Quantization for Spectral Super-Resolution

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 over-sampling 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 / 4}λ^{5 / 4}K^-λ/2) and O(M^{3 / 2}λ^{1 / 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^{- 1}K^{- 1}) only, regardless of the reconstruction algorithm.