Quantization for spectral super-resolution
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We show that the method of distributed noise-shaping beta-quantization offers superior performance for the problem of spectral super-resolution with quantization whenever there is redundancy in the number of measurements. More precisely, we define the oversampling ratio λ as the largest integer such that ⌊ M/λ⌋ - 1≥ 4/Δ, where M denotes the number of Fourier measurements and Δ is the minimum separation distance associated with the atomic measure to be resolved. We prove that for any number K≥ 2 of quantization levels available for the real and imaginary parts of the measurements, our quantization method combined with either TV-min/BLASSO or ESPRIT guarantees reconstruction accuracy of order O(M^1/2λ^1/2 K^- λ/2) and O(M^1/2λ^3/2 K^- λ) respectively, where the implicit constants are independent of M, K and λ. In contrast, naive rounding or memoryless scalar quantization for the same alphabet offers a guarantee of order O(M^-1K^-1) only, regardless of the reconstruction algorithm.
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