Estimating the Frequency of a Clustered Signal
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We consider the problem of locating a signal whose frequencies are "off grid" and clustered in a narrow band. Given noisy sample access to a function g(t) with Fourier spectrum in a narrow range [f_0 - Δ, f_0 + Δ], how accurately is it possible to identify f_0? We present generic conditions on g that allow for efficient, accurate estimates of the frequency. We then show bounds on these conditions for k-Fourier-sparse signals that imply recovery of f_0 to within Δ + Õ(k^3) from samples on [-1, 1]. This improves upon the best previous bound of O( Δ + Õ(k^5) )^1.5. We also show that no algorithm can do better than Δ + Õ(k^2). In the process we provide a new Õ(k^3) bound on the ratio between the maximum and average value of continuous k-Fourier-sparse signals, which has independent application.
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