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 left perpendicular M/X right perpendicular - 1 >= 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 >= 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.