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 oversampling ratio $\lambda$ as the largest integer such that $\lfloor M\mathrm{/}\lambda \rfloor -1\ge 4\mathrm{/}\mathrm{\Delta }$, where $M$ denotes the number of Fourier measurements and $\mathrm{\Delta }$ 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.