\mu (\omega )$$f(t)=∑j=1kajeiωjt+∫ν(ω)eiωtdμ(ω); i.e., exponential polynomials with a noise term. The frequencies $$\{\omega _j\}$${ωj} satisfy $$\omega _j\in [\eta ,2\pi -\eta ]$$ωj∈[η,2π-η] and $$\min _{i\ne j} |\omega _i-\omega _j|\ge \eta $$mini≠j|ωi-ωj|≥η for some $$\eta > 0$$η>0. We design a sublinear time randomized algorithm which, for any $$\epsilon \in (0,\eta /k]$$ϵ∈(0,η/k], which takes $$O(k\log k\log (1/\epsilon )(\log k+\log (\Vert a\Vert _1/\Vert \nu \Vert _1))$$O(klogklog(1/ϵ)(logk+log(‖a‖1/‖ν‖1)) samples of $$f(t)$$f(t) and runs in time proportional to number of samples, recovering $$\omega _j'\approx \omega _j$$ωj′≈ωj and $$a_j'\approx a_j$$aj′≈aj such that, with probability $$\varOmega (1)$$Ω(1), the approximation error satisfies $$|\omega _j'-\omega _j|\le \epsilon $$|ωj′-ωj|≤ϵ and $$|a_j-a_j'|\le \Vert \nu \Vert _1/k$$|aj-aj′|≤‖ν‖1/k for all $$j$$j with $$|a_j|\ge \Vert \nu \Vert _1/k$$|aj|≥‖ν‖1/k. We apply our model and algorithm to bearing estimation or source localization and discuss their implications for receiver array processing.