What’s the Frequency, Kenneth?: Sublinear Fourier Sampling Off the Grid
Algorithmica(2014)
摘要
We design a sublinear Fourier sampling algorithm for a case of sparse off-grid frequency recovery. These are signals with the form f(t) = ∑ _j=1^k a_j e^iω _j t + ∫ν (ω )e^iω tdμ (ω ) ; 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, for any ϵ∈ (0,η /k] , which takes O(klog klog (1/ϵ )(log k+log (‖ a‖ _1/‖ν‖ _1)) samples of f(t) and runs in time proportional to number of samples, recovering ω _j'≈ω _j and a_j'≈ a_j such that, with probability (1) , the approximation error satisfies |ω _j'-ω _j|≤ϵ and |a_j-a_j'|≤‖ν‖ _1/k for all j with |a_j|≥‖ν‖ _1/k . We apply our model and algorithm to bearing estimation or source localization and discuss their implications for receiver array processing.
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关键词
Sparse signal recovery,Fourier sampling,Sublinear algorithms,94A20,68W20
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