The effect of quantization on the performance of sampling designs

The most common form of quantization is rounding-off, which occurs in all digital systems. A general quantizer approximates an observed value by the nearest among a finite number of representative values. In estimating weighted integrals of a time series with no quadratic mean derivatives, by means of samples at discrete times, it is known that the rate of convergence of the mean-square error is reduced from n^-2 to n^-1.5 when the samples are quantized. For smoother time series, with k=1, 2, ... quadratic mean derivatives, it is now shown that the rate of convergence is reduced from n^-2k-2 to n^-2 when the samples are quantized, which is a very significant reduction. The interplay between sampling and quantization is also studied, leading to (asymptotically) optimal allocation between the number of samples and the number of levels of quantization