Quantization error and resolution in ensemble averaged data with noise

We investigate the properties of ensemble averaged data from a uniform quantizer, when the quantizer input signal is noisy. An expression for the mean-square error (MSE) MSE(/spl sigma/,N) of the ensemble averaged data, accounting for an ensemble of finite length N, and noise RMS /spl sigma/, is obtained. Previously published results for N=1 and N/spl rarr//spl infin/ are recovered. For intermediate N, we show that there is an optimal noise RMS, /spl sigma//sub opt/(N), which minimizes the MSE. Such a minimum point exists regardless of the type of noise probability distribution function. Conditions on /spl sigma/ and N for achieving a smaller MSE than in the noise-free case (/spl Delta//sup 2//12) are discussed. The convergence properties of MSE(/spl sigma/,N) for increasing N, and the effect of applying uniformly distributed dither, is established.