Linear Reconstruction of Quantized and Sampled Random Signals

Techniques for optimal mean-square linear reconstruction from quantized samples of a random signal and the resulting errors are discussed in this paper. The signal is assumed to be wide-sense stationary with Gaussian statistics. The shape of the sample pulse is arbitrary. It is shown that the pulse shape has no effect upon the minimum mean-square error. An optimal linear filter for reconstruction from quantized samples and the resulting error are obtained. The mean-square error that arises when the optimal filter for unquantized samples reconstructs from quantized samples is also obtained. The errors of the above-mentioned filters are then compared. It is shown that, for sufficiently high sampling rates, the filter that takes quantizing into account can achieve a significant reduction in quantizing error relative to the filter for unquantized samples. However, when a constraint of constant channel capacity is imposed, there is essentially no difference between the optimal performances of the two filters.