Quantization effects in filtering of stationary Gaussian processes

The performance lost to data quantization is considered in the context of minimum-mean-square error (MMSE) filtering of stationary Gaussian processes. It is seen that, for data uniformly partitioned into intervals of length Δ, the optimum MMSE estimator produces an increase in mean-square error of Δ2/12 Σk=0 λ hk 2 + O( Δ4) over that of the optimum estimator based on the original data, where {hk}k=0 ∞is the impulse response of the estimator based on the original data (which, of course, is linear). It is also seen that the same increase in MSE (to second order in Δ) is caused by applying the Linear estimation filter {hk}k=0 Δ directly to uniformly quantized data. Thus, for small Δ, the performance gained by using an optimum post-quantization estimator rather than by simply using the unquantized filter on this quantized data is negligible. Also, the second-order term in the expression for the increased MSE due to quantization is the same as the increased MSE that would be produced in the optimum filter {hk}k=0 ∞ if an additional orthogonal i.i.d. sequence with zero mean and variance Δ2/12 were added to the unquantized data. This behavior supports the use in this application of the common approximation used else where that errors due to uniform quantization are white with variance Δ2/12 and are orthogonal to the sequence quantized.