Information rates in sampling and quantization

A computer-aided analysis of a digital communications system has been conducted. Salient results compare the performance of approximately optimum PCM systems with the rate-distortion function, under the assumptions of Gaussian inputs and mean-square-error distortion measure. The calculated distortion is greater than the theoretical minimum by two or three decibels in most cases. The results presented also illuminate the tradeoff between sampling rate and number of quantization levels in a channel of fixed capacity. The presampling and reconstruction filters are represented in the computer by their impulse responses; that is, filtering of signals is accomplished by evaluating the convolution integrals numerically. Impulse responses with durations of 20 sample times or fewer are long enough to model most known results in optimum filtering of sampled stationary random functions. Procedures are described for computing the autocorrelation function of the quantizer output, and the cross correlation between the quantizer output and a signal correlated with the input. The procedure adopted is the naive one of integrating numerically the appropriate bivariate normal distribution.