Quantization for Spectral Super-Resolution

Abstract

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 \(\lambda \) as the largest integer such that \(\lfloor M/\lambda \rfloor - 1\ge 4/\Delta \), where M denotes the number of Fourier measurements and \(\Delta \) is the minimum separation distance associated with the atomic measure to be resolved. We prove that for any number \(K\ge 2\) 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 \(O(M^{1/4}\lambda ^{5/4} K^{- \lambda /2})\) and \(O(M^{3/2} \lambda ^{1/2} K^{- \lambda })\), respectively, where the implicit constants are independent of M, K and \(\lambda \). 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.