High-performance 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, if the (integer) oversampling ratio $\lambda$ is such that $\lfloor M/\lambda\rfloor - 1\geq 4/\Delta$, where $M$ denotes the number of Fourier measurements and $\Delta$ is the minimum separation distance associated with the atomic measure to be resolved, then for any number $K\geq 2$ of quantization levels available for the real and imaginary parts of the measurements, our quantization method guarantees reconstruction accuracy of order $O(\lambda K^{- \lambda/2})$, up to constants which are independent of $K$ and $\lambda$. In contrast, memoryless scalar quantization offers a guarantee of order $O(K^{-1})$ only.