A Study of Rough Amplitude Quantization by Means of Nyquist Sampling Theory

Quantization is an operation that takes place when a physical quantity is represented numerically. It is the assignment of an integral value to a physical quantity corresponding to the nearest number of units contained in it. Quantization is like "sampling-in-amplitude," which should be distinguished from the usual "sampling-in-time." The probability density distribution of the signal is sampled in this case, rather than the signal itself. Quantized signals take on only discrete levels and have probability densities consisting of uniformly-spaced impulses. If the quantization is fine enough so that a Nyquist-sampling restriction upon the probability density is satisfied, statistics are recoverable from the grouped statistics in a way similar to the recovery of a signal from its samples. When statistics are recoverable, the nature of quantization "noise" is understood. As a matter of fact, it is known to be uniformly distributed between plus and minus half a unit, and it is entirely random (first order) even though the signal may be of a higher-order process, provided that a multidimensional Nyquist restriction on the high-order distribution density is satisfied. This simple picture of quantization noise permits an understanding of round-off error and its propagation in numerical solution, and of the effects of analog-to-digital conversion in closed-loop control systems. Application is possible when the grain size is almost two standard deviations. Here the dynamic range of a variable covers about three quantization boxes.