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(σ,N) of the ensemble averaged data, accounting for an ensemble of finite length N, and noise RMS σ, is obtained. Previously published results for N=1 and N→∞ are recovered. For intermediate N, we show that there is an optimal noise RMS, σ_opt(N), which minimizes the MSE. Such a minimum point exists regardless of the type of noise probability distribution function. Conditions on σ and N for achieving a smaller MSE than in the noise-free case (Δ²/12) are discussed. The convergence properties of MSE(σ,N) for increasing N, and the effect of applying uniformly distributed dither, is established.